On homotopy groups of the suspended classifying spaces
In this paper, we determine the homotopy groups \pi_4(\Sigma K(A,1)) and \pi_5(\Sigma K(A,1)) for abelian groups A by using different facts and methods from group theory and homotopy theory: derived functors, the Carlsson simplicial construction, the…
Authors: Roman Mikhailov, Jie Wu
ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES R OMAN MIKHAILO V AND JIE WU Abstra t. In this pap er, w e determine the homotop y groups π 4 (Σ K ( A, 1)) and π 5 (Σ K ( A, 1)) for ab elian groups A b y using dieren t fats and metho ds from group theory and ho- motop y theory: deriv ed funtors, the Carlsson simpliial onstrution, the Baues-Go erss sp etral sequene, homotop y deomp ositions and the metho ds of algebrai K-theory . As the appliations, w e also determine π i (Σ K ( G, 1)) with i = 4 , 5 for some non-ab elian groups G = Σ 3 and SL( Z ) , and π 4 (Σ K ( A 4 , 1)) for the 4 -th alternating group A 4 . 1. Intr odution It is w ell-kno wn that the susp ension funtor applied to a top ologial spae shifts ho- mology groups, but " haotially" hanges homotop y groups. F or example, one an tak e a irle S 1 , whose homotop y t yp e is v ery simple. Its susp ension Σ S 1 = S 2 has ob vious homology groups, ho w ev er the problem of in v estigating the homotop y groups of S 2 is one of the deep est problems of algebrai top ology . Consider the follo wing funtors from the ategory of groups to the ategory of ab elian groups: π n (Σ m K ( − , 1)) : Gr → Ab , n ≥ 1 , m ≥ 1 dened b y A 7→ π n (Σ m K ( A, 1)) , where Σ m is the m -fold susp ension. It is lear that π n (Σ m K ( Z , 1)) = π n ( S m +1 ) , that is the homotop y groups of spheres app ear as the simplest ase of a general theory of homotop y groups of susp ensions of lassifying spaes. F or the ase m = 1 , 2 and n = 3 , 4 there is the follo wing natural omm utativ e diagram with exat ro ws [ 6℄: 0 / / π 3 (Σ K ( G, 1)) / / G ⊗ G / / [ G, G ] / / 1 0 / / π 4 (Σ 2 K ( G, 1)) / / G e ⊗ G / / [ G, G ] / / 1 0 / / H 2 ( G ) / / G ∧ G / / [ G, G ] / / 1 (1.1) 1991 Mathematis Subje t Classi ation. Primary 55Q52; Seondary 55P20, 55P40, 55P65, 55Q35. Key wor ds and phr ases. homotop y groups, Whitehead exat sequene, sp etral sequenes, Mo ore spaes, susp ensions of K ( G, 1) spaes, simpliial group. The resear h of the rst author is partially supp orted b y RFBR (gran t 08-01-91300 I N D a ). The resear h of the seond author is partially supp orted b y the A ademi Resear h F und of the National Univ ersit y of Singap ore R-146-000-101-112 . 1 ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 2 where G ⊗ G is the non-ab elian square of G in the sense of Bro wn-Lo da y [6 ℄, G e ⊗ G (resp. G ∧ G ) is the quotien t of G ⊗ G b y the normal subgroup generated b y elemen ts g ⊗ h + h ⊗ g (resp. g ⊗ g , g ∈ G ). In partiular, for an ab elian group A , there are natural isomorphisms π 3 (Σ K ( A, 1)) ≃ A ⊗ A π 4 (Σ 2 K ( A, 1)) ≃ π S 2 K ( A, 1) ≃ A e ⊗ A. The purp ose of this artile is to determine the homotop y groups π 4 (Σ K ( A, 1)) and π 5 (Σ K ( A, 1)) for ab elian groups A . In order to in v estigate the struture of these homotop y groups, w e use dieren t fats and metho ds of group theory and homotop y theory: deriv ed funtors, the Carlsson simpliial onstrution, the Baues-Go erss sp etral sequene [4℄, homotop y deomp ositions and the metho ds of algebrai K-theory . The om bination of these dieren t metho ds pro vides an eetiv e w a y for determining these homotop y groups. As reader will see, some our omputations use omm utator tri ks in simpliial groups. The homotop y group π 4 (Σ K ( A, 1)) as a funtor on A an b e giv en as follo ws: Theorem 1.1 (Theorem 3.1) . L et A b e any ab elian gr oup. Then ther e is a natur al short exat se quen e (Λ 2 ( A ) ⊗ A ) ⊕ 2 ⊕ A ⊗ A ⊗ Z / 2 ⊂ ✲ π 4 (Σ K ( A, 1)) ✲ ✲ T or( A, A ) . Mor e over (Λ 2 ( A ) ⊗ A ) ⊕ 2 is an (unnatur al) summand of π 4 (Σ K ( A, 1)) . An in teresting p oin t of this theorem is that the funtor π 4 (Σ K ( A, 1)) has T or( A, A ) as a natural quotien t. F or determining the struture of the group π 4 (Σ K ( A, 1)) , one has to solv e the group extension problem in Theorem 1.1 . F or nitely generated ab elian groups A , w e are able to solv e this problem. Giv en a nitely generated ab elian group A , let A = A 1 ⊕ M r ≥ 1 p is a prime A p r b e the primary deomp osition of A , where A 1 is torsion free and A p r is a free Z /p r -mo dule. Theorem 1.2 (Theorem 3.2) . L et A b e any nitely gener ate d ab elian gr oup. L et A = A 2 ⊕ B with B = A 1 ⊕ L p r 6 =2 A p r . Then π 4 (Σ K ( A, 1)) ∼ = 1 2 ( A 2 ⊗ A 2 ) ⊕ ( A 2 ⊗ B ) ⊕ 2 ⊕ B ⊗ 2 ⊗ Z / 2 ⊕ ( A ⊗ Λ 2 ( A )) ⊕ 2 ⊕ T or( A 2 , B ) ⊕ 2 ⊕ T or( B , B ) , wher e 1 2 ( A 2 ⊗ A 2 ) is a fr e e Z / 4 -mo dule with r ank of dim Z / 2 ( A 2 ⊗ A 2 ) . One p oin t of this theorem is that the (maximal) elemen tary 2 -group summand A 2 of A pla ys a k ey role in the group extension problem. Roughly sp eaking A 2 ⊗ A 2 is half do wn in the group π 4 (Σ K ( A, 1)) . As the appliations of Theorems 1.1 and 1.2 , w e are able to ompute π 4 ( M ( Z / 2 r , 2)) and their onnetions with π 4 (Σ K ( Z / 2 r , 1)) . As the diret onsequenes, the homotop y groups π 4 (Σ R P n ) and π 4 (Σ K (Σ 3 , 1)) are determined. (See subsetion 3.2 for the omputations of these homotop y groups.) F or the homotop y group π 5 (Σ K ( A, 1)) , as a funtor, it an b e desrib ed b y t w o exat sequenes giv en in diagram (4.1 ). Unfortunately it seems to o ompliated to pro due a anonial funtorial short exat sequene desription for the funtor π 5 (Σ K ( A, 1)) from ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 3 diagram (4.1). F or an y nitely generated ab elian group A , w e determine π 5 (Σ K ( A, 1)) in an un-funtorial w a y b y the follo wing steps: 1) F rom the Hopf bration, π 5 (Σ K ( A, 1)) ∼ = π 5 (Σ K ( A, 1) ∧ K ( A, 1)) ; 2) T ak e a primary deomp osition of A and write K ( A, 1) as a pro dut of opies of S 1 = K ( Z , 1) and K ( Z /p r , 1) ; 3) By using the fat that Σ X × Y ≃ Σ X ∨ Σ Y ∨ Σ X ∧ Y , write Σ K ( A, 1) ∧ K ( A, 1) as a w edge of the spaes in the form X = Σ m K ( Z /p r 1 1 , 1) ∧ K ( Z /p r 2 2 , 1) ∧ · · · ∧ K ( Z /p r t t , 1) with m + t ≥ 3 and m ≥ 1 ; 4) By applying the Hilton-Milnor Theorem, π 5 (Σ K ( A, 1)) b eomes a summation of π 5 ( X ) for some X in the ab o v e form. F or the spaes X in the ab o v e form, it is on tratible if p i 6 = p j for some i 6 = j and π 5 ( X ) an b e determined in Prop osition 4.2 for an o dd prime p . The only diult part is to ompute π 5 ( X ) for X giv en in the form X = Σ m K ( Z / 2 r 1 , 1) ∧ · · · ∧ K ( Z / 2 r t , 1) with m + t ≥ 3 and m ≥ 1 . Our omputations are then giv en ase-b y-ase (Prop osi- tions 4.4-4.6 and Theorems 4.1 -4.4 ), in whi h dieren t metho ds are in v olv ed. An instru- tional example is as follo ws: Let A = Z ⊕ Z / 2 . A ording to 1), π 5 (Σ K ( A, 1)) ∼ = π 5 (Σ K ( A, 1) ∧ K ( A, 1)) . As in 3), Σ K ( A, 1) ∧ K ( A, 1) ≃ Σ ( S 1 × R P ∞ ) ∧ ( S 1 × R P ∞ ) ≃ Σ ( S 1 ∨ R P ∞ ∨ Σ R P ∞ ) ∧ ( S 1 ∨ R P ∞ ∨ Σ R P ∞ ) = S 3 ∨ 2 W Σ 2 R P ∞ ∨ 2 W Σ 3 R P ∞ ∨ Σ R P ∞ ∧ R P ∞ ∨ ∨ 2 W Σ 2 R P ∞ ∧ R P ∞ ∨ Σ 3 R P ∞ ∧ R P ∞ . By applying the Hilton-Milnor Theorem as in 4), π 5 (Σ K ( A, 1) ∧ K ( A, 1)) is a summation of π 5 ( S 3 ) , π 5 (Σ 2 R P ∞ ) , π 5 (Σ R P ∞ ∧ R P ∞ ) , π 5 (Σ( R P ∞ ) ∧ 3 ) , · · · with m ultipliities. F rom Theorem 4.4 , w e ha v e π 5 (Σ 2 R P ∞ ) = Z / 8 and b y Prop osition 4.6 and Theorem 4.1, w e ha v e π 5 (Σ R P ∞ ∧ R P ∞ ) = π 5 (Σ( R P ∞ ) ∧ 3 ) = Z / 2 ⊕ 2 . The group π 5 (Σ K ( Z ⊕ Z / 2) ) will b e determined b y lling all p ossible summands with m ultipliities. As the appliations of our omputations on π 5 (Σ K ( A, 1)) , w e are able to determine π 5 (Σ R P n ) (Prop osition 4.8 ) and π 5 (Σ K (Σ 3 , 1)) (Prop osition 4.9 ). In setion 2 w e reall ertain fats from the homotop y theory , su h as the Whitehead exat sequene, the Carlsson simpliial onstrution and desrib e a sp etral sequene (2.9 ), whi h on v erges to π ∗ (Σ m K ( A, 1)) for an y ab elian group A , with E 2 -terms are ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 4 giv en b y the deriv ed funtors of ertain p olynomial funtors. W e illustrate ho w it w orks in Theorem 4.4 for omputing π 5 (Σ 2 K ( Z / 2 r , 1)) = Z / 8 if r = 1 , Z / 2 r +1 ⊕ Z / 2 if r > 1 . The in teresting p oin t is of ourse ho w Z / 8 sho ws up in the ase r = 1 while it b eomes Z / 2 r +1 ⊕ Z / 2 for r > 1 . The pro of is also based on the omputations of the deriv ed funtors of the an tisymmetri square e ⊗ 2 . There is a natural relation b et w een the problem onsidered and algebrai K-theory . Sine the plus-onstrution K ( G, 1) → K ( G, 1) + is a homologial equiv alene, there is a natural w eak homotop y equiv alene Σ K ( G, 1) → Σ( K ( G, 1) + ) This denes the natural susp ension map: π n ( K ( G, 1) + ) → π n +1 (Σ( K ( G, 1) + )) = π n +1 (Σ K ( G, 1)) for n ≥ 1 . This map w as studied in [3℄ in the ase of a p erfet group G . W e onsider the ase G = E ( R ) , i.e. the group of elemen tary matries o v er a ring R . In this ase the natural map K 3 ( R ) = π 3 ( K ( E ( R ) , 1) + ) → π 4 (Σ K ( E ( R ) , 1) ) is an isomorphism (Theorem 5.1). The natural relation to K-theory giv es a w a y ho w to ompute homotop y groups π i (Σ K ( E ( R ) , 1) ) for i = 4 , 5 for some rings. F or example, the ase G = S L ( Z ) is onsidered. As an appliation of our metho ds, w e also determine that π 4 (Σ K ( A 4 , 1)) = Z / 4 for the 4 -th alternating group A 4 . The artile is organized as follo ws. W e giv e a brief review for the quadrati funtors and the simpliial resolutions in Setion 2. The determination of π 4 (Σ K ( A, 1)) is giv en in setion 3, where the pro ofs of Theorems 1.1 and 1.2 are also giv en. In setion 4, w e giv e ase-b y-ase omputations for π 5 (Σ K ( A, 1)) . In Setion 5, w e giv e some relations to K -theory . 2. The Quadra ti Funtors and the Simpliial Resolutions 2.1. Whitehead Quadrati F untor. In [19, Chapter I I℄, J. H. C Whitehead in tro due the universal quadr ati funtor Γ 2 from ab elian groups to ab elian groups as follo ws: Let A b e an y ab elian group. Then Γ 2 ( A ) is the group generated b y the sym b ols γ ( x ) , one for ea h x ∈ A , sub jet to the dening relations (1) γ ( − x ) = γ ( x ) ; (2) γ ( x + y + z ) − γ ( x + y ) − γ ( y + z ) − γ ( x + z ) + γ ( x ) + γ ( y ) + γ ( z ) = 0 . Note. A ording to [19, p. 61℄, the group Γ 2 ( A ) is ab elian and so the m ultipliation in Γ 2 ( A ) is denoted b y + . Dene γ ( x, y ) = γ ( x + y ) − γ ( x ) − γ ( y ) . The follo wing prop osition helps for determining the group Γ 2 ( A ) . Prop osition 2.1. [19 , Theorem 5℄ L et A b e an ab elian gr oup with a b asis { a i | i ∈ I } for a wel l-or der e d index set I , and the dening r elations { b λ ≡ 0 } . Then the gr oup Γ 2 ( A ) is ombinatorial dene d by the set of symb oli gener ators γ ( a i ) , i ∈ I , and γ ( a i , a j ) , i, j ∈ I with i < j with dening r elations γ ( b λ ) ≡ 0 and γ ( a i , b λ ) ≡ 0 . ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 5 Example 2.1. W e list some examples of the group Γ 2 ( A ) . The rst t w o examples are diret onsequenes of the ab o v e prop osition. (1) Let A b e a free ab elian group with a basis { a i | i ∈ I } for a w ell-ordered index set I . Then Γ 2 ( A ) i s the free ab elian group with a basis giv en b y γ ( a i ) , i ∈ I , and γ ( a i , a j ) , i, j ∈ I with i < j . (2) If A is a yli group of nite order m generated b y a 1 , then Γ 2 ( A ) is yli of order m or 2 m , aording as m is o dd or ev en, generated b y γ ( a 1 ) . (3) Let A = L i ∈ I A i for a w ell-ordered index set I . Then [19 , Theorem 7℄ Γ 2 ( A ) ∼ = M i ∈ I Γ 2 ( A i ) ⊕ M i, j ∈ I i < j A i ⊗ A j . (4) F or a general ab elian group A , there is a short exat sequene [10 , form ula (13.8), p.93℄ A ⊗ A ⊂ t ✲ Γ 2 ( A ) ✲ ✲ A ⊗ Z / 2 , where t ( a ⊗ b ) = γ ( a, b ) = γ ( a + b ) − γ ( a ) − γ ( b ) . 2.2. Lo w er Homology of K ( A, 2) . The homology of Eilen b erg-MaLane spaes K ( A, n ) has b een studied in the lassial referene [10℄ and other pap ers. See also [ 5℄ for the funtorial desription of homology groups of K ( A, 2) in all dimensions. Lemma 2.1. [10 , Theorems 20.5 and 21.1℄ L et A b e any ab elian gr oup. Then (1) H 2 ( K ( A, 2)) = A ; (2) H 3 ( K ( A, 2)) = 0 ; (3) H 4 ( K ( A, 2)) = Γ 2 ( A ) . The homology H 5 ( K ( A, 2)) b eomes a sp eial funtor on A . Let R 2 ( A ) = H 5 ( K ( A, 2)) . The group R 2 ( A ) for nitely generated ab elian group A an b e omputed as follo ws [10 , Setion 22℄: 1) If A is a yli group of order innite or o dd, then R 2 ( A ) = 0 ; 2) If A = Z / 2 r Z with r ≥ 1 , then R 2 ( A ) ∼ = Z / 2 . 3) Let A = A 1 ⊕ A 2 . Then K ( A, 2) ≃ K ( A 1 , 2) × K ( A 2 , 2) . By using K unneth theorem together with the fat that H 1 ( K ( A, 2)) = H 3 ( K ( A, 2)) = 0 from Lemma 2.1, w e ha v e H 5 ( K ( A, 2)) ∼ = H 5 ( K ( A 1 , 2)) ⊕ H 5 ( K ( A 2 , 2)) ⊕ T or( H 2 ( K ( A 1 , 2)) , H 2 ( K ( A 2 , 2))) . Th us R 2 ( A 1 ⊕ A 2 ) ∼ = R 2 ( A 1 ) ⊕ R 2 ( A 2 ) ⊕ T or( A 1 , A 2 ) . (2.1) Reall the denition of the deriv ed funtors in the sense of Dold-Pupp e [9℄. Let F b e an endofuntor in the ategory of ab elian groups and A an ab elian group. T ak e a pro jetiv e resolution P ∗ → A . Let N − 1 b e the in v erse map to the normalization map due to Dold-Kan. Then N − 1 P ∗ is a free simpliial resolution of A . Then, the i -th deriv ed funtor of F applied to the ab elian group A , is dened as follo ws: L i F ( A ) = π i ( F ( N − 1 P ∗ )) , i ≥ 0 . ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 6 It is a w ell-kno wn fat that this denition do es not dep end on a hoie of a pro jetiv e resolution. In these notations, one has a natural isomorphism: R 2 ( A ) = L 1 Γ 2 ( A ) . 2.3. Whitehead exat sequene. Let X b e a ( r − 1) -onneted C W -omplex, r ≥ 2 . There is the follo wing long exat sequene of ab elian groups [19 , Theorem 1℄: · · · → H n +1 ( X ) → Γ n ( X ) → π n ( X ) h n → H n ( X ) → Γ n − 1 ( X ) → . . . , (2.2) where Γ n ( X ) = Im( π n (sk n − 1 ( X )) → π n (sk n ( X )) ) (here sk i ( X ) is the i -th sk eleton of X ), h n is the n th Hurewiz homomorphism. The Hurewiz theorem is equiv alen t to the statemen t Γ i ( X ) = 0 , i ≤ r. J. H. C. Whitehead omputed the term Γ r +1 ( X ) : In the follo wing theorem, assertion (1) w as giv en in [19, Theorem 14℄ and assertion (2) w as giv en the earlier pap er [18℄. A ording to the remarks in the end of [19 , Setion 14℄, assertion (2) has b een disussed b y G. W. Whitehead [20 ℄ as w ell. Theorem 2.1. L et X b e a ( r − 1) - onne te d C W - omplex with r ≥ 2 . Then (1) If r = 2 , then Γ 3 ( X ) ∼ = Γ 2 ( π 2 ( X )) . (2) If r > 2 , then Γ r +1 ( X ) ∼ = π r ( X ) ⊗ Z / 2 . The isomorphism Γ 2 ( π 2 ( X )) → Γ 3 ( X ) is onstruted as follo ws: Let η : S 3 → S 2 b e the Hopf map and let x ∈ π 2 ( X ) b e written as the the omp osite S 2 ˜ x ✲ sk 2 ( X ) ⊂ ✲ sk 3 ( X ) . Then the omp osite S 3 η ✲ S 2 ˜ x ✲ sk 2 ( X ) ⊂ ✲ sk 3 ( X ) denes an elemen t η ∗ ( x ) ∈ Γ 3 ( X ) . A ording to [19, Setion 13℄, the mapping η 1 : Γ 2 ( π 2 ( X )) → Γ 3 ( X ) , γ ( x ) 7→ η ∗ ( x ) , (2.3) is a w ell-dened isomorphism of groups. The onstrution of the isomorphism π r ( X ) ⊗ Z / 2 → Γ r +1 ( X ) in assertion (2) is similar. Reall the desription of the funtors Γ r +2 ( X ) due to H.-J. Baues [ 2℄. Consider the third sup er-Lie funtor L 3 s : Ab → Ab dened as L 3 s ( A ) = im { A ⊗ A ⊗ A l → A ⊗ A ⊗ A } where l ( a ⊗ b ⊗ c ) = { a, b, c } := a ⊗ b ⊗ c + b ⊗ a ⊗ c − c ⊗ a ⊗ b − c ⊗ b ⊗ a, a, b, c ∈ A. Observ e that L 3 s ( A ) = k er { A ⊗ Λ 2 ( A ) r → Λ 3 ( A ) } , where Λ i ( A ) is the i th exterior p o w er of A and the map r is giv en as r ( a ⊗ b ∧ c ) = a ∧ b ∧ c, a, b, c ∈ A. Let the omplex X b e simply onneted. Giv en an ab elian group A , dene the map q : Γ 2 ( A ) ⊗ A → L 3 s ( A ) ⊕ Γ 2 ( A ) ⊗ Z / 2 b y setting q ( γ 2 ( a ) ⊗ b ) = −{ b, a, a } + ( γ 2 ( a + b ) − γ 2 ( a ) − γ 2 ( b )) ⊗ 1 , a, b ∈ A. ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 7 Dene the group Γ 2 2 X = Γ 2 2 (Γ 2 ( π 2 X ) → π 3 X ) as the pushout: Γ 2 ( π 2 ( X )) ⊗ ( π 2 ( X ) ⊕ Z / 2) q ⊕ id / / η 1 ⊗ id L 3 s ( π 2 ( X )) ⊕ Γ 2 ( π 2 ( X )) ⊗ Z / 2 π 3 ( X ) ⊗ ( π 2 ( X ) ⊕ Z / 2) / / Γ 2 2 ( X ) (2.4) Theorem 2.2. [2 , Theorem 3.1℄ L et X b e a ( r − 1) - onne te d C W - omplexes with r ≥ 2 . 1) If r = 2 , then ther e is a natur al short exat se quen e 0 → Γ 2 2 ( X ) → Γ 4 ( X ) → R 2 ( π 2 ( X )) → 0 . 2) If r = 3 , then ther e is a natur al exat se quen e 0 → π 4 ( X ) ⊗ Z / 2 ⊕ Λ 2 ( π 3 ( X )) → Γ 5 ( X ) → T o r( π 3 ( X ) , Z / 2) → 0 . 3) If r ≥ 4 , ther e is a natur al exat se quen e 0 → π r +1 ( X ) ⊗ Z / 2 → Γ r +2 ( X ) → T or( π r ( X ) , Z / 2) → 0 . Let A b e an ab elian group. Consider the Hurewiz homomorphism h ∗ : π ∗ (Σ K ( A, 1)) → ˜ H ∗ (Σ K ( A, 1)) = ˜ H ∗− 1 ( K ( A, 1)) = ˜ H ∗− 1 ( A ) . Sine H ∗ ( K ( A, 1)) is graded omm utativ e ring, the inlusion A = H 1 ( K ( A, 1)) ⊂ ✲ H ∗ ( K ( A, 1)) indues a ring homomorphism λ : Λ( A ) − → H ∗ ( K ( A, 1)) . By [10, Theorem 19.3℄, λ is a monomorphism and so w e ma y onsider Λ n ( A ) ⊆ H n ( K ( A, 1)) = H n ( A ) . Lemma 2.2. F or every ab elian gr oup A , the Hur ewiz image Im( h n +1 : π n +1 (Σ K ( A, 1)) → H n ( A )) ontains the sub gr oup Λ n ( A ) . Pr o of. F rom the naturalit y , it sues to sho w that the statemen t holds for a free ab elian group A . When A is a free ab elian group, then K ( A, 1) is a (w eak) Cartesian pro dut of the irles. Th us Σ K ( A, 1) is a w edge of spheres from the susp ension splitting that Σ X × Y ≃ Σ X ∨ Σ Y ∨ Σ X ∧ Y and so the Hurewiz homomorphism indues an epimorphism h ∗ : π n +1 (Σ K ( A, 1)) ✲ ✲ H n ( A ) = Λ n ( A ) for a free ab elian group A . This nishes the pro of. ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 8 2.4. Carlsson onstrution. Let G ∗ b e a simpliial group and X a p oin ted simpliial set with a base p oin t ∗ . Consider the simpliial group F G ∗ ( X ) dened as F H ( X ) n = a x ∈ X n ( G n ) x , i.e. in ea h degree F G ∗ ( X ) n is the free pro dut of groups G n n umerated b y elemen ts of X n mo dulo ( G n ) ∗ , with the anonial hoie of fae and degeneray morphisms. It is pro v ed in [7℄ that the geometri realization | F G ∗ ( X ) | is homotop y equiv alen t to the lo op spae Ω( | X | ∧ B | G | ) . The main example w e will onsider is the simpliial irle X = S 1 with S 1 0 = {∗} , S 1 1 = {∗ , σ } , S 1 2 = {∗ , s 0 σ , s 1 σ } , . . . , S 1 n = {∗ , x 0 , . . . , x n } , where x i = s n . . . ˆ s i . . . s 0 σ and the simpliial group G ∗ with G n = G for a giv en group G , with iden tit y homomorphisms as all fae and degeneray maps. In this ase w e use the notation F G ( X ) = F G ∗ ( X ) . One has a homotop y equiv alene | F G ( S 1 ) | ≃ ΩΣ K ( G, 1) . The group F G ( S 1 ) n is the n -fold free pro dut of G : F G ( S 1 ) 1 = G, F G ( S 1 ) 2 = G ∗ G, F G ( S 1 ) 3 = G ∗ G ∗ G, . . . W e an formally iden tify G ∗ G with s 0 G ∗ s 1 G , G ∗ G ∗ G with s 1 s 0 G ∗ s 2 s 0 G ∗ s 2 s 1 G , et, and to dene naturally the fae and degeneray maps: F G ∗ ( S 1 ) : . . . − → . . . − → ← − . . . ← − G ∗ G ∗ G − → − → − → − → ← − ← − ← − G ∗ G − → − → − → ← − ← − G. Remark. Consider the seond term F G ( S 1 ) 2 = G ∗ G and fae morphisms d 0 , d 1 , d 2 : G ∗ G = s 0 ( G ) ∗ s 1 ( G ) → G dened as d 0 : ( s 0 ( g ) 7→ g s 1 ( g ) 7→ 1 , d 1 : ( s 0 ( g ) 7→ g s 1 ( g ) 7→ g , d 2 : ( s 0 ( g ) 7→ 1 s 1 ( g ) 7→ g . There is a natural omm utativ e diagram π 3 (Σ K ( G, 1)) ≃ / / G ⊗ G / / f G / / / / ≃ G ab ≃ π 3 (Σ K ( G, 1)) / / ( k er ( d 1 ) ∩ ker ( d 2 )) / B 2 / / G / / / / G ab (2.5) where B 2 is the 2-b oundary subgroup of G ∗ G and the map f is dened as 1 f ( g ⊗ h ) = [ s 0 ( g ) s 1 ( g ) − 1 , s 0 ( h )] . B 2 . There is a natural desription of the 2-b oundary (see [11℄, for example): B 2 = [ k er ( d 0 ) , k er ( d 1 ) ∩ ker ( d 2 )][ k er ( d 1 ) , k er ( d 2 ) ∩ k er ( d 0 )][ k er ( d 2 ) , k er ( d 0 ) ∩ ker ( d 1 )] . Diagram (2.5 ) implies that f is a natural isomorphism. 1 W e use the standard omm utator relations: [ g , h ] = g − 1 h − 1 g h ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 9 In the ase G = Z , the simpliial group F G ( S 1 ) is iden tial to the Milnor onstrution F ( S 1 ) , with F ( S 1 ) n a free group of rank n , for n ≥ 1 : F ( S 1 ) : . . . − → . . . − → ← − . . . ← − F 3 − → − → − → − → ← − ← − ← − F 2 − → − → − → ← − ← − Z . In this ase there is a homotop y equiv alene | F ( S 1 ) | ≃ Ω S 2 and the onstrution F ( S 1 ) pro vides a om binatorial mo del for the omputation of ho- motop y groups of the 2-sphere S 2 . The onstrution F ( S 1 ) w as studied from the group- theoretial p oin t of view in [21 ℄. It is easy to nd the simpliial generators of the homotop y lasses of π i ( F ( S 1 )) = π i +1 ( S 2 ) for i = 3 , 4 , 5 . In order to nd these simpliial generators, onsider the sequene of maps b et w een Milnor simpliial onstrutions F ( S 4 ) → F ( S 3 ) → F ( S 2 ) → F ( S 1 ) su h that the indued homomorphisms Z = π 2 ( F ( S 2 )) → π 2 ( F ( S 1 )) = Z and Z = π 3 ( F ( S 3 )) → π 3 ( F ( S 2 )) = Z / 2 are epimorphisms and dene the homotop y lasses of π 3 ( S 2 ) and π 4 ( S 3 ) resp etiv ely . F ( S 3 ) 4 − → . . . − → ← − . . . ← − Z ↓ ↓ η 2 F ( S 2 ) 4 − → . . . − → ← − . . . ← − F ( S 2 ) 3 − → − → − → − → ← − ← − ← − Z ↓ ↓ ↓ η F ( S 1 ) 4 − → . . . − → ← − . . . ← − F ( S 1 ) 3 − → − → − → − → ← − ← − ← − F ( S 1 ) 2 − → − → − → ← − ← − Z F or n ≥ 3 , the homotop y lass of π n ( S n − 1 ) dened as π n − 1 ( F ( S n − 2 )) is generated b y [ s 0 ( σ n − 2 ) , s 1 ( σ n − 2 )] in F ( S n − 2 ) n − 1 (see [21℄), where σ n − 2 is a generator of F ( S n − 2 ) n − 2 = Z . That is, w e an dene the simpliial susp ension maps η i : F ( S i +1 ) i +1 → F ( S i ) i +1 b y η i : σ i +1 → [ s 0 ( σ i ) , s 1 ( σ i )] , i ≥ 1 . Sine the generators of π i ( S 2 ) are presen ted b y susp ensions o v er Hopf bration for i = 3 , 4 , 5 , the simpliial generators of π i ( F ( S 1 )) , i = 2 , 3 , 4 are giv en b y the follo wing ele- men ts: w 2 ( x 0 , x 1 ) = [ x 0 , x 1 ] (2.6) w 3 ( x 0 , x 1 , x 2 ) = [[ x 0 , x 2 ] , [ x 0 , x 1 ]] (2.7) w 4 ( x 0 , x 1 , x 2 , x 3 ) = [[[ x 0 , x 3 ] , [ x 0 , x 1 ]] , [[ x 0 , x 2 ] , [ x 0 , x 1 ]]] . (2.8) Here w e use the natural notations x j := s i . . . ˆ s j . . . s 0 ( σ 1 ) , j = 0 , . . . , i for the basis elemen ts in F ( S 1 ) i +1 2 . 2 One an on tin ue the pro ess of onstrution of elemen ts w n +1 ( x 0 , . . . , x n ) b y the follo wing la w: w n +1 ( x 0 , . . . , x n ) = [ w n ( x 0 , . . . , ˆ x n − 1 , x n ) , w n ( x 0 , . . . , x n − 1 )] . In this ase, the 16-omm utator bra k et w 5 ( x 0 , . . . , x 4 ) orresp onds to the elemen t of order 2 in π 6 ( S 2 ) , but the 32-omm utator bra k et ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 10 2.5. Sp etral sequene. Consider an ab elian group A and its t w o-step at resolution 0 → A 1 → A 0 → A → 0 . By Dold-Kan orresp ondene, w e obtain the follo wing free ab elian simpliial resolution of A : N − 1 ( A 1 ֒ → A 0 ) : . . . − → − → − → ← − ← − A 1 ⊕ s 0 ( A 0 ) − → − → ← − A 0 . Applying Carlsson onstrution to the resolution N − 1 ( A 1 ֒ → A 0 ) , w e obtain the follo w- ing bisimpliial group: F N − 1 ( A 1 ֒ → A 0 ) 2 ( S n ) 3 − → − → − → − → ← − ← − ← − F N − 1 ( A 1 ֒ → A 0 ) 2 ( S n ) 2 − → − → − → ← − ← − N − 1 ( A 1 ֒ → A 0 ) 2 ↓↓↓↑↑ ↓↓↓↑↑ ↓↓↓↑↑ F A 1 ⊕ s 0 ( A 0 ) ( S n ) 3 − → − → − → − → ← − ← − ← − F A 1 ⊕ s 0 ( A 0 ) ( S n ) 2 − → − → − → ← − ← − A 1 ⊕ s 0 ( A 0 ) ↓↓↑ ↓↓↑ ↓↓↑ F A 0 ( S n ) 3 − → − → − → − → ← − ← − ← − F A 0 ( S n ) 2 − → − → − → ← − ← − A 0 Here the m th horizon tal simpliial group is Carlsson onstrution F N − 1 ( A 1 ֒ → A 0 ) m ( S n ) . By the result of Quillen [15 ℄, w e obtain the follo wing sp etral sequene: E 2 p,q = π q ( π p (Σ n K ( N − 1 ( A 1 ֒ → A 0 ) , 1)) = ⇒ π p + q (Σ n K ( A, 1)) . (2.9) Consider no w a non-ab elian analog of this sp etral sequene, for n = 1 . Supp ose no w that a group G is arbitrary , not neessary ab elian. Consider a simpliial resolution of G : G • → G, i.e. G • is a simpliial group with π 0 ( G • ) = G, π i ( G • ) = 0 , i > 0 . Consider the follo wing bisimpliial group G 2 ∗ G 2 ∗ G 2 − → − → − → − → ← − ← − ← − G 2 ∗ G 2 − → − → − → ← − ← − G 2 ↓↓↓↑↑ ↓↓↓↑↑ ↓↓↓↑↑ G 1 ∗ G 1 ∗ G 1 − → − → − → − → ← − ← − ← − G 1 ∗ G 1 − → − → − → ← − ← − G 1 ↓↓↑ ↓↓↑ ↓↓↑ G 0 ∗ G 0 ∗ G 0 − → − → − → − → ← − ← − ← − G 0 ∗ G 0 − → − → − → ← − ← − G 0 Again, b y the result of Quillen [15 ℄, w e obtain the follo wing sp etral sequene: E 2 p,q = π q ( π p (Σ K ( G • , 1))) = ⇒ π p + q (Σ K ( G, 1)) . (2.10) w 6 ( x 0 , . . . , x 5 ) lies in the simpliial b oundary subgroup B F ( S 1 ) 6 (see [10 ℄). The onstrution of a simpli- ial generator of the 3-torsion in π 6 ( S 2 ) is more tri ky: it is p ossible to nd its simpliial represen tativ e whi h is a pro dut of six bra k ets of the omm utator w eigh t six. ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 11 If G • is a free simpliial resolution, the sp etral sequene (2.10 ) on tains a lot of anonial dieren tials of a ompliated nature: H 4 ( G ) π 2 ( L 3 s (( G • ) ab ) ⊕ Γ 2 (( G • ) ab ) ⊗ Z / 2) π 2 (Γ 2 (( G • ) ab )) H 3 ( G ) π 1 ( L 3 s (( G • ) ab ) ⊕ Γ 2 (( G • ) ab ) ⊗ Z / 2) π 1 (Γ 2 (( G • ) ab )) H 2 ( G ) L 3 s ( G ab ) ⊕ Γ 2 ( G ab ) ⊗ Z / 2 Γ 2 ( G ab ) G ab 3. On gr oup π 4 (Σ K ( A, 1)) 3.1. The group π 4 (Σ K ( A, 1)) for an ab elian group A . Let A b e an ab elian group. Consider the homotop y omm utativ e diagram of bre sequenes Σ K ( A, 1) ∧ K ( A, 1) H ✲ Σ K ( A, 1) ✲ K ( A, 2) = B K ( A, 1) pull Σ K ( A, 1) ∧ K ( A, 1) w w w w w w w w w f ✲ K ( A, 2) ∨ K ( A, 2) ❄ ⊂ ✲ K ( A, 2) × K ( A, 2) , ∆ ❄ (3.1) where H is the Hopf bration. Th us w e ha v e the follo wing lemma: Lemma 3.1. Ther e ar e isomorphisms π n (Σ K ( A, 1) ∧ K ( A, 1)) ∼ = π n (Σ K ( A, 1)) ∼ = π n ( K ( A, 2) ∨ K ( A, 2)) for n ≥ 3 . In p artiular, π 3 (Σ K ( A, 1)) ∼ = π 3 (Σ K ( A, 1) ∧ K ( A, 1)) ∼ = A ⊗ A . By Lemma 2.1 , the lo w er homology of the w edge K ( A, 2) ∨ K ( A, 2) are the follo wing: Lemma 3.2. H 2 ( K ( A, 2) ∨ K ( A, 2)) = A ⊕ A, H 3 ( K ( A, 2) ∨ K ( A, 2)) = 0 , H 4 ( K ( A, 2) ∨ K ( A, 2)) = Γ 2 ( A ) ⊕ Γ 2 ( A ) , H 5 ( K ( A, 2) ∨ K ( A, 2)) = R 2 ( A ) ⊕ R 2 ( A ) . Lemma 3.3. The Hur ewiz image h n : π n ( K ( A, 2) ∨ K ( A, 2) ) − → H n ( K ( A, 2) ∨ K ( A, 2)) is zer o for n ≥ 3 . ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 12 Pr o of. The assertion follo ws from the omm utativ e diagram π n ( K ( A, 2) ∨ K ( A, 2)) h n ✲ H n ( K ( A, 2) ∨ K ( A, 2) ) π n ( K ( A, 2) × K ( A, 2) ) = 0 ❄ h n ✲ H n ( K ( A, 2) × K ( A, 2) ) . ❄ ∩ Theorem 3.1. L et A b e any ab elian gr oup. Then ther e is a natur al short exat se quen e (Λ 2 ( A ) ⊗ A ) ⊕ 2 ⊕ A ⊗ A ⊗ Z / 2 ⊂ ✲ π 4 (Σ K ( A, 1)) ✲ ✲ T or( A, A ) . Mor e over (Λ 2 ( A ) ⊗ A ) ⊕ 2 is an (unnatur al) summand of π 4 (Σ K ( A, 1)) . Pr o of. Let X = K ( A, 2) ∨ K ( A, 2) and let Y = K ( A, 2) × K ( A, 2) = K ( A ⊕ A, 2) . F rom Lemmas 3.2 and 3.3, there is a short exat sequene R 2 ( A ) ⊕ R 2 ( A ) ⊂ ✲ Γ 4 ( X ) ✲ ✲ π 4 ( X ) . The inlusion j : X = K ( A, 2) ∨ K ( A, 2) ⊂ ✲ Y = K ( A, 2) × K ( A, 2) indues a omm u- tativ e diagram H 5 ( X ) = R 2 ( A ) ⊕ R 2 ( A ) ⊂ ✲ Γ 4 ( X ) ✲ ✲ π 4 ( X ) H 5 ( Y ) = R 2 ( A ⊕ A ) j ∗ ❄ ∩ ∼ = φ ✲ Γ 4 ( Y ) j ∗ ❄ ✲ π 4 ( Y ) = 0 . j ∗ ❄ (3.2) By form ula (2.1 ), H 5 ( K ( A, 2) × K ( A, 2)) = H 5 ( K ( A ⊕ A, 2)) = R 2 ( A ) ⊕ R 2 ( A ) ⊕ T o r ( A, A ) and so the ok ernel of j ∗ : H 5 ( X ) → H 5 ( Y ) is T or( A, A ) . On the other hand, from Theorem 2.2 (1), there is a omm utativ e diagram of short exat sequenes Γ 2 2 ( X ) ⊂ ✲ Γ 4 ( X ) ✲ ✲ R 2 ( π 2 ( X )) = R 2 ( A ⊕ A ) Γ 2 2 ( Y ) ❄ ⊂ ✲ Γ 4 ( Y ) j ∗ ❄ ψ ✲ ✲ R 2 ( π 2 ( Y )) = R 2 ( A ⊕ A ) . ∼ = ❄ The omp osite ψ ◦ φ : R 2 ( A ⊕ A ) ∼ = φ ✲ Γ 4 ( Y ) ψ ✲ ✲ R 2 ( A ⊕ A ) is a natural self epimorphism for an y ab elian group A . It is an isomorphism for an y nitely generated ab elian A and so an isomorphism for an y ab elian group A b y onsidering the diret limit. Th us j ∗ : Γ 4 ( X ) → Γ 4 ( Y ) is an epimorphism and, from diagram (3.2 ), there is a short exat sequene Γ 2 2 ( X ) ⊂ ✲ π 4 ( X ) ✲ ✲ T or( A, A ) . (3.3) ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 13 Let Z = Σ K ( A, 1) ∧ K ( A, 1 ) and let f : Z → X b e the map in diagram ( 3.1). Consider the omm utativ e diagram π 3 ( Z ) ⊗ Z / 2 = Γ 2 2 ( Z ) ⊂ ∼ = ✲ Γ 4 ( Z ) ✲ π 4 ( Z ) ✲ H 4 ( Z ) ✲ Γ 3 ( Z ) = 0 Γ 2 2 ( X ) f ∗ ❄ ⊂ ✲ Γ 4 ( X ) f ∗ ❄ ✲ ✲ π 4 ( X ) , f ∗ ∼ = ❄ (3.4) where Γ 2 2 ( Z ) → Γ 4 ( Z ) is an isomorphism b eause its ok ernel R 2 ( π 2 ( Z )) = 0 . F rom the denition (2.4) of the funtor Γ 2 2 , f ∗ : Γ 2 2 ( Z ) − → Γ 2 2 ( X ) is a monomorphism with retrating homomorphism φ ′ : Γ 2 2 ( X ) → Γ 2 2 ( Z ) . By the short exat sequene (3.3 ), Γ 4 ( Z ) → π 4 ( Z ) is a monomorphism and so there is a short exat sequene A ⊗ A ⊗ Z / 2 = Γ 4 ( Z ) ⊂ ✲ π 4 ( Z ) ✲ ✲ H 4 ( Z ) = H 3 ( K ( A, 1) ∧ K ( A, 1)) . (3.5) Note that H 1 ( K ( A, 1)) = A and H 2 ( K ( A, 1)) = Λ 2 ( A ) . By the K unneth theorem, there is a natural short exat sequene ( A ⊗ Λ 2 ( A )) ⊕ 2 ⊂ ✲ H 4 ( Z ) ψ ′ ✲ ✲ T or( A, A ) . Consider the omp osite θ A : Γ 2 2 ( X ) = Γ 2 2 (Γ 2 ( A ⊕ A ) → A ⊗ A ) ⊂ ✲ π 4 ( X ) f − 1 ∗ ✲ π 4 ( Z ) ✲ ✲ H 4 ( Z ) ψ ′ ✲ ✲ T or( A, A ) , whi h is natural on an y ab elian group A . If A is a free ab elian group, then θ A = 0 . F or an y ab elian group A , ho ose an y free ab elian group A 0 with an epimorphism g : A 0 ։ A . F rom the denition (2.4) of Γ 2 2 , Γ 2 2 ( g ) : Γ 2 2 (Γ 2 ( A 0 ⊕ A 0 ) → A 0 ⊗ A 0 ) − → Γ 2 2 (Γ 2 ( A ⊕ A ) → A ⊗ A ) is an epimorphism. By the naturalit y of θ A , w e ha v e θ A = 0 b eause θ A 0 = 0 . No w, from diagram (3.4), there is a omm utativ e diagram of natural short exat sequenes A ⊗ A ⊗ Z / 2 = = = A ⊗ A ⊗ Z / 2 Γ 2 2 ( X ) ❄ ∩ ⊂ ✲ π 4 (Σ K ( A, 1)) ❄ ∩ ✲ ✲ T or( A, A ) ( A ⊗ Λ 2 ( A )) ⊕ 2 ❄ ❄ ⊂ ✲ H 4 ( Z ) ❄ ❄ ✲ ✲ T or( A, A ) . ∼ = ❄ It follo ws that there is a natural (on A ) isomorphism Γ 2 2 ( X ) ∼ = A ⊗ A ⊗ Z / 2 ⊕ ( A ⊗ Λ 2 ( A )) ⊕ 2 . ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 14 Sine ( A ⊗ Λ 2 ( A )) ⊕ 2 is an (unnatural) summand of H 4 ( Z ) , it is an (unnatural) summand of π 4 (Σ K ( A, 1)) . The pro of is nished. Corollary 3.1. L et p b e an o dd prime inte ger. Then π 4 (Σ K ( Z /p r , 1)) = Z /p r and the Hur ewiz homomorphism π 4 (Σ K ( Z /p r , 1)) → H 4 (Σ K ( Z /p r , 1)) is an isomorphism. Pr o of. In this ase, A ⊗ A ⊗ Z / 2 = 0 . Sine Z /p r is yli, Λ 2 ( A ) ⊗ A = 0 and hene the result. F or ompletely determining the group π 4 (Σ K ( A, 1)) , w e ha v e to onsider the divisibilit y problem of the elemen ts in the subgroup A ⊗ A ⊗ Z / 2 = Γ 4 ( Z ) ⊆ π 4 (Σ K ( A, 1)) = π 4 ( Z ) . W e solv e this problem for an y nitely generated ab elian group A . Lemma 3.4. L et A b e any ab elian gr oup and let j : M ( A, 1) → K ( A, 1) b e a map suh that j ∗ : H 1 ( M ( A, 1)) → H 1 ( K ( A, 1)) is an isomorphism. Then ther e is an (unnatur al) splitting exat se quen e π 4 (Σ M ( A, 1) ∧ M ( A, 1)) ⊂ x (Σ j ∧ j ) ∗ ✲ π 4 (Σ K ( A, 1) ∧ K ( A, 1)) x ✲ ✲ ( A ⊗ Λ 2 ( A )) ⊕ 2 . Pr o of. Let X = Σ M ( A, 1) ∧ M ( A, 1) and let Z = Σ K ( A, 1) ∧ K ( A, 1) . The assertion follo ws from the omm utativ e diagram of short exat sequenes Γ 4 ( X ) ⊂ ✲ π 4 ( X ) ✲ ✲ H 4 ( X ) = T or( A, A ) Γ 4 ( Z ) ∼ = ❄ ⊂ ✲ π 4 ( Z ) ❄ ✲ ✲ H 4 ( Z ) , ❄ ∩ where the b ottom ro w is short exat b y equation (3.5 ). Giv en a nitely generated ab elian group A , let A = A 1 ⊕ M r ≥ 1 p is a prime A p r b e the primary deomp osition of A , where A 1 is torsion free and A p r is a free Z /p r -mo dule. Theorem 3.2. L et A b e any nitely gener ate d ab elian gr oup. L et A = A 2 ⊕ B with B = A 1 ⊕ L p r 6 =2 A p r . Then π 4 (Σ K ( A, 1)) ∼ = 1 2 ( A 2 ⊗ A 2 ) ⊕ ( A 2 ⊗ B ) ⊕ 2 ⊕ B ⊗ 2 ⊗ Z / 2 ⊕ ( A ⊗ Λ 2 ( A )) ⊕ 2 ⊕ T or( A 2 , B ) ⊕ 2 ⊕ T or( B , B ) , wher e 1 2 ( A 2 ⊗ A 2 ) is a fr e e Z / 4 -mo dule with r ank of dim Z / 2 ( A 2 ⊗ A 2 ) . ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 15 Pr o of. Let X = Σ M ( A, 1) ∧ M ( A, 1) . By Lemma 3.4, it sues to sho w that π 4 ( X ) ∼ = 1 2 ( A 2 ⊗ A 2 ) ⊕ ( A 2 ⊗ B ) ⊕ 2 ⊕ B ⊗ 2 ⊗ Z / 2 ⊕ T o r( A 2 , B ) ⊕ 2 ⊕ T or( B , B ) . Observ e that there is a homotop y deomp osition X ≃ _ r , s ≥ 0 p, q prime Σ M ( A p r , 1) ∧ M ( A q s , 1) , where w e allo w r , s to b e 0 for ha ving the fator A 1 to b e app eared. Th us there is a deomp osition π 4 ( X ) ∼ = M r , s ≥ 0 p, q prime π 4 (Σ M ( A p r , 1) ∧ M ( A q s , 1)) . (3.6) Let r , s ≥ 1 and let p and q b e p ositiv e prime in tegers. F rom [ 13 , Corollary 6.6℄, there is a homotop y deomp osition Σ M ( Z /p r , 1) ∧ M ( Z /q s , 1) ≃ ∗ if p 6 = q , M ( Z /p min { r,s } , 3) ∨ M ( Z /p min { r,s } , 4) if p = q and max { p r , q s } > 2 . (3.7) By taking π 4 to ab o v e deomp osition, w e ha v e π 4 (Σ M ( Z /p r , 1) ∧ M ( Z /q s , 1)) ∼ = Z /p r ⊗ Z /q s ⊗ Z / 2 ⊕ T or( Z / 2 r , Z / 2 s ) (3.8) if max { p r , q s } > 2 . Clearly this form ula also holds for the ase where p r = 1 or q s = 1 . F or the ase p r = q s = 2 , w e laim that π 4 (Σ M ( Z / 2 , 1) ∧ M ( Z / 2 , 1 ))) = Z / 4 . (3.9) Let Y = Σ M ( Z / 2 , 1) ∧ M ( Z / 2 , 1) . F rom the short exat sequene Γ 4 ( Y ) = Z / 2 ⊂ ✲ π 4 ( Y ) ✲ ✲ H 4 ( Y ) = Z / 2 , the group π 4 ( Y ) = Z / 4 or Z / 2 ⊕ Z / 2 . Supp ose that π 4 ( Y ) = Z / 2 ⊕ Z / 2 . Then there exists an elemen t α ∈ π 4 ( Y ) of order 2 whi h has the non trivial Hurewiz image. Sine α is of order 2 , the map α : S 4 → Y extends to a map ˜ α : M ( Z / 2 , 4) → Y with ˜ α ∗ : H 4 ( M ( Z / 2 , 4)) − → H 4 ( Y ) an isomorphism. Let j : M ( Z / 2 , 3) − → Y b e the anonial inlusion. Then j ∗ : H 3 ( M ( Z / 2 , 3)) → H 3 ( Y ) is an isomorphism. Then ( j, ˜ α ) : M ( Z / 2 , 3) ∨ M ( Z / 2 , 4) − → Y is a homotop y equiv alene b eause it indues an isomorphism on homology , whi h on- tradits that the Steenro d op eration S q 2 : H 3 ( Y ; Z / 2 ) → H 5 ( Y ; Z / 2 ) is an isomorphism. Th us π 4 ( Y ) = Z / 4 . No w the assertion follo ws from deomp osition (3.6) and the omputational form u- lae (3.8 ) and (3.9). ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 16 Corollary 3.2. L et A 2 b e any elementary 2 -gr oup. Then ther e is a natur al short exat se quen e ( A 2 ⊗ Λ 2 ( A 2 )) ⊕ 2 ⊂ ✲ π 4 (Σ K ( A 2 , 1)) ✲ ✲ 1 2 ( A 2 ⊗ A 2 ) , wher e 1 2 ( A 2 ⊗ A 2 ) is a fr e e Z / 4 -mo dule. Mor e over this splits o unnatur al ly. Pr o of. By Theorem 3.1 , there is a natural short exat sequene ( A 2 ⊗ Λ 2 ( A 2 )) ⊕ 2 ⊂ ✲ π 4 (Σ K ( A 2 , 1)) ✲ ✲ π 4 (Σ K ( A 2 , 1)) / ( A 2 ⊗ Λ 2 ( A 2 )) ⊕ 2 . By Theorem 3.2 , the quotien t group π 4 (Σ K ( A 2 , 1)) / ( A 2 ⊗ Λ 2 ( A 2 )) ⊕ 2 is free Z / 4 -mo dule for an y nite dimensional elemen tary 2 -groups. The assertion follo ws b y taking diret limits. Remark 3.1. The summand 1 2 ( A 2 ⊗ A 2 ) is sub-quotient funtor of π 4 (Σ K ( A, 1)) on A in the fol lowing sense. F or any ab elian gr oup A , the Z / 2 - omp onent A 2 is given by the image of S q 1 ∗ : H 2 ( A ; Z / 2) − → H 1 ( A ; Z / 2) . Thus A 7→ A 2 is a sub funtor of the identity funtor on ab elian gr oups. Then π 4 (Σ K ( A 2 , 1)) is a sub funtor of π 4 (Σ K ( A, 1)) on A and so 1 2 ( A 2 ⊗ A 2 ) = π 4 (Σ K ( A 2 , 1)) / ( A 2 ⊗ Λ 2 ( A 2 )) ⊕ 2 is a sub-quotient funtor of π 4 (Σ K ( A, 1)) on A . 3.2. Appliations. As an appliation, w e ompute π i ( M ( Z /p r , 2)) for i ≤ 4 . By the Hurewiz Theorem, π 2 ( M ( Z /p r , 2)) = Z /p r . F rom the Whitehead exat sequene (2.2 ), w e ha v e Γ n ( M ( Z /p r , 2)) = π n ( M ( Z /p r , 2)) (3.10) for r ≥ 3 b eause H i ( M ( Z /p r , 2) = 0 for i ≥ 3 . It follo ws diretly that π 3 ( M ( Z /p r , 2)) = Γ 3 ( M ( Z /p r , 2)) = Γ 2 ( Z /p r ) = Z /p r if p > 2 , Z / 2 r +1 if p = 2 , (3.11) where Γ 2 ( A ) is omputed in Example 2.1 . F rom Theorem 2.2 (1), there is a short exat sequene Γ 2 2 ( M ( Z /p r , 2)) ⊂ ✲ Γ 4 ( M ( Z /p r , 2)) ✲ ✲ R 2 ( π 2 ( M ( Z /p r , 2))) = R 2 ( Z /p r ) . F rom Subsetion 2.2, R 2 ( Z /p r ) = 0 if p > 2 , Z / 2 if p = 2 . By the denition (2.4) of the funtor Γ 2 2 , the group Γ 2 2 ( M ( Z /p r , 2)) is giv en b y the push-out Γ 2 ( Z /p r ) ⊗ ( Z /p r ⊕ Z / 2) q ⊕ id ✲ L 3 s ( Z /p r ) ⊕ Γ 2 ( Z /p r ) ⊗ Z / 2 π 3 ( M ( Z /p r , 2)) ⊗ ( Z /p r ⊕ Z / 2) ∼ = η 1 ⊗ id ❄ ✲ Γ 2 2 ( M ( Z /p r , 2)) . ❄ Th us Γ 2 2 ( M ( Z /p r , 2)) ∼ = L 3 s ( Z /p r ) ⊕ Γ 2 ( Z /p r ) ⊗ Z / 2 . ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 17 Sine Z /p r is yli and L 3 s ( A ) is isomorphi to the k ernel of A ⊗ Λ 2 ( A ) → Λ 3 ( A ) , w e ha v e L 3 s ( Z /p r ) = 0 and so Γ 2 2 ( M ( Z /p r , 2)) = 0 if p > 2 , Z / 2 if p = 2 . A diret onsequene is: π 4 (Σ M ( Z /p r , 2)) = 0 for p > 2 . (3.12) F or the ase p = 2 , w e ha v e the short exat sequene Z / 2 ⊂ ✲ π 4 ( M ( Z / 2 r , 2)) ✲ ✲ Z / 2 . The remaining problem is to deide whether π 4 ( M ( Z / 2 r , 2)) is equal to Z / 2 ⊕ Z / 2 or Z / 4 . It has b een omputed in [22 ℄ that π 4 ( M ( Z / 2 , 2)) = Z / 4 . F or r > 1 , the group π 4 ( M ( Z / 2 r , 2)) seems not reorded in referenes. W e are going to determine the group π 4 ( M ( Z / 2 r , 2)) using our metho ds. Lemma 3.5. L et j : M ( Z / 2 r , 2) − → Σ K ( Z / 2 r , 1) b e the anoni al map induing isomorphism on H 2 . Then j ∗ : Γ 4 ( M ( Z / 2 r , 2)) − → Γ 4 (Σ K ( Z / 2 r , 1)) is an isomorphism. Pr o of. By Theorem 2.2 (1), there is a omm utativ e diagram of short exat sequenes Γ 2 2 ( M ( Z / 2 r , 2)) = Z / 2 ⊂ ✲ Γ 4 ( M ( Z / 2 r , 2)) ✲ ✲ R 2 ( Z / 2 r ) = Z / 2 Γ 2 2 ( K ( Z / 2 r , 2)) j ∗ ❄ ⊂ ✲ Γ 4 ( K ( Z / 2 r , 2)) j ∗ ❄ ✲ ✲ R 2 ( Z / 2 r ) = Z / 2 ∼ = j ∗ ❄ F rom the Whitehead exat sequene Γ 3 ( Z / 2 r ) = Z / 2 r +1 → π 3 (Σ K ( Z / 2 r , 1)) = Z / 2 r ⊗ Z / 2 r = Z / 2 r → 0 , w e ha v e η 1 ⊗ id : Γ 2 ( Z / 2 r ) ⊗ ( Z / 2 r ⊕ Z / 2) − → π 3 ( K ( Z / 2 r , 2)) ⊗ ( Z / 2 r ⊕ Z / 2) is an isomorphism. Similar to the omputation of Γ 2 2 ( M ( Z / 2 r , 2)) , w e ha v e Γ 2 2 ( K ( Z / 2 r , 2)) = Z / 2 with an isomorphism j ∗ : Γ 2 2 ( M ( Z / 2 r , 2)) ∼ = Γ 2 2 ( K ( Z / 2 r , 2)) . The assertion then follo ws b y 5 -lemma. Lemma 3.6. The gr oup Γ 4 (Σ K ( Z / 2 r , 1)) = Z / 2 ⊕ Z / 2 if r > 1 , Z / 4 if r = 1 . ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 18 Pr o of. Let Z = Σ K ( Z / 2 r , 1) ∧ K ( Z / 2 r , 1) and let H : Z → Σ K ( Z / 2 r , 1) b e the Hopf map. F rom equation (3.5), there is a omm utativ e diagram of exat sequene Γ 4 ( Z ) = Z / 2 ⊂ ✲ π 4 ( Z ) ✲ ✲ H 4 ( Z ) = Z / 2 r Γ 4 (Σ K ( Z / 2 r , 1)) H ∗ ❄ ⊂ ✲ π 4 (Σ K ( Z / 2 r , 1)) ∼ = H ∗ ❄ ✲ H 4 (Σ K ( Z / 2 r , 1)) , H ∗ ❄ where the b ottom ro w is left exat b eause H 5 (Σ K ( Z / 2 r , 1)) = H 4 ( Z / 2 r ) = 0 . If r > 1 , then Γ 4 ( Z ) is a summand of π 4 ( Z ) ∼ = π 4 (Σ K ( Z / 2 r , 1)) b y Theorem 3.2. Th us Γ 4 ( Z ) = Z / 2 is also a summand of Γ 4 (Σ K ( Z / 2 r , 1)) . It follo ws that Γ 4 (Σ K ( Z / 2 r , 1)) = Z / 2 ⊕ Z / 2 if r > 1 . If r = 1 , b y Corollary 3.2 , π 4 (Σ K ( Z / 2 , 1)) = Z / 4 and so Γ 4 (Σ K ( Z / 2 , 1)) ∼ = π 4 (Σ K ( Z / 2 r , 1)) = Z / 4 . The pro of is nished. Sine Γ 4 (Σ K ( Z / 2 r , 1)) → π 4 (Σ K ( Z / 2 r , 1)) is a monomorphism, from Lemmas 3.5 and 3.6 , w e ha v e the follo wing: Corollary 3.3. L et j : M ( Z / 2 r , 2) − → Σ K ( Z / 2 r , 1) b e the anoni al map induing isomorphism on H 2 . Then (1) π 4 ( M ( Z / 2 , 2)) = Z / 4 and j ∗ : π 4 ( M ( Z / 2 , 2)) → π 4 (Σ K ( Z / 2 , 1)) is an isomorphism. (2) F or r > 1 , π 4 ( M ( Z / 2 r , 2)) = Z / 2 ⊕ Z / 2 and j ∗ : π 4 ( M ( Z / 2 r , 2)) = Z / 2 ⊕ Z / 2 → π 4 (Σ K ( Z / 2 r , 1)) = Z / 2 ⊕ Z / 2 r is a monomorphism. Note that M ( Z / 2 , 2) = Σ R P 2 and Σ K ( Z / 2 , 1) = Σ R P ∞ with the anonial inlusion j : Σ R P 2 ֒ → Σ R P ∞ . A onsequene of Corollary 3.3 (1) on the susp ended pro jetiv e spaes are as follo ws. Corollary 3.4. L et j : Σ R P 2 → Σ R P n b e the anoni al inlusion with 3 ≤ n ≤ ∞ . (1) F or 4 ≤ n ≤ ∞ , j ∗ : π 4 (Σ R P 2 ) = Z / 4 → π 4 (Σ R P n ) is an isomorphism. (2) F or n = 3 , j ∗ : π 4 (Σ R P 2 ) = Z / 4 → π 4 (Σ R P 3 ) is a splitting monomorphism. Mor e over π 4 (Σ R P 3 ) ∼ = π 4 (Σ R P 2 ) ⊕ Z = Z / 4 ⊕ Z . Pr o of. Assertion (1) and the rst part of assertion (2) are diret onsequenes of Corol- lary 3.3. F or the seond part of assertion (2), notie that R P 3 = S O (3) . F rom the ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 19 omm utativ e diagram π 4 (Σ S O (3) ∧ S O (3) ) ∼ = ✲ π 4 (Σ R P ∞ ∧ R P ∞ ) π 4 (Σ S O (3)) H ∗ ❄ ✲ π 4 (Σ R P ∞ ) , ∼ = H ∗ ❄ w e ha v e π 4 (Σ S O (3)) ∼ = π 4 (Σ S O (3) ∧ S O (3) ) ⊕ π 4 ( B S O (3)) ∼ = π 4 (Σ R P ∞ ) ⊕ π 3 ( S O (3)) ∼ = Z / 4 ⊕ Z and hene the result. Another onsequene is as follo ws: Corollary 3.5. L et Σ 3 b e the thir d symmetri gr oup. Then π 4 (Σ K (Σ 3 , 1)) = Z / 12 . Pr o of. Reall that the in tegral homology groups of Σ 3 are 4-p erio di with the follo wing initial terms: H 1 (Σ 3 ) = Z / 2 , H 2 (Σ 3 ) = 0 , H 3 (Σ 3 ) = Z / 6 , H 4 (Σ 3 ) = 0 . Let X = Σ K (Σ 3 , 1) . The Whitehead exat sequene has the follo wing form: Γ 4 ( X ) ⊂ ✲ π 4 ( X ) ✲ H 3 (Σ 3 ) = Z / 6 ✲ Γ 3 ( X ) = Z / 4 ✲ ✲ π 3 ( X ) = Z / 2 . The inlusion Σ 2 = Z / 2 → Σ 3 indues an isomorphism π i (Σ K ( Z / 2 , 1)) = Z / 2 ∼ = ✲ π i (Σ K (Σ 3 , 1)) = Z / 2 for i = 2 , 3 . By Theorem 2.2 (1) together with Lemma 3.6 , the inlusion Σ 2 = Z / 2 → Σ 3 indues an Γ 4 (Σ K ( Z / 2 , 1)) = Z / 4 ∼ = ✲ Γ 4 (Σ K (Σ 3 , 1)) and hene the result. 4. On gr oup π 5 (Σ K ( A, 1)) 4.1. Some Prop erties of the F untor A 7→ π 5 (Σ K ( A, 1)) . F rom Hopf bration Σ K ( A, 1) ∧ K ( A, 1) ✲ Σ K ( A, 1) ✲ K ( A, 2) , it sues to ompute π 5 (Σ K ( A, 1) ∧ K ( A, 1)) . Let Z = Σ K ( A, 1) ∧ K ( A, 1) . Sine Z is 2 -onneted, from Theorem 2.2(2), there are natural exat sequenes π 4 (Σ K ( A, 1)) ⊗ Z / 2 ⊕ Λ 2 ( A ⊗ A ) _ H 6 ( Z ) / / Γ 5 ( Z ) / / π 5 ( Z ) / / / / H 5 ( Z ) , T or( A ⊗ A, Z / 2) (4.1) ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 20 where π 5 ( Z ) → H 5 ( Z ) is on to b y equation (3.5). The group π 4 (Σ K ( A, 1)) has b een determined b y Theorems 3.1 and 3.2 . Prop osition 4.1. L et A b e a fr e e ab elian gr oup. Then ther e is a natur al short exat se quen e (Λ 2 ( A ) ⊗ A ⊗ Z / 2) ⊕ 2 ⊕ A ⊗ 2 ⊗ Z / 2 ⊕ Λ 2 ( A ⊗ A ) ֒ → π 5 (Σ K ( A, 1)) ։ (Λ 3 ( A ) ⊗ A ) ⊕ 2 ⊕ Λ 2 ( A ) ⊗ 2 . Pr o of. Sine A is a free ab elian group, the Hurewiz homomorphism h ∗ : π ∗ ( Z ) → ˜ H ∗ ( Z ) is on to b eause Z is a w edge of spheres. Th us there is a short exat sequene Γ 5 ( Z ) ⊂ ✲ π 5 ( Z ) ✲ ✲ H 5 ( Z ) . By Theorem 3.1, π 4 ( Z ) ∼ = (Λ 2 ( A ) ⊗ A ) ⊕ 2 ⊕ A ⊗ A ⊗ Z / 2 for a free ab elian group A . The assertion follo ws from diagram (4.1). Prop osition 4.2. If A is a torsion ab elian gr oup with the pr op erty that 2 : A → A is an isomorphism, then ther e is a natur al short exat se quen e Λ 2 ( A ⊗ A ) ⊂ ✲ π 5 (Σ K ( A, 1)) ✲ ✲ H 4 ( K ( A, 1) ∧ K ( A, 1)) . Pr o of. It sues to sho w that the Hurewiz homomorphism h ∗ : π 6 ( Z ) → H 6 ( Z ) is on to. W e ma y assume that A is nitely generated b eause w e an tak e diret limit for general ase whene the nitely generated ase is pro v ed. Then A is a diret sum of the primary p -torsion groups Z /p r for some r ≥ 1 and o dd primes p . A ording to [ 12℄, there is homotop y deomp osition Σ K ( Z /p r , 1) ≃ X 1 ∨ · · · ∨ X p − 1 , where ¯ H q ( X i ; Z ) 6 = 0 if and only if q ≡ 2 i mo d 2 p − 2 . T ogether with the deomp osition form ula (3.7) for the smash pro dut of Mo ore spaes, up to 6 -sk eleton, Σ K ( A, 1) ∧ K ( A, 1) is homotop y equiv alen t to a w edge of spheres and Mo ore spaes. It follo ws that the Hurewiz homomorphism π 6 (Σ K ( A, 1) ∧ K ( A, 1)) − → H 6 (Σ K ( A, 1) ∧ K ( A, 1)) is on to and hene the result. F rom the ab o v e pro of, w e also ha v e the follo wing: Prop osition 4.3. L et A b e any ab elian gr oup. L et Z 1 2 = { m 2 r ∈ Q | m ∈ Z , r ≥ 0 } . Then ther e is natur al short exat se quen e Λ 2 ( A ⊗ A ) ⊗ Z 1 2 ⊂ ✲ π 5 (Σ K ( A, 1)) ⊗ Z 1 2 ✲ ✲ H 4 ( K ( A, 1) ∧ K ( A, 1)) ⊗ Z 1 2 . F or omputing the group π 5 (Σ K ( A, 1)) , as one see from the ab o v e, the tri ky part is the 2 -torsion. Whene A on tains 2 -torsion summands, the Hurewiz homomorphism π 6 ( Z ) → H 6 ( Z ) is no longer epimorphism in general and so Γ 5 ( Z ) → π 5 ( Z ) is not a monomorphism in general. Also the group π 5 ( Z ) in diagram (4.1) admits non-trivial extension. The omputation of the group π 5 (Σ K ( A, 1)) for nitely generated ab elian groups A an b e giv en b y the follo wing steps: Step 1. T ak e a primary deomp osition of A and write K ( A, 1) as a pro dut of opies of S 1 = K ( Z , 1) and K ( Z /p r , 1) . ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 21 Step 2. By using the fat that Σ X × Y ≃ Σ X ∨ Σ Y ≃ Σ X ∧ Y for an y spaes X and Y , one gets Σ( X 1 × X 2 ) ∧ ( X 1 × X 2 ) ≃ Σ( X 1 ∨ X 2 ∨ X 1 ∧ X 2 ) ∧ ( X 1 ∨ X 2 ∨ X 1 ∧ X 2 ) ≃ Σ ( X ∧ 2 1 ∨ X ∧ 2 2 ∨ X ∧ 2 1 ∧ X ∧ 2 2 ∨ 2 W X 1 ∧ X 2 ∨ 2 W X ∧ 2 1 ∧ X 2 ∨ 2 W X 1 ∧ X ∧ 2 2 ) . F rom this, Σ K ( A, 1) ∧ K ( A, 1) is then homotop y equiv alen t to a w edge of the spaes in the form X = Σ m K ( Z /p r 1 1 , 1) ∧ K ( Z /p r 2 2 , 1) ∧ · · · ∧ K ( Z /p r t t , 1) with m + t ≥ 3 and m ≥ 1 . Step 3. By applying the Hilton-Milnor Theorem, w e ha v e Ω(Σ X ∨ Σ Y ) ≃ ΩΣ X × ΩΣ Y × ΩΣ((ΩΣ X ) ∧ (ΩΣ Y )) ≃ ΩΣ X × ΩΣ Y × ΩΣ W ∞ i,j = 1 X ∧ i ∧ Y ∧ j . Th us π n (Σ X ∨ Σ Y ) ∼ = π n (Σ X ) ⊕ π n (Σ Y ) ⊕ π n ∞ _ i,j = 1 Σ X ∧ i ∧ Y ∧ j ! . Note that the onnetivit y of X ∧ i ∧ Y ∧ j tends to ∞ as i, j → ∞ . By rep eating the ab o v e pro edure, π n (Σ K ( A, 1) ∧ K ( A, 1)) is isomorphism to a diret sum of the groups π n ( X ) with X giv en in the form ab o v e. Notie that Σ m K ( Z /p r 1 1 , 1) ∧ K ( Z /p r 2 2 , 1) ∧ · · · ∧ K ( Z /p r t t , 1) ≃ ∗ if the primes p i 6 = p j for some i 6 = j . Th us w e only need to ompute π 5 (Σ m K ( Z /p r 1 , 1) ∧ K ( Z /p r 2 , 1) ∧ · · · ∧ K ( Z /p r t , 1)) for a prime p . If t = 0 , the homotop y group π 5 ( S m ) is kno wn b y π 5 ( S 3 ) = π 5 ( S 4 ) = Z / 2 and π 5 ( S 5 ) = Z . F or an o dd prime p , this homotop y group an b e determined b y Prop osition 4.2 . The rest w ork in this setion is of ourse to ompute π 5 (Σ m K ( Z / 2 r 1 , 1) ∧ K ( Z / 2 r 2 , 1) ∧ · · · ∧ K ( Z / 2 r t , 1)) with m + t ≥ 3 . When m + t ≥ 5 , w e ha v e π 5 ( X ) = 0 if m + t > 5 Z / 2 min { r 1 ,...,r t } if m + t = 5 with t ≥ 1 for X = Σ m K ( Z / 2 r 1 , 1) ∧ K ( Z / 2 r 2 , 1) ∧ · · · ∧ K ( Z / 2 r t , 1) . The rst less ob vious ase is m + t = 4 , whi h will b e disussed in the next subsetion. 4.2. The Group π 5 (Σ m K ( Z / 2 r 1 , 1) ∧ K ( Z / 2 r 2 , 1) ∧ · · · ∧ K ( Z / 2 r t , 1)) for m + t = 4 and m, t ≥ 1 . W e rst onsider the ase t = 1 . Lemma 4.1. The Hur ewiz homomorphism h 5 : π 5 (Σ 2 K ( Z / 2 r , 1)) → H 5 (Σ 2 K ( Z / 2 r , 1)) is onto for any r ≥ 1 . ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 22 Pr o of. Let X = Σ 2 K ( Z / 2 r , 1) . Consider the Whitehead exat sequene π 5 ( X ) h 5 ✲ H 5 ( X ) → Γ 4 ( X ) = Z / 2 → π 4 ( X ) → H 4 ( X ) = 0 . Th us the Hurewiz homomorphism h 5 is on to if and only if π 4 ( X ) 6 = 0 . Let f : S 3 → X b e a map represen ting the generator for π 3 ( X ) = Z / 2 r . F rom the remark to Theorem 2.1, π 4 ( X ) = 0 if and only if the omp osite S 4 η ✲ S 3 f ✲ X is n ull homotopi, if and only if the map f : S 3 → X extends to a map ˜ f : Σ C P 2 → X b eause Σ C P 2 is the homotop y obre of η : S 4 → S 3 . Supp ose that there exists a map ˜ f : Σ C P 2 → X su h that ˜ f | S 4 = f . By taking mo d 2 ohomology , there is omm utativ e diagram H 5 (Σ C P 2 ; Z / 2) ✛ ˜ f ∗ H 5 ( X ; Z / 2) = Z / 2 H 3 (Σ C P 2 ; Z / 2) = H 3 ( S 3 ; Z / 2) ∼ = S q 2 ✻ ✛ ˜ f ∗ = f ∗ ∼ = H 3 ( X ; Z / 2) = Z / 2 . S q 2 ✻ It follo ws that S q 2 : H 3 ( X ; Z / 2) − → H 5 ( X ; Z / 2) is an isomorphism. On the other hand, from the fat that X = Σ 2 K ( Z / 2 r , 1) and S q 2 : H 1 ( K ( Z / 2 r , 1); Z / 2) → H 3 ( K ( Z / 2 r , 1)) is zero, S q 2 : H 3 ( X ; Z / 2) → H 5 ( X ; Z / 2) is zero. This giv es a on tradition. The assertion follo ws. Prop osition 4.4. π 5 (Σ 3 K ( Z / 2 r , 1)) = Z / 2 for r ≥ 1 . Pr o of. Let X = Σ 3 K ( Z / 2 r , 1) . Consider the Whitehead exat sequene π 6 ( X ) h 6 ✲ H 6 ( X ) → Γ 5 ( X ) = Z / 2 → π 5 ( X ) → H 5 ( X ) = 0 . By Lemma 4.1, h 6 : π 6 ( X ) → H 6 ( X ) is on to. Th us π 5 ( X ) ∼ = Γ 5 ( X ) = Z / 2 . No w w e onsider the ase t = 2 . Prop osition 4.5. L et r 1 , r 2 ≥ 1 . Then π 5 (Σ 2 K ( Z / 2 r 1 , 1) ∧ K ( Z / 2 r 2 , 1)) = Z / 2 ⊕ Z / 2 min { r 1 ,r 2 } if max { r 1 , r 2 } > 1 , Z / 4 if r 1 = r 2 = 1 . Pr o of. Let X = Σ 2 K ( Z / 2 r 1 , 1) ∧ K ( Z / 2 r 2 , 1) . By Lemma 4.1, there exists a map f i : S 5 − → Σ 2 K ( Z / 2 r i , 1) , i = 1 , 2 , whi h indues an epimorphism f i ∗ : H 5 ( S 5 ) ✲ ✲ H 5 (Σ 2 K ( Z / 2 r i , 1) . Let j : Y = Σ 2 M ( Z / 2 r 1 , 1) ∧ M ( Z / 2 r 2 , 1) ֒ → X b e the anonial inlusion. Then the map f : Y ∨ S 5 ∧ K ( Z / 2 r 2 , 1) ∨ K ( Z / 2 r 1 , 1) ∧ S 5 ( j,f 1 ∧ id , id ∧ f 2 ) ✲ X indues an isomorphism on H j ( ; Z / 2) for j ≤ 6 . Th us f ∗ : π k Y ∨ S 5 ∧ K ( Z / 2 r 2 , 1) ∨ K ( Z / 2 r 1 , 1) ∧ S 5 − → π k ( X ) ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 23 is an isomorphism for k ≤ 5 . Note that π k ( S 5 ∧ K ( Z / 2 r 2 , 1)) = π k ( K ( Z / 2 r 1 , 1) ∧ S 5 ) = 0 for k ≤ 5 . Th us j ∗ : π k ( Y ) → π k ( X ) is an isomorphism for k ≤ 5 . In partiular, π 5 ( Y ) ∼ = π 5 ( X ) . If max r 1 , r 2 > 1 , from deomp osition ( 3.7), w e ha v e Σ 2 M ( Z / 2 r 1 , 1) ∧ M ( Z / 2 r 2 , 1) ≃ M ( Z / 2 min { r 1 ,r 2 } , 4) ∨ M ( Z / 2 min { r 1 ,r 2 } , 5) and so π 5 ( Y ) ∼ = π 5 ( M ( Z / 2 min { r 1 ,r 2 } , 4)) ⊕ π 5 ( M ( Z / 2 min { r 1 ,r 2 } , 5)) = Z / 2 ⊕ Z / 2 min { r 1 ,r 2 } . Consider the ase r 1 = r 2 = 1 . F rom form ula ( 3.9 ) and the F reuden thal Susp ension Theorem, π 5 (Σ 2 M ( Z / 2 , 1) ∧ M ( Z / 2 , 1)) ∼ = π 4 (Σ M ( Z / 2 , 1) ∧ M ( Z / 2 , 1)) ∼ = Z / 4 . The pro of is nished. The last ase is t = 3 . Prop osition 4.6. L et r 1 , r 2 , r 3 ≥ 1 and let r = min { r 1 , r 2 , r 3 } . Then π 5 (Σ K ( Z / 2 r 1 , 1) ∧ K ( Z / 2 r 2 , 1) ∧ K ( Z / 2 r 3 , 1)) = Z / 2 ⊕ Z / 2 r if max { r 1 , r 2 , r 3 } > 1 , Z / 2 ⊕ Z / 2 if r 1 = r 2 = r 3 = 1 . Pr o of. Let X = Σ K ( Z / 2 r 1 , 1) ∧ K ( Z / 2 r 2 , 1) ∧ K ( Z / 2 r 3 , 1) . Let f 1 b e the omp osite S 6 g ✲ Σ 3 K ( Z / 2 r 1 , 1) ∼ = Σ K ( Z / 2 r 1 , 1) ∧ S 1 ∧ S 1 ⊂ ✲ X , where g is a map whi h indues epimorphism on H 6 ( ) b y Lemma 4.1 . Similarly , w e ha v e the maps f i : S 6 − → Σ K ( Z / 2 r 1 , 1) ∧ K ( Z / 2 r 2 , 1) ∧ K ( Z / 2 r 3 , 1) , i = 2 , 3 , b y replaing K ( Z / 2 r 1 , 1) b y K ( Z / 2 r i , 1) . Let Y = Σ M ( Z / 2 r 1 , 1) ∧ M ( Z / 2 r 2 , 1) ∧ M ( Z / 2 r 3 , 1) and let j : Y ֒ → X b e the anonial inlusion. The map Y ∨ S 6 ∨ S 6 ∨ S 6 ( j,f 1 ,f 2 ,f 3 ) ✲ X indues an isomorphism on H k ( ; Z / 2) for k ≤ 6 and so ( j, f 1 , f 2 , f 3 ) ∗ : π 5 ( Y ∨ S 6 ∨ S 6 ∨ S 6 ) = π 5 ( Y ) − → π 5 ( X ) is an isomorphism. If max { r 1 , r 2 , r 3 } > 1 , from deomp osition ( 3.7), Y ≃ M ( Z / 2 r , 4) ∨ M ( Z / 2 r , 5) ∨ M ( Z / 2 r , 5) ∨ M ( Z / 2 r , 6) and so π 5 ( Y ) = Z / 2 ⊕ Z / 2 r ⊕ Z / 2 r . If r 1 = r 2 = r 3 = 1 , there is a homotop y deomp osition [ 22 , Corollary 3.7℄ Σ R P 2 ∧ R P 2 ∧ R P 2 ≃ Σ C P 2 ∧ R P 2 ∨ Σ 4 R P 2 ∨ Σ 4 R P 2 . By [22, Lemma 6.34 (2)℄, π 5 (Σ C P 2 ∧ R P 2 ) = 0 ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 24 and so π 5 (Σ R P 2 ∧ R P 2 ∧ R P 2 ) = Z / 2 ⊕ Z / 2 , whi h nishes the pro of. Remark 4.1. F or the ase X = Σ K ( Z / 2 , 1) ∧ K ( Z / 2 , 1) ∧ K ( Z / 2 , 1) , the Hurewiz homomorphism π 5 ( X ) = Z / 2 ⊕ Z / 2 − → H 5 ( X ) = Z / 2 ⊕ Z / 2 is an isomorphism and so, in the Whitehead exat sequene, H 6 ( X ) − → Γ 5 ( X ) = Z / 2 is on to. This giv es an example that the morphism H 6 ( Z ) → Γ 5 ( Z ) in diagram ( 4.1 ) ma y not b e zero, whi h is the only example in the ase m + t = 4 . More examples will b e sho wn up in the ase m + t = 3 in the next subsetions. 4.3. The Group π 5 (Σ K ( Z / 2 r , 1)) ∼ = π 5 (Σ K ( Z / 2 r , 1) ∧ K ( Z / 2 r , 1)) . Lemma 4.2. L et X = Σ K ( Z / 2 r , 1) ∧ K ( Z / 2 r , 1) with r ≥ 1 . Then mo d 2 Hur ewiz homomorphism π 6 ( X ) h 6 ✲ H 6 ( X ) ✲ H 6 ( X ; Z / 2) is zer o. Pr o of. Reall that the mo d 2 ohomology ring H ∗ ( K ( Z / 2 r , 1); Z / 2) ∼ = E ( u 1 ) ⊗ P ( u 2 ) with the r th Bo kstein β r ( u 1 ) = u 2 . Let x i (and y i ) denote the basis for H i ( K ( Z / 2 r ; Z / 2) . The Steenro d op erations and the Bo kstein on lo w er homology are giv en b y S q 2 ∗ x 4 = x 2 S q 2 ∗ y 4 = y 2 β r ( x 4 ) = x 3 β r y 4 = y 3 β r ( x 2 ) = x 1 β r y 2 = y 1 . The Z / 2 -v etor spae s − 1 ˜ H k ( X ; Z / 2) with k ≤ 6 has a basis giv en b y the table k = 6 x 1 y 4 x 2 y 3 x 3 y 2 x 4 y 1 5 x 1 y 3 x 2 y 2 x 3 y 1 4 x 1 y 2 x 2 y 1 3 x 1 y 1 Let α ∈ H 6 ( X ; Z / 2) b e a spherial lass. Then s − 1 α = ǫ 1 x 1 y 4 + ǫ 2 x 2 y 3 + ǫ 3 x 3 y 2 + ǫ 4 x 4 y 1 for some ǫ i ∈ Z / 2 . Observ e that for an y spherial lass, β s ( α ) = S q t ∗ ( α ) = 0 for an y s, t ≥ 1 . By applying S q 2 ∗ to α , w e ha v e 0 = S q 2 ∗ ( s − 1 α ) = ǫ 1 S q 2 ∗ ( x 1 y 4 ) + ǫ 2 S q 2 ∗ ( x 2 y 3 ) + ǫ 3 S q 2 ∗ ( x 3 y 2 ) + ǫ 4 S q 2 ∗ ( x 4 y 1 ) = ǫ 1 x 1 y 2 + 0 + 0 + ǫ 4 x 2 y 1 in s − 1 H 4 ( X ; Z / 2) . Th us ǫ 1 = ǫ 4 = 0 (4.2) ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 25 By applying the Bo kstein β r to α , w e ha v e 0 = β r ( s − 1 α ) = ǫ 1 β r ( x 1 y 4 ) + ǫ 2 β r ( x 2 y 3 ) + ǫ 3 β r ( x 3 y 2 ) + ǫ 4 β r ( x 4 y 1 ) = ǫ 1 x 1 y 3 + ǫ 2 x 1 y 3 + ǫ 3 x 3 y 1 + ǫ 4 x 3 y 1 = ( ǫ 1 + ǫ 2 ) x 1 y 3 + ( ǫ 3 + ǫ 4 ) x 3 y 1 and so ǫ 1 + ǫ 2 = ǫ 3 + ǫ 4 = 0 . T ogether with equation ( 4.2 ), w e ha v e ǫ i = 0 for 1 ≤ i ≤ 4 . Th us α = 0 and hene the result. Theorem 4.1. π 5 (Σ K ( Z / 2 , 1)) ∼ = π 5 (Σ K ( Z / 2 , 1) ∧ K ( Z / 2 , 1)) = Z / 2 ⊕ Z / 2 . Pr o of. Let X = Σ K ( Z / 2 , 1) ∧ K ( Z / 2 , 1) . Notie that H 6 ( X ) = Z / 2 ⊕ Z / 2 ∼ = H 6 ( X ; Z / 2) . F rom diagram (4.1), there is an exat sequene H 6 ( X ) = Z / 2 ⊕ Z / 2 ⊂ ✲ Γ 5 ( X ) ✲ π 4 ( X ) ✲ ✲ H 5 ( X ) = Z / 2 ⊕ Z / 2 . By Corollary 3.2 , π 4 ( X ) ∼ = π 4 (Σ K ( Z / 2 , 1)) ∼ = Z / 4 . F rom Theorem 2.2 (2), there is a short exat sequene π 4 ( X ) ⊗ Z / 2 ⊕ Λ 2 ( π 3 ( X )) = Z / 2 ⊂ ✲ Γ 5 ( X ) ✲ ✲ T or( π 3 ( X ) , Z / 2) = Z / 2 . Th us the group Γ 5 ( X ) is of order 4 . It follo ws that the monomorphism H 6 ( X ) = Z / 2 ⊕ Z / 2 ⊂ ✲ Γ 4 ( X ) is an isomorphism and hene the result. Lemma 4.3. L et r 1 , r 2 ≥ 1 with max { r 1 , r 2 } > 1 . Then ther e is a short exat se quen e Z / 2 ⊕ Z / 2 ⊂ ✲ Γ 5 (Σ K ( Z / 2 r 1 , 1) ∧ K ( Z / 2 r 2 , 1)) ✲ ✲ Z / 2 . Pr o of. Let A = Z / 2 r 1 ⊕ Z / 2 r 2 . Let X = Σ K ( Z / 2 r 1 , 1) ∧ K ( Z / 2 r 2 , 1) . Then X is a retrat of Σ K ( A, 1) ∧ K ( A, 1) . F rom Theorem 3.2, Γ 4 ( X ) = Z / 2 r 1 ⊗ Z / 2 r 2 ⊗ Z / 2 = Z / 2 is summand of π 4 ( X ) and so π 4 ( X ) ∼ = Γ 4 ( X ) ⊕ H 4 ( X ) = Γ 4 ( X ) ⊕ H 3 ( K ( Z / 2 r 1 , 1) ∧ K ( Z / 2 r 2 , 1)) ∼ = Γ 4 ( X ) ⊕ T or( Z / 2 r 1 , Z / 2 r 2 ) ∼ = Z / 2 ⊕ Z / 2 min { r 1 ,r 2 } . The assertion follo ws from Theorem 2.2(2). There is a anonial hoie of sk eleton sk n ( K ( Z / 2 r , 1)) with sk n ( K ( Z / 2 r , 1) = sk n − 1 ( K ( Z / 2 r , 1) ∪ e n . This indues a hoie of sk eleton sk n (Σ K ( Z / 2 r 1 , 1) ∧ K ( Z / 2 r 2 , 1)) = Σ [ i + j ≤ n sk i ( K ( Z / 2 r 1 , 1)) ∧ sk j ( K ( Z / 2 r 2 , 1)) . Lemma 4.4. L et r 1 , r 2 ≥ 1 . L et r = min { r 1 , r 2 } . L et X = Σ K ( Z / 2 r 1 , 1) ∧ K ( Z / 2 r 2 , 1) . Then ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 26 (1) sk 4 ( X ) ≃ M ( Z / 2 r , 3) ∨ S 3 . (2) If r 1 = r 2 = 1 , then sk 5 ( X ) ≃ S 5 ∨ S 5 ∨ Σ R P 2 ∧ R P 2 . (3) If max { r 1 , r 2 } > 1 , then sk 5 ( X ) ≃ S 5 ∨ S 5 ∨ M ( Z / 2 r , 3) ∨ M ( Z / 2 r , 4) . (4) Γ 5 ( X ) ∼ = π 5 (Σ M ( Z / 2 r 1 , 1) ∧ M ( Z / 2 r 2 , 1)) = Z / 2 ⊕ Z / 2 if r 1 = r 2 = 1 , Z / 4 ⊕ Z / 2 if min { r 1 , r 2 } = 1 and max { r 1 , r 2 } > 1 , Z / 2 ⊕ Z / 2 ⊕ Z / 2 if r 1 , r 2 > 1 . Pr o of. W e ma y assume that r 1 ≤ r 2 and so r = r 1 . Let x i ( y i ) b e a basis for H i ( K ( Z / 2 r 1 , 1); Z / 2) ( H i ( K ( Z / 2 r 2 , 1); Z / 2) ), whi h represen ts the i -dimensional ell in the spae K ( Z / 2 r k , 1) . Then s − 1 ˜ H ∗ (sk n +1 ( X ); Z / 2) has a basis giv en b y x i y j with i + j ≤ n and i, j ≥ 1 . In partiular, s − 1 ˜ H ∗ (sk 4 ( X ); Z / 2) has a basis { x 1 y 1 , x 1 y 2 , x 2 y 1 } with the Bo kstein β r 1 ( x 2 y 1 ) = x 1 y 1 . There is (unique up to homotop y) 2 -lo al 3 -ell omplex with this homologial struture whi h is giv en b y S 4 ∨ M ( Z / 2 r , 3) . Th us sk 4 ( X ) ≃ S 4 ∨ M ( Z / 2 r , 3) , whi h is assertion (1). (2) and (3). Observ e that s − 1 ˜ H ∗ (sk 5 ( X ); Z / 2) has a basis { x 1 y 1 , x 1 y 2 , x 2 y 1 , x 1 y 3 , x 2 y 2 , x 3 y 3 } . Let j : Σ M ( Z / 2 r 1 , 1) ∧ M ( Z / 2 r 2 , 1) ⊂ ✲ sk 5 ( X ) b e the anonial inlusion. F or i = 1 , 2 , the omp osite S 5 g ✲ Σ 2 K ( Z / 2 r i , 1) ∼ = Σ K ( Z / 2 r i , 1) ∧ S 1 ⊂ ✲ Σ K ( Z / 2 r 1 , 1) ∧ K ( Z / 2 r 2 , 1) , in whi h g is map that induing isomorphism on H 5 ( ; Z / 2) as in Lemma 4.1 , indues a map f i : S 5 − → sk 5 ( X ) . By insp eting homology , the map ( f 1 , f 2 , j ) : S 5 ∨ S 5 ∨ Σ M ( Z / 2 r 1 , 1) ∧ M ( Z / 2 r 2 , 1) − → sk 5 ( X ) indues an isomorphism on mo d 2 homology and so it is a homotop y equiv alen t lo alized at 2 . If max { r 1 , r 2 } > 1 , then from deomp osition 3.7, Σ M ( Z / 2 r 1 , 1) ∧ M ( Z / 2 r 2 , 1) ≃ M ( Z / 2 r , 3) ∨ M ( Z / 2 r , 4) and so sk 5 ( X ) ≃ S 5 ∨ S 5 ∨ M ( Z / 2 r , 3) ∨ M ( Z / 2 r , 4) in this ase. Th us assertions (2) and (3) follo w. (4). Case I. max { r 1 , r 2 } > 1 . By the denition of the Whitehead's funtor Γ , Γ 5 ( X ) = Im ( π 5 (sk 4 ( X )) → π 5 (sk 5 ( X )) = Im( π 5 ( S 4 ∨ M ( Z / 2 r , 3)) → π 5 ( S 5 ∨ S 5 ∨ M ( Z / 2 r , 3) ∨ M ( Z / 2 r , 4))) = π 5 ( M ( Z / 2 r , 3) ∨ M ( Z / 2 r , 4)) b eause M ( Z / 2 r , 3) ∨ M ( Z / 2 r , 4))) ≃ Σ M ( Z / 2 r 1 , 1) ∧ M ( Z / 2 r 2 , 1) = ( S 4 ∨ M ( Z / 2 r , 3)) ∪ e 5 . No w it sues to ompute π 5 ( M ( Z / 2 r , 3) ∨ M ( Z / 2 r , 4)) = π 5 ( M ( Z / 2 r , 3)) ⊕ π 5 ( M ( Z / 2 r , 4)) . It is straigh t forw ard to see that π 5 ( M ( Z / 2 r , 4)) = Z / 2 represen ted b y the omp osite S 5 η ✲ S 4 ⊂ ✲ M ( Z / 2 r , 4) . ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 27 If r = min { r 1 , r 2 } = 1 , then π 5 ( M ( Z / 2 r , 3) = Z / 4 aording to [ 22, Prop osition 5.1℄. If r = min { r 1 , r 2 } > 1 , w e ompute π 5 ( M Z / 2 r , 3) . Observ e that this is in the stable range and so π 5 ( M ( Z / 2 r , 3)) ∼ = π s 5 ( M ( Z / 2 r , 3)) . No w w e are w orking in the stable homotop y ategory . Sine η : S 5 → S 4 is of order 2 , there is a map ˜ η : M ( Z / 2 , 5) − → S 4 su h that ˜ η | S 5 ≃ η . Sine the iden tit y map of M ( Z / 2 , 5) is of order 4 (see for instane [17 , Theorem 4.4℄), there is a omm utativ e diagram S 5 ⊂ j ✲ M ( Z / 2 , 5) M ( Z / 2 r , 3) pinc h ✲ ✛ ¯ η S 4 ˜ η ❄ [2 r ] ✲ S 4 , where the b ottom ro w is the obre sequene. The omp osite ¯ η : S 5 → M ( Z / 2 r , 3) repre- sen ts an elemen t in π s 5 ( M ( Z / 2 r , 3)) that maps do wn to π s 5 ( S 4 ) = Z / 2( η ) . Sine the map j : S 5 → M ( Z / 2 , 5) is of order 2 , the omp osite ¯ η ◦ j is of order 2 . It follo ws that π 5 ( M ( Z / 2 r , 3)) ∼ = π s 5 ( M ( Z / 2 r , 3)) ∼ = π s 5 ( S 4 ) ⊕ π s 5 ( S 3 ) ∼ = Z / 2 ⊕ Z / 2 . (4.3) Case I I. r 1 = r 2 = 1 . In this ase, similar to the ab o v e argumen ts, Γ 5 ( X ) = Γ 5 (Σ R P 2 ∧ R P 2 ) = π 5 (Σ R P 2 ∧ R P 2 ) . (4.4) W e ompute this homotop y group. Note that Σ R P 2 ∧ R P 2 = sk 4 ( X ) ∪ e 5 = ( S 4 ∨ M ( Z / 2 , 3) ∪ e 5 . There is a obre sequene S 4 f ✲ S 4 ∨ M ( Z / 2 , 3) g ✲ Σ R P 2 ∧ R P 2 , where the omp osite S 4 f ✲ S 4 ∨ M ( Z / 2 , 3) pro j . ✲ S 4 is of degree 2 b eause S q 1 ∗ : H 5 (Σ R P 2 ∧ R P 2 ; Z / 2) = Z / 2 − → H 4 (Σ R P 2 ∧ R P 2 ; Z / 2) = Z / 2 ⊕ Z / 2 is not zero, and the omp osite S 4 f ✲ S 4 ∨ M ( Z / 2 , 3) pro j . ✲ M ( Z / 2 , 3) is homotopi to the omp osite S 4 η ✲ S 3 ⊂ j ✲ M ( Z / 2 , 3) b eause π 4 ( M ( Z / 2 , 3)) = Z / 2 and S q 2 ∗ : H 5 (Σ R P 2 ∧ R P 2 ; Z / 2) = Z / 2 − → H 3 (Σ R P 2 ∧ R P 2 ; Z / 2) = Z / 2 is an isomorphism. Sine Γ 5 (Σ R P 2 ∧ R P 2 ) = π 5 (Σ R P 2 ∧ R P 2 ) , g ∗ : π 5 ( S 4 ∨ M ( Z / 2 , 3)) − → π 5 (Σ R P 2 ∧ R P 2 ) ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 28 is an epimorphism. By applying the Hilton-Milnor Theorem, π 5 ( S 4 ∨ M ( Z / 2 , 3)) ∼ = π 4 (Ω( S 4 ∨ M ( Z / 2 , 3))) ∼ = π 4 (Ω S 4 × Ω( M ( Z / 2 , 3)) × ΩΣ(Ω S 4 ∧ Ω M ( Z / 2 , 3))) ∼ = π 4 (Ω S 4 ) ⊕ π 4 (Ω( M ( Z / 2 , 3))) ∼ = π 5 ( S 4 ) ⊕ π 5 ( M ( Z / 2 , 3)) ∼ = Z / 2 ⊕ π 5 ( M ( Z / 2 , 3)) . F rom [22, Prop osition 5.1℄, π 5 ( M ( Z / 2 , 3)) = Z / 4 generated b y the homotop y lass of an y map φ : S 5 → M ( Z / 2 , 3) su h that the omp osite S 5 → M ( Z / 2 , 3) → S 4 is homotopi to η , the generator for π 5 ( S 4 ) = Z / 2 , and, for an y su h a hoie of map φ , the elemen t 2[ φ ] is giv en b y the homotop y lass of the omp osite S 5 η ✲ S 4 η ✲ S 3 ⊂ j ✲ M ( Z / 2 , 3) . F rom the fat that g ◦ f ≃ ∗ , the omp osite π 5 ( S 4 ) f ∗ ✲ π 5 ( π 5 ( S 4 ∨ M ( Z / 2 , 3))) g ∗ ✲ π 5 (Σ R P 2 ∧ R P 2 ) is zero. Observ e that f ∗ ( η ) = 2 η + [ j ◦ η ◦ η ] = 2[ φ ] . Th us g ∗ (2[ φ ]) = 0 and so π 5 (Σ R P 2 ∧ R P 2 ) is a quotien t group Z / 2 ⊕ Z / 2 . On the other hand, from Theorem 2.2(2), there is short exat sequene Z / 2 ⊂ ✲ Γ 5 (Σ R P 2 ∧ R P 2 ) = π 5 (Σ R P 2 ∧ R P 2 ) ✲ ✲ Z / 2 . It follo ws that π 5 (Σ R P 2 ∧ R P 2 ) = Z / 2 ⊕ Z / 2 . The pro of is nished. Let Len 3 (2 r ) = sk 3 ( K ( Z / 2 r , 1)) b e the 3 -dimensional lens spae. Lemma 4.5. L et r 1 , r 2 ≥ 1 . L et X 1 = Σ M ( Z / 2 r 1 , 1) ∧ M ( Z / 2 r 2 , 1) , X 2 = ΣLen 3 (2 r 1 ) ∧ Len 3 (2 r 2 ) , X = Σ K ( Z / 2 r 1 , 1) ∧ K ( Z / 2 r 2 , 1) . Then ther e is a ommutative diagr am Γ 5 ( X 1 ) ∼ = ✲ π 5 ( X 1 ) Γ 5 ( X 2 ) ∼ = ❄ ⊂ x ✲ π 5 ( X 2 ) ❄ ∩ x ✲ ✲ Z / 2 r 1 ⊕ Z / 2 r 2 H 6 ( X ) ✲ Γ 5 ( X ) ∼ = ❄ ✲ π 5 ( X ) ❄ ❄ ✲ ✲ ( Z / 2 min { r 1 ,r 2 } ) ⊕ 2 , ❄ ❄ wher e the r ows ar e exat and the midd le a splitting short exat se quen e. ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 29 Pr o of. As in the pro of in Lemma 4.4, s − 1 ˜ H k ( X ) for k ≤ 6 has a basis { x i y j | i + j ≤ 6 , i, j ≥ 1 } . Th us sk 4 ( X ) ⊆ X 1 ⊆ sk 5 ( X ) ⊆ X 2 ⊆ sk 7 ( X ) and so the omm utativ e diagram follo ws, where Γ 5 ( X 1 ) ∼ = Γ 5 ( X 2 ) ∼ = Γ 5 ( X ) are giv en b y Lemma 4.4 . Sine sk 5 ( X ) ⊆ X 2 , π 5 ( X 2 ) → π 5 ( X ) is on to. No w w e sho w that the middle ro w in the diagram splits o. By taking the susp ension, there is a omm utativ e diagram of short exat sequenes Γ 5 ( X 2 ) ⊂ ✲ π 5 ( X 2 ) ✲ ✲ H 5 ( X 2 ) Γ 6 (Σ X 2 ) ∼ = ❄ ⊂ ✲ π 6 (Σ X 2 ) ❄ ✲ ✲ H 6 ( X 2 ) , ∼ = ❄ where the left olumn is an isomorphism b eause Γ 5 ( X 2 ) ∼ = π 5 ( X 1 ) ∼ = π 6 (Σ X 1 ) ∼ = Γ 6 (Σ X 2 ) . Th us π 5 ( X 2 ) ∼ = π 6 (Σ X 2 ) (4.5) b y the 5 -Lemma. F rom Lemma 4.1 , there is a map g : S 5 − → Σ 2 K ( Z / 2 r , 1) induing an isomorphism on H 5 ( ; Z / 2) . It follo ws that Σ 2 Len 3 (2 r ) = Σ 2 sk 3 ( K ( Z / 2 r , 1)) ≃ S 5 ∨ Σ 2 M ( Z / 2 r , 1) (4.6) and so Σ X 2 = Σ 2 Len 3 (2 r 1 ) ∧ Len 3 (2 r 2 ) ≃ ( S 5 ∨ Σ 2 M ( Z / 2 r 1 , 1)) ∧ Len 3 (2 r 2 ) ≃ Σ 5 Len 3 (2 r 2 ) ∨ Σ 2 Len 3 (2 r 2 ) ∧ M ( Z / 2 r 1 , 1) ≃ S 8 ∨ M ( Z / 2 r 2 , 6) ∨ M ( Z / 2 r 1 , 6) ∨ M ( Z / 2 r 2 , 3) ∧ M ( Z / 2 r 1 , 1) . (4.7) Th us π 6 (Σ X 2 ) ∼ = Z / 2 r 1 ⊕ Z / 2 r 2 ⊕ Γ 6 (Σ X 2 ) and hene the result. Theorem 4.2. L et r > 1 . Then π 5 (Σ K ( Z / 2 r , 1)) ∼ = π 5 (Σ K ( Z / 2 r , 1) ∧ K ( Z / 2 r , 1)) ∼ = Z / 2 ⊕ Z / 2 r ⊕ Z / 2 r . Pr o of. Let X = Σ K ( Z / 2 r , 1) ∧ K ( Z / 2 r , 1) . By Lemma 4.5, π 5 ( X ) ∼ = Z / 2 r ⊕ Z / 2 r ⊕ Im(Γ 5 ( X ) → π 5 ( X )) . F rom Lemma 4.4, Γ 5 ( X ) = Z / 2 ⊕ 3 . ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 30 By Lemma 4.1, the omp osite π 6 ( X ) → H 6 ( X ) = Z / 2 r ⊕ Z / 2 r → H 6 ( X ; Z / 2) is zero. Th us H 6 ( X ) = Z / 2 r ⊕ Z / 2 r − → Γ 5 ( X ) = Z / 2 ⊕ Z / 2 ⊕ Z / 2 detets t w o opies of Z / 2 -summands in Γ 5 ( X ) . The pro of is nished. 4.4. The Group π 5 (Σ K ( Z / 2 r 1 , 1) ∧ K ( Z / 2 r 2 , 1)) with r 1 < r 2 . Our omputation is giv en b y analyzing the ell struture. Let x i b e a basis for ˜ H i ( K ( Z / 2 r 1 , Z / 2)) and let y i b e a basis for ˜ H i ( K ( Z / 2 r 2 ; Z / 2)) . Then s − 1 ˜ H k (Σ K ( Z / 2 r 1 , 1) ∧ K ( Z / 2 r 2 , 1) , k ≤ 6 , has a basis { x i y j | i + j ≤ 6 } . F rom the assumption that r 1 < r 2 , the Steenro d op eration and Bo kstein are indiated b y the follo wing diagram k = 6 x 2 y 3 x 1 y 4 x 3 y 2 x 4 y 1 k = 5 x 1 y 3 β r 1 ❄ ✛ β r 2 0 x 2 y 2 x 3 y 1 β r 2 ❄ ✛ β r 1 k = 4 x 1 y 2 S q 2 ∗ ❄ ✛ β r 1 x 3 y 1 S q 2 ∗ ❄ k = 3 x 1 y 1 , ✛ β r 1 (4.8) where the dash arro ws mean that the next Bo kstein β r 2 , whi h omes from H ∗ ( K ( Z / 2 r 2 , 1)) , do es not atually happ en in the Bo kstein sp etral sequene up to this range. Lemma 4.6. L et r 2 > r 1 ≥ 1 and let X = Σ K ( Z / 2 r 1 , 1) ∧ K ( Z / 2 r 2 , 1) . Then the susp ension E : π 5 ( X ) − → π 6 (Σ X ) is an isomorphism. Pr o of. F rom form ula ( 4.5 ) together with the fat that π n − 1 (sk n ( Y )) ∼ = π n − 1 ( Y ) , π 5 (sk 6 (ΣLen 3 (2 r 1 ) ∧ Len 3 (2 r 2 ))) − → π 6 (sk 7 (ΣLen 3 (2 r 1 ) ∧ Len 3 (2 r 2 ))) . Notie that sk 6 ( X ) = sk 6 (ΣLen 3 (2 r 1 ) ∧ Len 3 (2 r 2 )) ∪ e 6 ∪ e 6 ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 31 indiated b y the elemen ts x 1 y 4 and x 4 y 1 in diagram (4.8). Then there is a omm utativ e diagram of righ t exat sequenes π 5 ( S 5 ∨ S 5 ) f ∗ ✲ π 5 ( Z ) ✲ ✲ π 5 (sk 6 ( X )) π 6 ( S 6 ∨ S 6 ) ∼ = ❄ Σ f ∗ ✲ π 6 (Σ Z ) ∼ = ❄ ✲ ✲ π 6 (sk 7 (Σ X )) , E ❄ where Z = sk 6 (ΣLen 3 (2 r 1 ) ∧ Len 3 (2 r 2 )) and f : S 5 ∨ S 5 → Y is the atta hing map for sk 6 ( X ) . The assertion follo ws b y the 5 -lemma. Theorem 4.3. L et r 2 > r 1 ≥ 1 . Then π 5 (Σ K ( Z / 2 r 1 , 1) ∧ K ( Z / 2 r 2 , 1)) = Z / 2 ⊕ Z / 4 if r 1 = 1 , r 2 = 2 , Z / 2 ⊕ Z / 8 if r 1 = 1 , r 2 ≥ 3 , Z / 2 ⊕ Z / 2 r 1 ⊕ Z / 2 r 1 +1 if r 2 > r 1 > 1 . Pr o of. F rom Lemma 4.6 , it sues to ompute π 6 (Σ 2 K ( Z / 2 r 1 , 1) ∧ K ( Z / 2 r 2 , 1)) . Let X = sk 7 (Σ 2 K ( Z / 2 r 1 , 1) ∧ K ( Z / 2 r 2 , 1)) . F rom splitting form ula (4.7), sk 7 (Σ 2 Len 3 (2 r 1 ) ∧ Len 3 (2 r 2 )) ≃ M ( Z / 2 r 2 , 6) ∨ M ( Z / 2 r 1 , 6) ∨ M ( Z / 2 r 1 , 4) ∨ M ( Z / 2 r 1 , 5) . Let Y = sk 7 (Σ 2 Len 3 (2 r 1 ) ∧ Len 3 (2 r 2 )) . Then s − 2 ˜ H ∗ ( Y ; Z / 2 ) has a basis listed in dia- gram (4.8) exluding the elemen ts x 1 y 4 and x 4 y 1 . Let P n (2 r ) = M ( Z / 2 r , n − 1) . The mo d homology ˜ H ∗ ( P n (2 r ); Z / 2) has a basis u r n − 1 and v r n with degrees | u r n − 1 | = n − 1 , v r n | = n and the Bo kstein β r ( v r n ) = u r n − 1 . Sine X = Y ∪ e 7 ∪ e 7 , there is a obre sequene S 6 1 ∨ S 6 2 f ✲ P 7 (2 r 1 ) ∨ P 7 (2 r 2 ) ∨ P 5 (2 r 1 ) ∨ P 6 (2 r 1 ) g ✲ X q ✲ S 7 1 ∨ S 7 2 , where g ∗ ( u r 1 6 , v r 2 7 ; u r 2 6 , v r 2 7 ; u r 1 4 , v r 1 5 ; u r 1 5 , v r 1 6 ) = s 2 ( x 1 y 3 , x 2 y 3 ; x 3 y 1 , x 3 y 2 ; x 1 y 1 , x 2 y 1 ; x 1 y 2 , x 2 y 2 ) for at hing the orresp onding elemen ts in ˜ H ∗ ( X ; Z / 2) . Here the map f is the atta h- ing map with f | S 6 1 , f | S 6 2 orresp onding to the homologial lasses s 2 ( x 4 y 1 ) and s 2 ( x 1 y 4 ) , resp etiv ely . Namely , the indued b oundary map q : X → S 7 1 ∨ S 7 2 has the homologial prop ert y that q ∗ : H 7 ( X ; Z / 2) → H 7 ( S 7 1 ∨ S 7 2 ) is giv en b y q ∗ ( x 2 y 3 ) = q ∗ ( x 3 y 2 ) = 0 , q ∗ ( s 2 ( x 4 y 1 )) = ι 1 , q ∗ ( s 2 ( x 1 y 4 )) = ι 2 , where ι j is the basis for H 7 ( S 7 j ; Z / 2) . F or j = 1 , 2 , let X j b e the homotop y obre of f j . Then there is a omm utativ e diagram S 6 1 ∨ S 6 2 f ✲ P 7 (2 r 1 ) ∨ P 7 (2 r 2 ) ∨ P 5 (2 r 1 ) ∨ P 6 (2 r 1 ) g ✲ X q ✲ S 7 1 ∨ S 7 2 S 6 j ∪ ✻ f | S 6 j ✲ P 7 (2 r 1 ) ∨ P 7 (2 r 2 ) ∨ P 5 (2 r 1 ) ∨ P 6 (2 r 1 ) w w w w w w w w w g j ✲ X j θ j ✻ q j ✲ S 7 j . ∪ ✻ (4.9) Statemen t 1. θ j ∗ : ˜ H ∗ ( X j ; Z / 2) → ˜ H ∗ ( X ; Z / 2) is a monomorphism. Mor e over, Im( θ 1 ∗ : H 7 ( X 1 ; Z / 2) → H 7 ( X ; Z / 2)) ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 32 has the b asis given by { s 2 ( x 2 y 3 ) , s 2 ( x 3 y 2 ) , s 2 ( x 4 y 1 ) } and Im( θ 2 ∗ : H 7 ( X 2 ; Z / 2) → H 7 ( X ; Z / 2)) has the b asis given by { s 2 ( x 2 y 3 ) , s 2 ( x 3 y 2 ) , s 2 ( x 1 y 4 ) } . Thus a b asis for ˜ H ∗ ( X j ; Z / 2) an b e liste d in diagr am (4.8) by r emoving one element. The statemen t follo ws immediately b y applying mo d 2 homology to diagram (4.9 ), where the only simple omputation is giv en b y he king the image of θ j ∗ . Statemen t 2. The omp osite φ j : S 6 j f | S 6 j ✲ P 7 (2 r 1 ) ∨ P 7 (2 r 2 ) ∨ P 5 (2 r 1 ) ∨ P 6 (2 r 1 ) pro j . ✲ P 7 (2 r 1 ) is nul l homotopi for j = 1 , 2 . Consider the omm utativ e diagram of obre sequenes S 6 j f | S 6 j ✲ P 7 (2 r 1 ) ∨ P 7 (2 r 2 ) ∨ P 5 (2 r 1 ) ∨ P 6 (2 r 1 ) g j ✲ X j S 6 j w w w w w w w w w ✲ P 7 (2 r 1 ) pro j . ❄ ✲ Z . δ ❄ Then dim ˜ H ∗ ( Z ; Z / 2) = 3 and δ ∗ : H ∗ ( X j ; Z / 2) → H ∗ ( Z ; Z / 2) is on to. F rom diagram ( 4.8 ), the Bo kstein β t : H 7 ( Z ; Z / 2) → H 6 ( Z ; Z / 2) is 0 for t < r 1 with the rst non-trivial Bo kstein giv en b y β r 1 oming from β r 1 ( x 2 y 3 ) = x 1 y 3 in diagram ( 4.8 ). Note that π 6 ( P 7 (2 r 1 )) = Z / 2 r 1 generated b y the inlusion ¯ ι : S 6 ֒ → P 7 (2 r 1 ) . Then the homotop y lass [ φ j ] = k ¯ ι for some k ∈ Z . If k ≡ 1 mo d 2 , then dim ˜ H ∗ ( Z ; Z / 2) = 1 whi h on tradits to that dim ˜ H ∗ ( Z ; Z / 2) = 3 . Th us k m ust b e divisible b y 2 . Let k = 2 t k ′ with k ′ ≡ 1 mo d 2 for some t ≥ 1 . If t < r 1 , then there is a non trivial Bo kstein β t on ˜ H ∗ ( Z ; Z / 2) whi h is imp ossible from the ab o v e. Hene t ≥ r 1 and so [ φ j ] = 0 in π 6 ( P 7 (2 r 1 )) = Z / 2 r 1 . Statemen t 2 follo ws. Statemen t 3. The omp osite ψ : S 6 1 f | S 6 1 ✲ P 7 (2 r 1 ) ∨ P 7 (2 r 2 ) ∨ P 5 (2 r 1 ) ∨ P 6 (2 r 1 ) pro j . ✲ P 6 (2 r 1 ) is nul l homotopi. Consider the omm utativ e diagram of obre sequenes S 6 1 f | S 6 1 ✲ P 7 (2 r 1 ) ∨ P 7 (2 r 2 ) ∨ P 5 (2 r 1 ) ∨ P 6 (2 r 1 ) g 1 ✲ X 1 S 6 1 w w w w w w w w w ✲ P 6 (2 r 1 ) p pro j . ❄ g ′ ✲ W . δ ❄ ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 33 Then dim ˜ H ∗ ( W ; Z / 2) = 3 and δ ∗ : ˜ H ( X 1 ; Z / 2) → ˜ H ∗ ( W ; Z / 2) is on to. Moreo v er, H 7 ( W ; Z / 2) has a basis giv en b y δ ∗ ( s 2 ( x 4 y 1 )) . By Statemen t 1, a basis for ˜ H ∗ ( X 1 ; Z / 2) is listed in diagram (4.8) b y remo ving x 1 y 4 . The anonial pro jetion p : P 7 (2 r 1 ) ∨ P 7 (2 r 2 ) ∨ P 5 (2 r 1 ) ∨ P 6 (2 r 1 ) − → P 6 (2 r 1 ) has the prop ert y that p ∗ ( u r 1 5 ) = u r 1 5 , p ∗ ( v r 1 6 ) = v r 1 6 and p ∗ ( x ) = 0 for x to the other elemen ts in the basis for ˜ H ∗ ( P 7 (2 r 1 ) ∨ P 7 (2 r 2 ) ∨ P 5 (2 r 1 ) ∨ P 6 (2 r 1 ); Z / 2) . In partiular, p ∗ ( v r 1 5 ) = 0 . Note that S q 2 ∗ δ ∗ ( s 2 ( x 4 y 1 )) = δ ∗ ( S q 2 ∗ ( s 2 ( x 4 y 1 ))) = δ ∗ ( s 2 ( x 2 y 1 )) = δ ∗ ( g 1 ∗ ( v r 1 5 ) = g ′ ∗ ◦ p ∗ ( v r 1 5 ) = 0 . If follo ws that S q 2 ∗ : H 7 ( W ; Z / 2) → H 5 ( W ; Z / 2) is zero. F rom the exat sequene π 6 ( S 5 ) = Z / 2 2 r 1 0 ✲ π 6 ( S 5 ) = Z / 2 ✲ π 6 ( P 6 (2 r 1 ) ✲ π 5 ( S 5 ) = Z 2 r 1 ✲ Z , w e ha v e π 6 ( P 6 (2 r 1 )) = Z / 2 (4.10) generated b y the omp osite ¯ η : S 6 η ✲ S 5 ⊂ ✲ P 6 (2 r 1 ) . Th us the homotop y lass [ ψ ] = 0 or ¯ η . If [ ψ ] = ¯ η , then S q 2 : H 7 ( W ; Z / 2) → H 5 ( W ; Z / 2) is not zero, whi h is imp ossible from the ab o v e. Hene [ ψ ] = 0 . This nishes the pro of for Statemen t 3. Statemen t 4. Ther e is a homotopy de omp osition X ≃ P 7 (2 r 1 ) ∨ T 1 ∨ T 2 , wher e ˜ H ∗ ( T 1 ; Z / 2) and ˜ H ∗ ( T 2 ; Z / 2) have b asis liste d by the midd le and the right mo dules in diagr am (4.6), r esp e tively. F rom Statemen ts 2 and 3, the atta hing map f | S 6 1 maps in to the subspae P 7 (2 r 2 ) ∨ P 5 (2 r 1 ) up to homotop y b eause, in the range of π 6 , w e ha v e π 6 ( P 7 (2 r 1 ) ∨ P 7 (2 r 2 ) ∨ P 5 (2 r 1 ) ∨ P 6 (2 r 1 )) ∼ = π 6 ( P 7 (2 r 1 )) ⊕ π 6 ( P 7 (2 r 2 )) ⊕ π 6 ( P 5 (2 r 1 )) ⊕ π 6 ( P 6 (2 r 1 )) . ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 34 Th us there is a homotop y omm utativ e diagram of obre sequenes S 6 1 f ′ ✲ P 7 (2 r 2 ) ∨ P 5 (2 r 1 ) ✲ T 2 S 6 1 ∨ S 6 2 ❄ ∩ f ✲ P 7 (2 r 1 ) ∨ P 7 (2 r 2 ) ∨ P 5 (2 r 1 ) ∨ P 6 (2 r 1 ) ❄ ∩ g ✲ X i 1 ❄ S 6 1 pro j . ❄ f ′ ✲ P 7 (2 r 2 ) ∨ P 5 (2 r 1 ) pro j . ❄ ✲ T 2 . q 1 ❄ (4.11) F rom Statemen t 2, there is a homotop y omm utativ e diagram of obre sequenes S 6 1 ∨ S 6 2 f ✲ P 7 (2 r 1 ) ∨ P 7 (2 r 2 ) ∨ P 5 (2 r 1 ) ∨ P 6 (2 r 1 ) g ✲ X ∗ ❄ ✲ P 7 (2 r 1 ) pro j . ❄ = = = = = = = = = = = = = = = = P 7 (2 r 1 ) . q 2 ❄ No w the omp osite P 7 (2 r 1 ) ∨ T 2 ( g | P 7 (2 r 1 ) ,i 1 ) ✲ X ( q 2 ,q 1 ) ✲ P 7 (2 2 r 1 ) ∨ T 2 is a homotop y equiv alene b y insp eting the homology and hene the statemen t. Computation of the Homotopy Gr oup: F rom Statemen t 4, w e ha v e π 6 ( X ) ∼ = π 6 ( P 7 (2 r 1 ) ∨ T 1 ∨ T 2 ) ∼ = π 6 ( P 7 (2 r 1 )) ⊕ π 6 ( T 1 ) ⊕ π 6 ( T 2 ) ∼ = Z / 2 r 1 ⊕ π 6 ( T 1 ) ⊕ π 6 ( T 2 ) . F or omputing π 6 ( T 1 ) , sine T 1 = P 6 (2 r 1 ) ∪ e 7 , there is a righ t exat sequene π 6 ( S 6 ) = Z ✲ π 6 ( P 6 (2 r 1 )) = Z / 2 ✲ ✲ π 6 ( T 1 ) , where π 6 ( P 6 (2 r 1 )) = Z / 2 is giv en in form ula (4.10). F rom diagram (4.6), S q 2 ∗ : H 7 ( T 1 ; Z / 2) − → H 5 ( T 1 ; Z / 2) is an isomorphism and so the atta hing map S 6 → P 6 (2 r 1 ) of T 1 is non-trivial. It follo ws that π 6 ( T 1 ) = 0 . No w w e ompute π 6 ( T 2 ) . F rom diagram 4.11 , there is a righ t exat sequene π 6 ( S 6 ) = Z / f ′ ∗ ✲ π 6 ( P 7 (2 r 2 ) ∨ P 5 (2 r 1 )) = π 6 ( P 7 (2 r 2 )) ⊕ π 6 ( P 5 (2 r 1 )) ✲ ✲ π 6 ( T 2 ) . Note that a basis for ˜ H ∗ ( T 2 ) an b e listed in the righ t mo dule of diagram (4.6). The omp osite π 6 ( S 6 ) = Z f ′ ∗ ✲ π 6 ( P 7 (2 r 2 )) ⊕ π 6 ( P 5 (2 r 1 )) pro j . ✲ π 6 ( P 7 (2 r 2 )) = Z / 2 r 2 is of degree 2 r 1 b eause of the existene of the Bo kstein β r 1 . Mo v eo v er the omp osite S 6 f ′ ✲ P 7 (2 r 2 ) ∨ P 5 (2 r 1 ) pro j . ✲ P 5 (2 r 1 ) pinc h ✲ S 5 (4.12) is homotopi to η b eause of the existene of the Steenro d op eration S q 2 ∗ . ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 35 Case I. r 1 = 1 . A ording to [22, Prop osition 5.1℄, π 6 ( P 5 (2)) = Z / 4 generated b y the homotop y lass of an y ma y S 6 → P 5 (2) su h that the omp osite S 6 → P 5 (2) → S 5 is η . It follo ws that there is a righ t exat sequene Z f ′ ∗ =(2 r 1 ,λ ) ✲ Z / 2 r 2 ⊕ Z / 4 ✲ ✲ π 6 ( T 2 ) , where λ : Z → Z / 4 is an epimorphism. Th us π 6 ( T 2 ) = Z / 4 if r 2 = 2 , Z / 8 if r 2 ≥ 3 . (4.13) Case I I. r 1 > 1 . F rom form ula 4.3 , w e ha v e π 6 ( P 5 (2 r 1 )) = Z / 2 ⊕ Z / 2 . Sine the omp osite in (4.12 ) is essen tial, the omp osite π 6 ( S 6 ) = Z f ′ ∗ ✲ π 6 ( P 7 (2 r 2 )) ⊕ π 6 ( P 5 (2 r 1 )) pro j . ✲ π 6 ( P 5 (2 r 1 )) = Z / 2 ⊕ Z / 2 is non trivial and so there is righ t exat sequene Z (2 r 1 ,λ ) ✲ Z / 2 r 2 ⊕ ( Z / 2 ⊕ Z / 2) ✲ ✲ π 6 ( T 2 ) with λ : Z → Z / 2 ⊕ Z / 2 non trivial. It follo ws that π 6 ( T 2 ) = Z / 2 r 1 +1 ⊕ Z / 2 for r 1 > 1 . (4.14) The pro of is nished no w. 4.5. The Group π 5 (Σ 2 K ( Z / 2 r , 1)) . W e use the sp etral sequene indued from Carls- son's onstrution for omputing this group. Let A b e an ab elian group and 0 → A 1 δ → A 0 → A → 0 a t w o-step at resolution of A , i.e. A 0 is a free ab elian group. The diagram (1.1) implies that there is a natural isomorphism π 4 (Σ 2 K ( A, 1)) ≃ A e ⊗ A, where e ⊗ 2 : e ⊗ 2 ( A ) = A e ⊗ A := A ⊗ A/ ( a ⊗ b + b ⊗ a, a, b ∈ A ) . Giv en a free ab elian group ¯ A, theorem 2.2 (2) implies the follo wing natural exat sequene: Γ 5 (Σ 2 K ( ¯ A , 1)) / / π 5 (Σ 2 K ( ¯ A , 1)) / / / / H 5 (Σ 2 K ( ¯ A , 1)) ¯ A e ⊗ ¯ A ⊗ Z / 2 ⊕ Λ 2 ( ¯ A ) / / π 5 (Σ 2 K ( ¯ A, 1)) / / / / Λ 3 ( ¯ A ) ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 36 The sp etral sequene (2.9 ) for n = 2 , giv es the follo wing diagram of exat sequenes: L 1 Λ 3 ( A ) A e ⊗ A ⊗ Z / 2 ⊕ Λ 2 ( A ) / / π 5 (Σ 2 K ( A, 1)) / / / / L 1 e ⊗ 2 ( A ) π 0 ( π 5 (Σ 2 K ( N − 1 ( A 1 δ → A 0 ) , 1))) / / π 5 (Σ 2 K ( A, 1)) / / / / π 1 ( π 4 Σ 2 K ( N − 1 ( A 1 δ → A 0 ) , 1)) Λ 3 ( A ) (4.15) Consider the rst deriv ed funtor of the funtor e ⊗ 2 . The short exat sequene LS P 2 ( A ) → L ⊗ 2 ( A ) → L e ⊗ 2 ( A ) in the deriv ed ategory has the follo wing mo del: Λ 2 ( A 1 ) δ 2 / / _ A 1 ⊗ A 0 δ 1 / / _ S P 2 ( A 0 ) _ A 1 ⊗ A 1 δ ′ 2 / / ( A 1 ⊗ A 0 ) ⊕ ( A 0 ⊗ A 1 ) δ ′ 1 / / A 0 ⊗ A 0 S P 2 ( A 1 ) δ ′′ 2 / / A 1 ⊗ A 0 δ ′′ 1 / / A 0 e ⊗ A 0 with δ 2 ( a 1 ∧ a ′ 1 ) = a 1 ⊗ δ ( a ′ 1 ) − a ′ 1 ⊗ δ ( a 1 ) δ 1 ( a 1 ⊗ a 0 ) = a 0 δ ( a 1 ) δ ′ 2 ( a 1 ⊗ a ′ 1 ) = ( a 1 ⊗ δ ( a ′ 1 ) , − a ′ 1 ⊗ δ ( a 1 )) δ ′ 1 ( a 1 ⊗ a 0 , a ′ 1 ⊗ a ′ 0 ) = δ ( a 1 ) ⊗ a 0 + δ ( a ′ 1 ) ⊗ a ′ 0 δ ′′ 2 ( a 1 a ′ 1 ) = a 1 ⊗ δ ( a ′ 1 ) + a ′ 1 ⊗ δ ( a 1 ) δ ′′ 1 ( a 1 ⊗ a 0 ) = ∂ ( a 1 ) e ⊗ a 0 for a 0 , a ′ 0 ∈ A 0 , a 1 , a ′ 1 ∈ A 1 . F or n ≥ 2 , lo oking at the resolution Z n → Z of the yli group Z /n , w e obtain the follo wing represen tativ e of the elemen t L e ⊗ 2 ( Z /n ) in the deriv ed ategory: Z 2 n → Z n → Z / 2 ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 37 In partiular, L 1 e ⊗ 2 ( Z / 2 k ) = Z / 2 k +1 , k ≥ 1 . (4.16) Here L 1 e ⊗ 2 denotes the rst deriv ed funtor of e ⊗ 2 (see 2.2). W e will use the follo wing: Lemma 4.7. (L emma 2.1 fr om [ 21℄ ) L et G ∗ b e a simpliial gr oup and let n ≥ 0 . Supp ose that π 0 ( G ∗ ) ats trivial ly on π n ( G ∗ ) . Then the homotopy gr oup π n ( G ∗ ) is ontaine d in the enter of G n / B G n , wher e B G n is the n th simpliial b oundary sub gr oup of G n . Theorem 4.4. The homotopy gr oup π 5 (Σ 2 K ( Z / 2 r , 1)) = Z / 8 if r = 1 Z / 2 r +1 ⊕ Z / 2 if r > 1 . Pr o of. Case 1: r = 1 . The natural epimorphism Z → Z / 2 indues the homomorphisms π n ( S 3 ) = π n (Σ 2 K ( Z , 1)) → π n (Σ 2 K ( Z / 2 , 1)) = π n (Σ 2 R P ∞ ) , n ≥ 1 . The diagram (4.15 ) together with (4.16) implies the follo wing short exat sequenes: Z / 2 π 5 ( S 3 ) Z / 2 / / π 5 (Σ 2 R P ∞ ) / / / / Z / 4 (4.17) Consider this map simpliially , at the lev el of the natural map b et w een the Carlsson onstrutions F ( S 2 ) = F Z ( S 2 ) → F Z / 2 ( S 2 ) : F ( S 2 ) 4 − → . . . − → ← − . . . ← − F ( S 2 ) 3 − → − → − → − → ← − ← − ← − Z ↓ ↓ ↓ F Z / 2 ( S 2 ) 4 − → . . . − → ← − . . . ← − F Z / 2 ( S 2 ) 3 − → − → − → − → ← − ← − ← − F Z / 2 ( S 2 ) 2 Here F Z / 2 ( S 2 ) k is the free pro dut of k 2 opies of Z / 2 . In partiular F Z / 2 ( S 2 ) 4 = h s j s i ( σ ) 0 ≤ i < j ≤ 3 | ( s j s i ( σ )) 2 = 1 i Using the desription of the elemen t (2.7 ), w e see that the simpliial yle whi h denes the image of π 5 ( S 3 ) in π 5 (Σ 2 R P ∞ ) an b e hosen of the form [[ s 2 s 1 ( σ ) , s 1 s 0 ( σ )] , [ s 2 s 1 ( σ ) , s 2 s 0 ( σ )]] ∈ F Z / 2 ( S 2 ) 4 With the help of lemma 4.7, w e ha v e [[ s 2 s 1 ( σ ) , s 1 s 0 ( σ )] , [ s 2 s 1 ( σ ) , s 2 s 0 ( σ )]] = [[( s 2 s 1 ( σ ) , s 1 s 0 ( σ )] , ( s 2 s 1 ( σ ) s 2 s 0 ( σ )) 2 ] ≡ [[ s 2 s 1 ( σ ) , s 1 s 0 ( σ )] , ( s 2 s 1 ( σ ) , s 2 s 0 ( σ ))] 2 mo d B Z / 2 ( S 2 ) 4 sine [[ s 2 s 1 ( σ ) , s 1 s 0 ( σ )] , s 2 s 1 ( σ ) s 2 s 0 ( σ )] is a yle in F Z / 2 ( S 2 ) . That is, the image of the elemen t π 5 ( S 3 ) is divisible b y 2 in π 5 (Σ 2 R P ∞ ) . The diagram (4.17 ) implies the result. ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 38 Case 2: r > 1 . No w the diagram ( 4.15 ) together with (4.16 ) implies the follo wing short exat sequene 0 → Z / 2 → π 5 (Σ 2 K ( Z / 2 r , 1)) → Z / 2 r +1 → 0 (4.18) Therefore, π 5 (Σ 2 K ( Z / 2 r , 1)) is either Z / 2 r +2 or Z / 2 r +1 ⊕ Z / 2 . By theorem 2.2 (2), the Whitehead exat sequene for Σ 2 K ( A, 1) has the follo wing form: A e ⊗ A ⊗ Z / 2 ⊕ Λ 2 ( A ) _ H 4 ( A ) / / Γ 5 (Σ 2 K ( A, 1)) / / π 5 (Σ 2 K ( A, 1)) / / H 3 ( A ) T or ( A, Z / 2) (4.19) F or A = Z / 2 r it is of the follo wing form: Z / 2 _ Γ 5 (Σ 2 K ( Z / 2 r , 1)) / / π 5 (Σ 2 K ( Z / 2 r , 1)) / / / / Z / 2 r Z / 2 (4.20) The natural pro jetion Z / 2 r ։ Z / 2 indues the map Z / 2 _ ≃ / / Z / 2 _ Γ 5 (Σ 2 K ( Z / 2 r , 1)) / / Γ 5 (Σ 2 R P ∞ ) Z / 2 0 / / Z / 2 (4.21) where the lo w er map is zero sine the indued map T or ( Z / 2 r , Z / 2) → T or ( Z / 2 , Z / 2) is zero. The fat that π 5 (Σ 2 R P ∞ ) = Z / 8 together with diagram (4.19 ) implies that Γ 5 (Σ 2 R P ∞ ) = Z / 4 . Hene Γ 5 (Σ 2 K ( Z / 2 r , 1)) = Z / 2 ⊕ Z / 2 , sine there is no an y endomor- phism Z / 4 → Z / 4 with zero map on quotien ts Z / 2 → Z / 2 (as in diagram (4.21 )). The dia- gram (4.20 ) and exat sequene (4.18) implies that π 5 (Σ 2 K ( Z / 2 r , 1)) = Z / 2 r +1 ⊕ Z / 2 . ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 39 4.6. Appliations. Prop osition 4.7. The gr oup Γ 5 (Σ R P ∞ ) = Z / 2 ⊕ Z / 2 ⊕ Z / 2 . Pr o of. Consider the Whitehead exat sequene π 6 (Σ R P ∞ ) h 0 ✲ H 6 (Σ R P ∞ ) = Z / 2 ✲ Γ 5 (Σ R P ∞ ) ✲ π 5 (Σ R P ∞ ) = Z / 2 ⊕ Z / 2 ✲ H 5 (Σ R P ∞ ) = 0 , where π 5 (Σ R P ∞ ) = Z / 2 ⊕ Z / 2 b y Theorem 4.1 and the Hurewiz homomorphism h 6 : π 6 (Σ R P ∞ ) → H 6 (Σ R P ∞ ) is zero b eause, otherwise, it w ould indues a splitting of Σ R P 5 whi h is imp ossible b y insp eting the Steenro d op eration on mo d 2 homology . Th us the order of Γ 5 (Σ R P ∞ ) is 8 . W e ha v e to determine the group Γ 5 (Σ R P ∞ ) . By the denition, Γ 5 (Σ R P ∞ ) = Im( π 5 (Σ R P 3 ) → π 5 (Σ R P 4 )) . Th us the inlusion Σ R P 3 → Σ R P ∞ indues an epimorphism Γ 5 (Σ R P 3 ) ✲ ✲ Γ 5 (Σ R P ∞ ) . Note that R P 3 = S O (3) and so, b y the Hopf bration, π 5 (Σ S O (3)) ∼ = π 5 ( B S O (3)) ⊕ π 5 (Σ S O (3) ∧ S O (3)) ∼ = π 4 ( S O (3)) ⊕ π 5 (Σ R P 3 ∧ R P 3 ) ∼ = Z / 2 ⊕ π 5 (Σ R P 3 ∧ R P 3 ) . F rom Lemmas 4.4 and 4.5, π 5 (Σ R P 3 ∧ R P 3 ) ∼ = Γ 5 (Σ R P 3 ∧ R P 3 ) ⊕ Z / 2 ⊕ Z / 2 ∼ = π 5 (Σ R P 2 ∧ R P 2 ) ⊕ Z / 2 ⊕ Z / 2 ∼ = Z / 2 ⊕ Z / 2 ⊕ Z / 2 ⊕ Z / 2 . It follo ws that its quotien t Γ 5 (Σ R P ∞ ) m ust b e an elemen tary 2 -group and so hene the result. Prop osition 4.8. F or the susp ende d pr oje tive sp a es, π 5 (Σ R P n ) = Z / 2 if n = 1 , Z / 2 ⊕ 3 if n = 2 , Z / 2 ⊕ 5 if n = 3 , Z / 2 ⊕ 3 if n = 4 , Z / 2 ⊕ 2 if 3 ≤ n ≤ ∞ . Pr o of. When n = 1 , π 5 ( S 2 ) = Z / 2 from T o da's table[16 ℄. When n = 2 , π 5 (Σ R P 2 ) = Z / 2 ⊕ 3 is giv en in [22 , Theorem 6.36℄. When n = 3 , π 5 (Σ R P 3 ) has b een omputed in Prop osition 4.7 . F or n ≥ 4 , sine sk 6 (Σ R P ∞ ) = Σ R P 5 , π 5 (Σ R P n ) ∼ = π 5 (Σ R P ∞ ) = Z / 2 ⊕ Z / 2 b y Theorem 4.2 . The remaining ase is π 5 (Σ R P 4 ) . Let F b e the homotop y bre of the pin h map Σ R P 6 ✲ Σ R P 6 / R P 4 = M ( Z / 2 , 6) . By insp eting the Serre sp etral sequene to the bre sequene Ω M ( Z / 2 , 6) ✲ F ✲ ΩΣ R P 6 , ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 40 the anonial injetion j : Σ R P 4 → F indues an isomorphism on H k ( ; Z / 2) for k ≤ 6 and so j ∗ : π k (Σ R P 4 ) − → π k ( F ) is an isomorphism for k ≤ 5 . In partiular, π 5 (Σ R P 4 ) ∼ = π 5 ( F ) . F rom the exat sequene π 5 (Ω M ( Z / 2 , 6)) = Z / 2 ✲ π 5 ( F ) ✲ π 5 (Σ R P 6 ) = Z / 2 ⊕ Z / 2 , the group π 5 ( F ) is of order at most 8 and so is π 5 (Σ R P 4 ) . F rom Prop osition 4.7 , Γ 5 (Σ R P ∞ ) = Im( π 5 (Σ R P 3 ) → π 5 (Σ R P 4 )) = Z / 2 ⊕ 3 . It follo ws that π 5 (Σ R P 4 ) = Z / 2 ⊕ 3 and hene the result. Prop osition 4.9. π 5 (Σ K (Σ 3 , 1)) ≃ Z / 2 ⊕ Z / 2 . Pr o of. This follo ws from the analysis of the map b et w een the Whitehead exat sequenes (2.2 ) indued b y the natural map Z / 2 ֒ → Σ 3 : H 5 ( Z / 2) / / Γ 5 (Σ R P ∞ ) / / π 5 (Σ R P ∞ ) / / H 4 ( Z / 2) H 5 (Σ 3 ) / / Γ 5 (Σ K (Σ 3 , 1)) / / π 5 (Σ K (Σ 3 , 1)) / / H 4 (Σ 3 ) Here the natural isomorphism Γ 5 (Σ R P ∞ ) → Γ 5 (Σ K (Σ 3 , 1)) follo ws from the diagram L 2 Γ 2 2 ( Z / 4 ։ Z / 2) L 2 Γ 2 2 ( Z / 4 ։ Z / 2) Γ 3 2 ( Z / 4 ։ Z / 2 , Z / 2 ֒ → Z / 4) Γ 3 2 ( Z / 4 ։ Z / 2 , Z / 2 ֒ → Z / 12) Γ 5 (Σ R P ∞ ) / / Γ 5 (Σ K (Σ 3 , 1)) L 1 Γ 2 2 ( Z / 4 ։ Z / 2) L 1 Γ 2 2 ( Z / 4 ։ Z / 2) 5. Rela tion to K-theor y As w e men tioned in the in tro dution, there is a natural relation b et w een the problem onsidered and algebrai K-theory . Sine the plus-onstrution K ( G, 1) → K ( G, 1) + is a homologial equiv alene, there is a natural w eak homotop y equiv alene Σ K ( G, 1) → Σ( K ( G, 1) + ) This denes the natural susp ension map: π n ( K ( G, 1) + ) → π n +1 (Σ( K ( G, 1) + )) = π n +1 (Σ K ( G, 1)) for n ≥ 1 . ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 41 Giv en a group G and its maximal p erfet normal subgroup P ⊳ G, one has natural isomorphism π n ( K ( P , 1) + ) ≃ π n ( K ( G, 1) + ) , n ≥ 2 sine K ( P , 1) + is homotop y equiv alen t to the univ ersal o v ering spae of K ( G, 1) + . F or a p erfet group G , the Whitehead exat sequenes form the follo wing omm utativ e diagram: H 4 ( G ) / / Γ 2 ( H 2 ( G )) / / π 3 ( K ( G, 1) + ) / / / / H 3 ( G ) H 4 ( G ) / / H 2 ( G ) ⊗ Z / 2 / / π 4 (Σ K ( G, 1)) / / / / H 3 ( G ) (5.1) Here w e will lo ok at the appliations of the follo wing t w o lassial onstrutions: 1) Let R b e a ring and G = E ( R ) , the group of elemen tary matries. The group E ( R ) is p erfet and the plus-onstrution K ( E ( R ) , 1 ) + also denoted ˜ K ( R ) , denes the algebrai K-theory of R : K n ( R ) = π n ( K ( E ( R ) , 1) + ) , n ≥ 2 . 2) Let Σ ∞ b e the innite p erm utation groups and A ∞ is the innite alternating sub- group. There is the follo wing desription of stable homotop y groups of spheres [14 ℄: π S n = π n ( K (Σ ∞ , 1) + ) = π n ( K ( A ∞ , 1) + ) , n ≥ 2 . (5.2) 5.1. Let R b e a ring. In this ase, one has the natural homomorphisms: K n ( R ) → π n +1 (Σ K ( E ( R ) , 1) ) , n ≥ 2 . F or n = 2 , learly one has the natural isomorphism: K 2 ( R ) ≃ H 2 ( E ( R )) ≃ π 3 (Σ K ( E ( R ) , 1) ) . (5.3) It is sho wn in [1℄ that the map Γ 2 ( K 2 ( R )) → K 3 ( R ) fators as Γ 2 ( K 2 ( R )) ։ K 2 ( R ) ⊗ K 1 ( Z ) ⋆ → K 3 ( R ) , where ⋆ is the pro dut in algebrai K-theory: ⋆ : K i ( S ) ⊗ K j ( T ) → K i + j ( S ⊗ T ) . Hene the diagram (5.1) has the follo wing form: H 4 ( E ( R )) / / Γ 2 ( K 2 ( R )) / / K 3 ( R ) / / / / H 3 ( E ( R )) H 4 ( E ( R )) / / K 2 ( R ) ⊗ K 1 ( Z ) ⋆ 6 6 m m m m m m m m m m m m m m m / / π 4 (Σ K ( E ( R ) , 1) ) / / / / H 3 ( E ( R )) (5.4) and the natural map K 3 ( R ) → π 4 (Σ K ( E ( R ) , 1) ) (5.5) is an isomorphism. F rom equations (5.3) and ( 5.5 ) together with the fat that S L ( Z ) = E ( Z ) , w e ha v e the follo wing: Theorem 5.1. The natur al homomorphism K n ( R ) − → π n +1 (Σ K ( E ( R ) , 1) ) ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 42 is an isomorphism for n = 2 , 3 . In p artiular, π 3 (Σ K ( S L ( Z ) , 1)) ∼ = K 2 ( Z ) ∼ = Z / 2 and π 4 (Σ K ( S L ( Z ) , 1)) ∼ = K 3 ( Z ) ∼ = Z / 48 . Remark 5.1. The isomorphism (5.5 ) and Carlsson onstrution F E ( R ) ( S 1 ) giv es a w a y , for an elemen t of K 3 ( R ) , to asso iate an elemen t from F E ( R ) ( S 1 ) 3 = E ( R ) ∗ E ( R ) ∗ E ( R ) (uniquely mo dulo B F E ( R ) ( S 1 ) ): K 3 ( R ) / / /o /o /o ≃ * * E ( R ) ∗ E ( R ) ∗ E ( R ) Z F E ( R ) ( S 1 ) 3 B F E ( R ) ( S 1 ) 3 j J w w o o o o o o o o o o o o E ( R ) ∗ E ( R ) ∗ E ( R ) B F E ( R ) ( S 1 ) 3 section O O . It is in teresting to represen t in this w a y kno wn elemen ts from K 3 ( R ) for dieren t rings. F or R = Z , x ∈ S L ( Z ) = E ( Z ) , denote b y x (1) , x (2) , x (3) the orresp onden t elemen ts in the free ub e S L ( Z ) ∗ S L ( Z ) ∗ S L ( Z ) . T ak e the follo wing omm uting elemen ts of S L ( Z ) : u = 1 0 0 0 − 1 0 0 0 1 , v = 1 0 0 0 1 0 0 0 − 1 The struture of the elemen t (2.7), diagram (5.4) and w ell-kno wn fats ab out struture of K 2 ( Z ) imply that, using the ab o v e notations, the elemen t [[ u (2) , v (3) ] , [ u (1) , v (3) ]] orresp onds to the elemen t of order 2 in K 3 ( Z ) . It w ould b e in teresting to see an elemen t of S L ( Z ) ∗ S L ( Z ) ∗ S L ( Z ) whi h orresp onds to the generator of K 3 ( Z ) = Z / 48 . Consider the ase R = Z and n = 5 . In this ase, E ( Z ) = S L ( Z ) and w e ha v e the follo wing omm utativ e diagram with exat horizon tal sequenes: Z ⊕ ( Z / 2) 2 / / / / ( Z / 2) 3 / / 0 / / Z / 2 H 5 S L ( Z ) / / Γ 4 ( ˜ K ( Z )) / / K 4 ( Z ) / / Z / 2 / / Z / 4 H 5 ( S L ( Z )) / / Γ 5 (Σ K ( S L ( Z ) , 1)) / / π 5 (Σ K ( S L ( Z ) , 1)) / / Z / 2 0 / / Z / 2 ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 43 and the follo wing omm utativ e diagram: ( Z / 2) 2 / / ( Z / 2) 3 / / / / Z / 2 Γ 2 2 (Γ 2 ( K 2 ( Z )) → K 3 ( Z )) / / Γ 4 ( ˜ K ( Z )) / / / / R 2 ( K 2 ( Z )) π 4 (Σ K ( S L ( Z ) , 1)) ⊗ Z / 2 / / Γ 5 (Σ K ( S L ( Z ) , 1)) / / T or ( π 3 (Σ K ( S L ( Z )) , 1) , Z / 2) Z / 2 / / ( Z / 2) 2 / / / / Z / 2 Simple analysis sho ws that the susp ension map Γ 4 ( ˜ K ( Z )) → Γ 5 (Σ K ( S L ( Z ) , 1)) is an epimorphism and therefore w e ha v e the follo wing theorem: Theorem 5.2. The Hur ewiz homomorphism π 5 (Σ K ( S L ( Z ) , 1)) → H 4 ( S L ( Z )) = Z / 2 is an isomorphism. Remark. Sine K 4 ( Z ) = 0 , w e see that the natural homomorphism K 4 ( Z ) → π 5 (Σ K ( S L ( Z ) , 1)) is not an isomorphism. 5.2. Here w e will use (5.2 ) for ertain omputations. Theorem 5.3. L et A 4 b e the 4-th alternating gr oup. Then π 4 (Σ K ( A 4 , 1)) = Z / 4 . Pr o of. First reall that 3 H 1 ( A 4 ) = Z / 3 , H 2 ( A 4 ) = Z / 2 , H 3 ( A 4 ) = Z / 6 , H 4 ( A 4 ) = 0 H 2 ( A ∞ ) = Z / 6 , H 3 ( A ∞ ) = Z / 12 , π 3 (Σ K ( A 4 , 1)) = Z / 6 Consider the Whitehead exat sequene for the spae Σ K ( A 4 , 1) : Γ 3 (Σ K ( A 4 , 1)) / / π 4 (Σ K ( A 4 , 1)) / / H 3 ( A 4 ) / / Γ 2 ( H 1 ( A 4 )) / / π 3 (Σ K ( A 4 , 1)) / / / / H 2 ( A 4 ) Z / 6 / / Z / 3 / / Z / 6 / / / / Z / 2 3 These omputations w ere done with the help of HAP-system. The authors thank Graham Ellis for these omputations ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 44 Sine R 2 ( π 2 Σ K ( A 4 , 1)) = R 2 ( Z / 3) = 0 , w e ha v e Γ 3 (Σ K ( A 4 , 1)) = Γ 2 2 ( Z / 3 ֒ → Z / 6) . It follo ws from the denition of the funtor Γ 2 2 that it is isomorphi to the pushout Z / 3 ⊗ ( Z / 3 ⊕ Z / 2) / / 0 Z / 6 ⊗ ( Z / 3 ⊕ Z / 2) / / Γ 2 2 ( Z / 3 ֒ → Z / 6) That is, Γ 2 2 ( Z / 3 ֒ → Z / 6) = Z / 2 and there is the follo wing short exat sequene: 0 → Z / 2 → π 4 (Σ K ( A 4 , 1)) → Z / 2 → 0 . W e ome to the extension problem: is it Z / 2 2 or Z / 4 ? Consider the monomorphism A 4 ֒ → A ∞ and the map b et w een orresp onding Whitehead sequenes: Γ 3 (Σ K ( A 4 , 1)) / / π 4 (Σ K ( A 4 , 1)) / / H 3 ( A 4 ) / / Γ 2 ( H 1 ( A 4 )) Γ 3 (Σ K ( A ∞ , 1)) / / π 4 (Σ K ( A ∞ , 1)) / / H 3 ( A ∞ ) / / Γ 2 ( H 1 ( A ∞ )) whi h is Z / 2 / / π 4 (Σ K ( A 4 , 1)) / / Z / 6 / / Z / 6 Γ 2 2 (0 → Z / 6) / / π 4 (Σ K ( A ∞ , 1)) / / Z / 12 / / 0 (5.6) It is easy to see that Γ 2 2 (0 → Z / 6) = Z / 2 and that Γ 2 2 ( Z / 3 ֒ → Z / 6 ) → Γ 2 2 (0 → Z / 6) is an isomorphism. W e obtain the follo wing diagram: Z / 2 / / π 4 (Σ K ( A 4 , 1)) / / / / Z / 2 _ Z / 2 / / π 4 (Σ K ( A ∞ , 1)) / / / / Z / 12 (5.7) No w w e use the isomorphism (5.2 ). Consider the susp ension: K ( A ∞ , 1) + → ΩΣ K ( A ∞ , 1) + ≃ ΩΣ K ( A ∞ , 1) and the orresp onding map b et w een Whitehead sequenes: H 4 ( A ∞ ) / / Γ 2 ( π S 2 ) / / π S 3 / / / / H 3 ( A ∞ ) H 4 ( A ∞ ) / / π S 2 ⊗ Z / 2 / / π 4 (Σ K ( A ∞ , 1)) / / H 3 ( A ∞ ) ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 45 Sine π S 3 = Z / 24 , π S 4 = 0 , w e onlude that the Whitehead sequene for K ( A ∞ , 1) + has the follo wing form: H 4 ( A ∞ ) / / Γ 2 ( π S 2 ) / / π S 3 / / / / H 3 ( A ∞ ) Z / 2 / / Z / 4 / / Z / 24 / / / / Z / 12 W e onlude that the map π S 3 → π 4 (Σ K ( A ∞ , 1)) is an isomorphism and that the map H 4 ( A ∞ ) → Γ 2 2 (0 → Z / 6) is the zero map. The diagram (5.7) has the follo wing form: Z / 2 / / π 4 (Σ K ( A 4 , 1)) / / / / Z / 2 _ Z / 2 / / Z / 24 / / / / Z / 12 The result follo ws. Referenes [1℄ D. Arlettaz: Algebrai K-theory of rings from a top ologial viewp oin t, Publ. Math. 44 (2008), 3-84. [2℄ H.-J. Baues: Homotopy typ e and homolo gy , Oxford Math. Monographs (1996), Oxford Univ ersit y Press. [3℄ H.-J. Baues and D. Condu h e: On the tensor algebra of a non-ab elian group and appliation, K-The ory, 5 (1992), 531554. [4℄ H.-J. Baues and P . Go erss: A homotop y op eration sp etral sequene for the omputation of homotop y groups, T op olo gy 39 (2000), 161-192. [5℄ L. Breen: On the funtorial homology of ab elian groups, J. Pur e Appl. A lg. 142 (1999), 199-237. [6℄ R. Bro wn and J.-L. Lo da y: V an Kamp en theorems for diagrams of spaes, T op olo gy 26 (1987), 311335. [7℄ G. Carlsson: A simpliial group onstrution for balaned pro duts, T op olo gy 23 (1985), 8589. [8℄ F. R. Cohen, J. C. Mo ore, and J. A. Neisendorfer: T orsion in homotop y groups, A nn. of Math. (2) 109 (1979), 121168. [9℄ A. Dold and D. Pupp e: Homologie ni h t-additiv er F un toren; An w endugen. A nn. Inst. F ourier 11 (1961) [10℄ S. Eilen b erg and S. Ma Lane: On the groups H ( π , n ) , I I: Metho ds of omputation, A nn. Math. 60 , (1954), 49139. [11℄ G. Ellis and R. Mikhailo v: A olimit of lassifying spaes, preprin t h .35 81 [12℄ J. C. Harris and N. J. Kuhn: Stable deomp ositions of lassifying spaes of nite ab elian p -groups. Math. Pr o . Cambridge Philos. So . 103 (1988) 427449. [13℄ J. A. Neisendorfer: Primary homotop y theory . Mem. A mer. Math. So . 25 (1980), no. 232, iv+67 pp. [14℄ S. Priddy: On Ω ∞ S ∞ and the innite symmetri group, A lgebr ai T op olo gy, Pr o . Symp os. Pur e Math. 22 , 217-220 (1971). ON HOMOTOPY GR OUPS OF THE SUSPENDED CLASSIFYING SP A CES 46 [15℄ D. G. Quillen: Sp etral sequenes of a double semi-simpliial group. T op olo gy 5 (1966) 155157. [16℄ H. T o da: Comp osition metho ds in homotop y groups of spheres, Prineton Univ. Press, 1962. [17℄ H. T o da: Order of the iden tit y lass of a susp ension spae. A nn. Math. 78 (1963) 300325. [18℄ J. H. C. Whitehead: The homotop y t yp e of sp eial kind of p olyhedron, A nnals de la So . Polon. de Math. 21 (1948), 51110. [19℄ J. H. C. Whitehead: A ertain exat sequene, A nn. Math. 52 (1950), 51110. [20℄ G. W. Whitehead: On spaes with v anishing lo w-dimensional homotop y groups, pr o . Nat. A d. Si. 34 (1948), 207211. [21℄ J. W u: Com binatorial desription of homotop y groups of ertain spaes, Math. Pr o . Camb. Phyl. So . 130, (2001), 489513. [22℄ J. W u: Homotop y theory of the susp ensions of the pro jetiv e plane Memoirs AMS 162 (2003), No. 769. Steklo v Ma thema tial Institute, Gubkina 8, Moso w, R ussia 119991 E-mail addr ess : romanvmmi.ras.r u Dep ar tment of Ma thema tis, Na tional University of Singapore, 2 Siene Drive 2 Sin- gapore 117542 E-mail addr ess : matwujnus.edu.s g URL : www.math.nus.ed u. sg/ m at wuj ie
Original Paper
Loading high-quality paper...
Comments & Academic Discussion
Loading comments...
Leave a Comment