Equivariant homotopy and deformations of diffeomorphisms

EQUIV ARIANT HOMOTOPY AND DEF ORMA TIONS OF DIFFEOMORPHISMS C. DUR ´ AN AND A. RIGAS Abstract. W e presen t a wa y of constructing and deform ing diffeomorphisms of manifolds endo wed with a Lie group action. This is applied to the study of exotic diffeomorphisms and inv olutions of spheres and to the equiv arian t homotop y of Lie groups. 1. Introduction In this pap er we inv estigate the geometric and algebraic relation b etw een tw o areas: On the one ha nd, the study of the symmetries of geo metric generator s of homo- topy gr oups: it has long b een a theme in homotopy theor y to pro duce gener ators and elemen ts of homotopy groups that a re “ nice” with resp ect to symmetr y a nd geometric prop erties (e.g. [Bo], and, for mor e recent r esults, [P ¨ u] and the refer- ences therein). On the other hand, we have the co nstruction of deforma tio ns of diffeomorphisms, or the non-existence of such, i.e., orientation-preserving diffeo- morphisms in different isotopy cla sess, which give rise for example to exotic spheres ([KM, Du]). Let us note that b oth a reas hav e imp or tant apllications in physics, [KR] fo r geometric generato rs of homotopy gr oups, and [AB] for exotic phenomena . W e noticed the link b etw een the two in a particular e xample, the rela tionship betw een a distiguished g enerator of π 6 ( S 3 ) and an exotic diffeomor phism o f S 6 . In this pape r w e abstra c t this principle and r e a ch what we call “equiv aria nt J-pr o cess”, since it is analogous to the construction of the J-homomo rphism in to p o logy; we wan t to r emark right a way tha t in the pr esent work w e ar e ma inly interested in the algebr a o f the J-pro cess, instead of the top olog y . Our main r esult says that e quivaria nt ly symmetry-preser ving deformations of elements of homotopy gro ups provide in a ca nonical wa y deforma tions of diffeomor phisms; and we give examples in which the lack of equiv a riance of the homotopy deformatio ns implies that the deformed maps ceas e to b e diffeomorphisms at some p oint (or , conversely , tha t no equiv ariant deforma tion exists). This pro ce s s b egins the distillation and abs tr ac- tion o f the phenomena that app ear s in the authors’ resea rch in exotic maps and inv olutions ([Du, DMR, ADPR , DPR]), in o rder to sea r ch for a constructive and algebraic theor y of exotic phenomena. The pap er is organized as follows: in section 2 we describ e the main technique, which is a c tually quite easy to pr ove; we b elieve that the relev ant issue here is the abstraction o f the principle and the conseq uences of its a pplication in co nc r ete ex- amples. In section 3 we explain the example that in fact motiv ated the main result: 1 2 C. DUR ´ AN AND A. RIGAS the equiv ariant differential geometry of a Blakers-Massey element (a g enerator of π 6 ( S 3 )) and its rela tion to the exotic diffeomo r phisms (i.e., not deforma ble to the ident ity through diffeomorphis ms ) o f the 6-sphere, and exotic in volutions of the 6-sphere and 5-s pher e. This setting will provide several applications o f the main results, first de fo rming this exotic diffeomorphism of S 6 to a rational (still exo tic) diffeomorphism, deforming the exo tic inv olutions to rational ma ps, a nd then show- ing the non-e xistence of equiv ariant deformatio ns o f maps even though it is known that non-equiv ariant deformatio ns exist. These applica tions a r e done in section 4. W e fina lly comment on some future directions in section 5. Ac kno wledgm e n ts. T he a uthors would lik e to thank T. P ¨ uttmann for helpful discussions. 2. The equiv ariant J- process Let us de s crib e the equiv ariant J-pro cess , which has tw o ing redients: • A Lie group G acting differen tiably on a manifold M (from now on w e assume that all gro ups , actions, ma ps are in the differen tiable ca tegory). W e deno te the a ction of G on M by a dot g · m , g ∈ G, m ∈ M . • A map α : M → G . Note that in the cas e where M is a n -sphere, the homotopy clas s of the ma p α then represents an element in the homoto py group π n ( G ). In this situa tion, we have Definition. Define J α : M → M , the J-pro cess self map of M asso cia ted to α , by J α ( m ) = α ( m ) · m . W e have then Theorem 2. 1. L et J α : M → M b e a J -pr o c ess asso ciate d to a map α : M → G and an action of G on M . If α is G -e quivariant with r esp e ct to the c onjugation action of G on itself, then J α is a bije ct ion with inverse ( J α ) − 1 = J α − 1 and the p owers ar e given by J α n = J α ◦ · · · ◦ J α = J α n Pr o of. The equiv ariance hypo thesis on α transla tes to α ( g · m ) = g α ( m ) g − 1 . Then, we just compute: J α − 1 ( J α ( m )) = J α − 1 ( α ( m ) · m ) = α − 1 ( α ( m ) · m ) · ( α ( m ) · m ) = (( α ( m ) α − 1 ( m ) α − 1 ( m )) · ( α ( m ) · m ) b y equiv ariance = α − 1 ( m ) · ( α ( m ) · m ) = m . The pro o f of the p ow er formula is similar.  Consider now a one pa rameter family α t : M → G of differentiable maps. If this deforma tio n satisfies the equiv ariance prop er ty for all t , then there is a one parameter family J α t of J-pr o cesses of M ; theor em 2.1 then guara ntees that this deformation is thr ough diffe omorphisms . W e shall see in sec tion 4 that sometimes EQUIV ARIANT HOMOTOPY AN D DEFORMA TIONS OF DIFFE OMORPHISMS 3 a deformation through J-pr o cesses is g uarateed to fail, that is, the ma p J α t m ust cease to b e a diffeomorphism at so me p oint in the parameter t (of course in this case the deformatio n is not e quiv ariant). W e now endow the J-pro c ess construction with an additional structure: suppo se that in a ddition to the data in theorem 2.1, we have an inv olution δ : M → M . Then we have Theorem 2. 2 . L et M b e a G -manifold, α : M → G satisfying t he hyp othesis of the or em 2.1. If in addition ther e is an involution δ of M such that α ( δ ( m )) = α − 1 ( m ) , and δ c ommutes with the G -action (thus pr o ducing a G × Z 2 -action on M ), then the J-pr o c ess δ ◦ J α is another involution of M . Pr o of. Co mpute: δ J α ( δ J α ( m )) = δ J α ( δ ( α ( m ) · m )) = δ [ α ( δ ( α ( m ) · m )) · ( δ α ( m ) · m )] = δ [ α − 1 ( α ( m ) · m ) · ( δ α ( m ) · m )] , by the equiv aria nc e of δ and the inv erse map, = δ [ α − 1 ( m ) · ( δ α ( m ) · m )] , by the G -eq uiv aria nce of α , = δ δ [ α − 1 ( m ) · ( α ( m ) · m )] , since α and δ commute , = m, by δ 2 = 1 and the group action prop erty .  In the next sec tio ns we apply these results to an imp orta nt sp ecia l case . 3. Blakers-Massey elements and exotic diffeomorphisms 3.1. An equiv arian t generator of π 6 ( S 3 ) . The ob jective of this section is to study the geometry – in particular, the equiv ariant geometry – o f a disting uished generator of π 6 ( S 3 ) ∼ = Z 12 . A map f : S 6 → S 3 is called b y us a Blakers-Massey element if its clas s [ f ] ∈ π 6 ( S 3 ) genera tes π 6 ( S 3 ). Recall that π 6 ( S 3 ) classifies the principal S 3 -bundles ov er S 7 , each element of π 6 ( S 3 ) corresp o nding to an equiv- alence class of S 3 -bundles ov er S 7 . F o r example, the bundle S 3 · · · S p (2) → S 7 corres p o nds to a genera to r of π 6 ( S 3 ); the Gromoll-Meyer exotic sphere ([GM]) of non-negative curv ature is a given as a Riemannian quotien t o f the to tal space. Also, Grov e and Ziller ([GZ]) co nstructed cohomog e neit y one metrics o f non-nega tive cur- v ature on principa l S O (4 )-bundles ov er S 4 , allowing the cons truction of metr ics o f non-negative curv ature on the exo tic 7-spheres that are bundles over S 4 ; some of these bundles ar e covered by S 3 -bundles over S 7 . It is a classic result [Ja, T o] that π 6 ( S 3 ) ∼ = Z 12 ; the traditiona l genera tor is obtained fro m the commutator of quaternions as follows: consider the ma p ˆ h : S 3 × S 3 → S 3 given b y h ( x, y ) = xy x − 1 y − 1 , w her e we consider S 3 as the gr oup o f unit quater nions. Since h (1 , y ) = h ( x, 1 ) = 1, h factor s thro ug h a ma p h : S 3 ∧ S 3 ∼ = S 6 → S 3 which generates π 6 ( S 3 ) ([T o]). In [DMR], by means of studying the geometry of g eo desics of cer tain metrics on bundles ov er S 7 , there is a constr uc tio n 4 C. DUR ´ AN AND A. RIGAS of a differentiable gener ator b of π 6 ( S 3 ) that is directly defined o n S 6 : c onsider the map b : S 6 → S 3 defined a s follows: the spher e S 6 is expr essed a s the set S 6 = { ( p, w ) ∈ H × H / ℜ ( p ) = 0 , | p | 2 + | w | 2 = 1 } , where H denotes the qua ternions. Define b : S 6 → S 3 by b ( p, w ) = ( w | w | e π p ¯ w | w | , w 6 = 0 − 1 , w = 0 where e x = c o s( | x | ) + sin( | x | ) x | x | denotes the exp onential map of the Lie group S 3 of unit qua ternions. The map b , which is a pr iori not even contin uous a t p o int s of the for m ( p, 0), is in fact analytic, and it g enerates π 6 ( S 3 ) (see [DMR] and section 4.1). Having a differen tiable map as a repr e sentativ e, obtained by g eometrical meth- o ds, invites the study of its geometry , in particula r with r esp ect to group actions; we shall presently se e that this distinguis he d generator has many symmetry pr op- erties and is a co ho mogeneity one map. W rite S O (4) = S 3 × S 3 / ∼ = , where ( q , r ) ∼ = ( − q , − r ), and S O (3) = S 3 / ± 1 . W e represent S O (4) and S O (3) as follows Define ( q , r ) · ( p, w ) = ( q p ¯ q , r w ¯ q ). This repr e sentation embeds S O (4) in the exceptio nal Lie gro up G 2 ; the gro up S O (3) is represented by the standard qua ternionic conjugatio n C q ( x ) = q x ¯ q . The map ( q , r ) 7→ ± q provides an epimorphism φ : S O (4 ) → S O (3). Then, a simple computation shows that b (( q , r ) · ( p , w )) = q b ( p, w ) ¯ q . Th us, w e have the following commutativ e diagra m, S O (4) × S 6 → S 6 ( φ, b ) ↓ ↓ b S O (3) × S 3 → S 3 And thus the Bla kers-Massey element b is e quiv ariant. Note tha t b oth actions are c o homogeneity o ne: the co njugation ac tion o n S 3 has as quotient the in terv al [0 , 1], r e alized b y the map R : S 3 → [ − 1 , 1], R ( θ ) = Re ( θ ); the regular orbits are diffeomorphic to S 2 and ± 1 a re the t wo singula r orbits. The S O (4) action on S 6 has as the inv aria nt function S : S 6 → [ − 1 , 1], S ( p, w ) = | p | 2 − | w | 2 (= 2 | p | 2 − 1 = 1 − 2 | w | 2 ); the singular orbits ar e S 2 ( w = 0) and S 3 ( p = 0). The re g ular or bits are diffeomorphic to S 2 × S 3 , that is, the set { ( p, w ) / | p | = r 1 , | w | = r 2 } when r 1 and r 2 are bo th no n-zero. The following r e ma rk will b e imp ortant in the s equel: R emark 3.1 . Consider the comp osition β = P ◦ b : S 6 → S O (3), where P is the cano nical double cov er pro jection S 3 → S O (3 ). Then taking the ± equiv alence classes in the formula b (( q , r ) · ( p, w )) = q b ( p, w ) ¯ q we a ls o get a commutativ e diagram S O (4) × S 6 → S 6 ( φ, β ) ↓ ↓ β S O (3) × S O (3 ) → S O (3) EQUIV ARIANT HOMOTOPY AN D DEFORMA TIONS OF DIFFE OMORPHISMS 5 Let us a lso study at the powers o f b : b n ( p, w ) = ( w | w | e nπ p ¯ w | w | , w 6 = 0 ( − 1) n , w = 0 Then a ll the p ow er s of the Bla kers-Massey element hav e the sa me equiv ariance prop erties. 3.2. The Blak ers-Mass e y element and exotic maps. This concrete map b is a fundamen tal building blo ck of exotic maps (degree o ne diffeomor phis ms of spher es not isotopic to the identit y , and free involutions not conjugate to the antipo dal map); see [Du, DMR, ADPR]. Define σ ( p, w ) = ( b ( p, w ) pb ( p, w ) − 1 , b ( p, w ) pb ( p, w ) − 1 ) then σ : S 6 → S 6 is a degree o ne diffeomorphism not isoto pic to the ident ity ([Du, DMR]), a nd is a gener ator of the groups Γ 7 of isotopy classes of diffeomor phisms of S 6 ; ([DMR]); it is k non ([KM]) that Γ 7 ∼ = Z 28 . The map σ k represents k ∈ Z 28 , and th us in particular any diffeomorphism of S 6 can b e deformed to σ k for infinitely many k . Tw o maps σ k and σ ℓ are is otopic to each other if and only if k ≡ ℓ mo d 28 , howev er let us remark that no explicit isotopy is known b etw een σ k and σ k +28 r , r 6 = 0. Note that the map σ is S O (3) (and not S O (4)) equiv ariant. It is not obvious tha t σ is a diffeomo r phism; in [Du] this fact is e stablished by indirect geometric metho ds , and in [DMR ] it is obser ved that the inv erse of σ is given by σ ( p, w ) = ( b ( p , w ) − 1 pb ( p, w ) , b ( p, w ) − 1 pb ( p, w ). Note that if b w ere con- stant, this inv erse is immediate; but since b depends on ( p, w ), in g eneral such a result would not be true . How ever, w e shall see that the equiv ariance is the s truc- tural reason for σ b eing a diffeomo rphism b y applying the main theor em (co mpare 3.2 o f [DMR]); in order to do this, let us ma ke prec ise how the J- pro cess works in this ca se: let B : S 6 → S O (7) b e given by B ( x ) = ∆ ◦ P ◦ b ( x ) = ∆ ◦ β , wher e ∆ : S O (3) → S O (7) is the diago nal embedding of S O (3) × S O (3) in S O (7) (with a 1 in the middle), i.e. ∆( T ) =   T 0 0 0 1 0 0 0 T   Note that the imag e o f B fa lls in to a subgroup of S O (7 ) that is iso morphic to S O (3). Then we are in the setup of theorem 2 .1: B : S 6 → S O (3) is e quiv ariant with res pe ct to the conjugatio n action o f S O (3) in itself, a nd σ can b e simply written as the J-pr o cess σ ( x ) = J B ( x ) ( x ) = B ( x ) x , since the pro jection of S 3 to S O (3) is realized b y the standa rd quaternionic co njugation q 7→ T q , T q ( x ) = q x ¯ q . Thus we recov er the results of [DMR] in a mo re s tr uctural wa y: σ is a diffeomorphism, and the p owers o f σ are given b y σ k ( x ) = B k ( x ) x . Also, σ allows the construction o f exotic involutions: ( p, w ) 7→ − σ ( p, w ) is a free in volution of S 6 that is not conjugate to the antipo dal inv olution; since σ acts on ( p, w ) by a quaternionic conjugatio n, it preser ves the real part o f w and th us this in volution restricts to an (also exotic) involution o f the S 5 defined by Re( w ) = 0 , where it has a simple pictorial de s cription ([ADPR, ADPR-1]). Aga in, 6 C. DUR ´ AN AND A. RIGAS the fact that − σ is an inv olution follows immediately now by theor e m 2 .2, taking δ to b e the antipo dal in volution of S n and α the B lakers-Massey elemen t, since b ( − p, − w ) = b ( p, w ). 4. Applica tions 4.1. Deform atio ns of di ffeomorphi sms and in volutions. As mentioned in sec- tion 2 , the eq uiv aria nt J -pro cess technique provides a “canonical” way o f defor m- ing diffeomorphisms J α ( m ) through e quivariant deformations o f the corr esp onding maps α . W e will use this technique to deform the diffeomorphis ms σ k : S 6 → S 6 , which represent any isotopy class of diffeomorphisms of S n , to r ational maps; this could pav e the wa y to the alg ebro-g e ometric study of such ex o tic diffeomorphisms. The steps in the deformation are as follows: we will constr uct a homotopy H ( s, p, w ) : [0 , 1] × S 6 → S 6 betw een the Blakers-Massey element and a rational map; this ho motopy will b e equiv ariant fo r a ll v alues of the deformation parameter s . By theo rem 2.1, the maps J H s : S 6 → S 6 will b e diffeomor phisms; and there fo re this pro cedure furnishe s a deformation b etw een the exotic diffeomorphis m σ and a rational map R , in the same isotopy cla ss of σ and therefore also a genera tor of the group Γ 7 . Then p ow ers of σ r epresenting all other iso to py classes ar e als o taken care of by theorem 2.1, since J H k s = J k H s = R k . In order to construct this defor mation H s , all we need to do is to reconsider prop osition 1 of [DMR] car efully and equiv ariantly . Spelling out the exp o nent ial in the B la kers-Massey ele ment, we hav e b ( p, w ) = cos( π | p | ) + sin( π | p | ) | p | (1 − | p | 2 ) wp ¯ w . The functions x 7→ sin( π x ) and x 7→ x (1 − x 2 ) a re b oth o dd, p ositive on (0 , 1) a nd hav e a zer o of or der 1 a t x = 0 a nd x = 1; therefore g ( x ) = sin( π x ) / ( x (1 − x 2 )) is a n even, p ositive, differentiable function on [0 , 1], (in particula r this expla ins the analiticity of b ). W e will homotop g affinely to the co nstant function 1 ; in order to deal with cos( π | p | ), we use the function c ( x ) = 1 − 4 x 2 . The function c is the simplest even function satisfying the pro p erty that it has the same sign as co s( πx ) on [0 , 1]. Now co nsider the r ( p, w ) = (1 − 4 | p | 2 + w p ¯ w ) and the affine homotopy ˆ H ( s, p, w ) = ˆ H s ( p, w ) = (1 − s ) b ( p, w ) + sr ( p, w ) . F or any s , the map ˆ H s is equiv ariant with r e sp ect to conjugatio n since b is equiv ariant and r is a p olynomia l in | p | , p and w . Also , the expressions in | p | are all even, and therefore these ma ps can be written in ter ms of | p | 2 , p, w , but since p is purely imag inary , | p | 2 = − p 2 and all expressio ns inv olved are a nalytic expr essions in the non-commuting quaternio nic v ariables p , w and ¯ w . Rewriting ˆ H , we have ˆ H s ( p, w ) = [(1 − s ) cos( π | p | ) + sc ( | p | )] + [(1 − s ) g ( | p | ) + s ] wp ¯ w . EQUIV ARIANT HOMOTOPY AN D DEFORMA TIONS OF DIFFE OMORPHISMS 7 Then the sign prop er ties of c ( x ) and g ( x ) imply that ˆ H ( s, p, w ) is never zero. Then H ( s, p, w ) = ˆ H ( s, p, w ) / | ˆ H ( s, p, w ) | furnishes a n equiv ariant ho motopy be- t ween the Blakers-Massey element and the map Q : S 6 → S 3 . Q ( p, w ) = 1 + 4 p 2 + w p ¯ w p (1 + 4 p 2 ) 2 − | w | 4 p 2 Now the map R ( p, w ) = ( Q ( p, w ) p ¯ Q ( p, w ) , Q ( p, w ) p ¯ Q ( p, w )) is a ra tional dif- feomorphism o f S 6 that is not isotopic to the identit y; its p owers are r ational diffeomorphisms representing all iso topy classes of diffeomor phisms of S 6 . W riting R e xplicitly , we have R ( p, w ) =  (1 + 4 p 2 + w p ¯ w ) p (1 + 4 p 2 − w p ¯ w ) (1 + 4 p 2 ) 2 − | w | 4 p 2 , (1 + 4 p 2 + w p ¯ w ) w (1 + 4 p 2 − w p ¯ w ) (1 + 4 p 2 ) 2 − | w | 4 p 2  . F or each v alue s of the deformatio n parameter, the map H s satisfies the hypoth- esis of theor em 2 .2 with resp ect to the antipo dal in volution o f S n . T he r efore − J H s is a deformation of the exotic in volution σ o f S 6 , thro ugh inv olutions that also restrict to S 5 , a nd, since close enough inv olutions a re ea sily see n to b e conjugate, these inv olutiona s are all exo tic. At the end of the deformatio n we reach the inv o- lution − R ( p, w ), which is a ra tional inv olution o f S 6 . Note that whe n restric ted to S 5 , defined by Re( w ) = 0, ¯ w = − w and the map R ( p, w ) = −  (1 + 4 p 2 − w pw ) p (1 + 4 p 2 + w pw ) (1 + 4 p 2 ) 2 − w 4 p 2 , (1 + 4 p 2 − w pw ) w (1 + 4 p 2 + w pw ) (1 + 4 p 2 ) 2 − w 4 p 2  . is an exotic inv olution o f S 5 defined by a rational map in the non-commuting quaternionic v ariables p and w . Let us r emark that deforming α through plain (not neces sarily equiv aria nt ) ho- motopies pro duces a defo rmation of J α that is not nece ssarily through diffeomor- phisms. W e shall take adv an tage o f this in the next section. 4.2. The equiv ariant Serre problem. It is know [Ja, T o ] that π 6 ( S 3 ) ∼ = Z 12 . How ever, no explict deformation of tw elve times a generator is known. The authors call this the Serr e pr oblem : to find an explicit homotopy betw een the 12th p ow er of the Blakers-Massey element and the iden tity , or, in other terms, to under s tand how the quaternio ns are homotopy commutativ e in the 12 p ow er. This problem is still op en, altough sig nificant a dv ances hav e b een made recent ly ([P¨ u]); a solution to the Serre pr oblem has far- reaching consequences, for example, the writing of explicit non-cancellatio n phenomena and new mo dels for exotic spheres (see, e.g., [Ri97]). W e show that, altough the genera tor a nd all its p ow ers are represented by equiv ari- ant maps, ther e is no e quivaria nt solution to the Ser re pr o blem (cf Theor em 4.1). W e b elieve that, in additio n to the statement o f the theorem (which can proba bly be prov en using sta nda rd metho ds o f equiv aria nt homotopy theory), the metho d o f pro of by using the relationship betw een e q uiv ariant ho motopy and isotopy through explicit formulas is of indep endent interest: an equiv ariant homotopy would imply that the order o f the gr oup of homotopy 7 -spheres divides 12 and we know it is isomorphic to Z 28 ([KM, EK]); therefor e such a homoto py is not p o ssible. W e also extend this result to homo topies of the Bla kers-Massey element inside o ther gro ups. 8 C. DUR ´ AN AND A. RIGAS Theorem 4.1 . Ther e exists no differ entiable homotopy φ : [0 , 1] × S 6 → S 3 b etwe en b 12 and a c onstant map su ch that for e ach t ∈ [0 , 1] , φ ( t, · ) : S 6 → S 3 is S O (3) - e quivaria nt . Pr o of. W e first ada pt the Blakers-Massey element to the prop ositio n a b ove, by considering lifting to S 3 and considering b : S 6 → S 3 as an S 3 -equiv ariant map. An S O (3) equiv ariant homo topy b etw een b 12 and the constant map then lifts to an S 3 -equiv ariant homotopy b t ( p, w ) such that b 0 ( p, w ) = 1 and b 1 ( p, w ) = b ( p, w ). If such a homoto py exits, the maps σ 12 t ( p, w ) = ( b t ( p, w ) 12 pb t ( p, w ) − 12 , b t ( p, w ) 12 pb t ( p, w ) − 12 ) , furnish an isotopy b etw een σ 12 and the identit y diffeomorphism. But by sta nda rd differential to p o logy metho ds (see, fo r exa mple, [Ko]), the map φ : π 0 (Diff ( S 6 )) → Γ 7 , φ ( f ) = D 7 ∪ f D 7 , is an is omorphism ([Mi]), wher e Γ 7 is the g roup o f differen- tiable structure s on S 7 under the connected sum o p er ation. Thus σ 12 represents 12 in the group Γ 7 ≡ Z 28 ([KM]), a nd we ge t a contradiction.  Note that the imag e of B k is contained in the chain of inclusions S O (3) ⊂ S U (3) ⊂ G 2 ⊂ S O (7). Theorem 4.1 s tates that, even though B 12 is homoto pic (inside o f S O (3)) to the co nstant map, no equiv aria nt homotopy can ex ist. Now the “Serr e pr oblem” for all the other groups has b een s olved: there exist explicit generator s γ of π 6 ( S U (3)) ∼ = Z 6 , δ of π 6 ( G 2 ) ∼ = Z 3 and explicit homotopies b etw een γ 6 and the constant map [PR, P ¨ u] and δ 3 and the constant map [Ri92]. Also π 6 ( S O (7)) = 0. E x plicit ho motopies b etw een b a nd γ , δ and the constant map inside of the resp ective groups ca n b e constructed using the geo metry of the chain S O (3) ⊂ S U (3) ⊂ G 2 ⊂ S O (7 ) [P ¨ u]. Thus, not only we hav e that b 6 , b 3 and b ar e homotopic to the c o nstant map inside of S U (3) , G 2 and S O (7); ex plicit homotopies can b e written. Theorem 4.2. The maps b 6 , b 3 , b ar e homotopic to the c onstant map in S U (3) , G 2 , S O (7) , r esp e ctively, thr ough explicit homotopie s. However, no S O (3) -e quivariant homotopy exists. Pr o of. All the groups in the chain S O (3) ⊂ S U (3) ⊂ G 2 ⊂ S O (7) act on S 6 through the ca nonical action o f S O (7 ). Mimic the pro o f of Theor em 1 with σ 6 , σ 3 and σ in place of σ 12 .  The symmetry- br eaking mechanism of these homo to pies is b eautifully illustrated in the s tructure of the group π 6 ( S U (3)) [PR]: one way o f determining the structure of the homoto py of topo logical groups is by finding a gener ator A such that A k = e , the iden tity o f the group; this is the w ay that this was done for π 6 ( G 2 ) ∼ = Z 3 , by finding a gener ator such that A 3 = e ([Ri92]). The p ow er map A 7→ A k of matr ices is of c o urse equiv ariant under co njugation a nd this pro cess pr ovides equiv ariant deformations. Ho wev er, in the cas e of S U (3), the ho motopy uses a deformation of the pro duct of matric e s thr ough the Car ta n suba lgebra of S U (3 ) a nd the symmetry is bro ken; s ee [PR] for details. EQUIV ARIANT HOMOTOPY AN D DEFORMA TIONS OF DIFFE OMORPHISMS 9 5. Concluding remarks First we wan t to note that all the construc tio ns in [DMR, ADPR] related to the B la kers-Massey element and the exotic diffeo mo rphisms can b e generalized by substituting a ll the quaternions in volv ed by Cayley num b ers and mo difying the relev ant dimensions. Howev er, this passage involv es no n-trivial modificatio ns of the techniques in the pro ofs. The unit quaternions ar e a g r oup, and thus the equiv ariance pr op erties make sense; the unit Cayley num b ers ar e not a group and what is the right extension of theorem 2.1 and its applications r emains to b e seen. The co nstruction in the main theorem, the applications g iven and the compu- tations in sectio n 3 of [DMR ] suggests that there exists an “algebra o f exo ticity”, which is yet to be des c rib ed. It would als o b e interesting to follow the known, non- e quiv ariant homotopies of the res p e ctive p owers o f the B lakers-Massey element and the identit y in S U (3), G 2 and S O (7) and study the asso cia ted J - pro cess self-maps of M , which at so me point m ust cease to be diffeo morphisms a nd determine the structure of the singularities that a pp ear, which could shed some lig ht in the gener al ques tion of what makes a diffeomorphism exotic, o r, b etter, how does o ne detect the exo ticity o f a degree one diffeomorphism tha t is given by a for mula. References [ADPR] U. Abr esc h, C. E. Dur´ an, T. P¨ uttmann , A. Ri gas, Wie dersehen metrics and exotic invo- lutions of Euclide an spher es , J. Reine Angew. M ath. 605 (2007) , 1-21. [ADPR-1] U. Abresch , C. E. Dur´ an, T. P¨ uttmann, A. Rigas, An exotic involution of the 5-sph er e , quic ktime mov ie, h ttp://homepage.ruhr-uni-b och um.de/Thomas.Puettmann/XIn v olution.html [AB] T. Asselmeyer-Maluga, C . H. Brans Exoti c smo othness and Physics: Differential T op ol- o gy and Sp ac etime Mo dels, W or l d Scient ific, 2007. [Bd] G. E. Bredon, Intr o duction t o c omp act tr ansformation g r oups , Pure and Applied Math- ematics, V ol. 46, Academic Pr ess, New Y ork-London 1972. [Bo] R. Bott, The stable homotopy of the classic al gr oups. Ann. of Math. (2) 70 (1959) 313- 337. [Du] C. E. Dur´ an, Pointe d Wie dersehen metrics on exotic spher es and diffe omorphism s of S 6 , Geom. Dedicata 88 (2001), 199-210. [DPR] C. E. Dur´ an, T. P ¨ uttmann, A. Rigas, Some ge ometric formulas and c anc el lations in algebr aic and differ ential top olo gy , Mat. Contemp. 28 (2005), 133-149. [DMR] C. E. Dur´ an, A. Mendoza, A. Ri gas, Blakers-Massey elements and exotic diffe omor- phisms of S 6 and S 14 , T rans. 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