A Counterexample to a Conjecture about Positive Scalar Curvature

PROCEEDINGS OF THE AMERICAN MA THEMA TICAL SOCIETY V olume 00, Nu mber 0, Pa ges 00 0–000 S 0002-9939(XX)000 0-0 A COUNTEREXAMPLE TO A CONJECTURE ABOUT POSITIVE SCALAR CUR V A TURE DANIEL P APE AND THOMAS SCHICK (Comm unicated by Daniel Ruberman) Abstract. [1, Conjecture 1] a sserts that a closed smo oth manifold M with non-spin universal cov ering admits a metric of positive scalar curv ature if and only if a certain homological condition is s atisfied. W e present a counterexam- ple to this conjecture, based on the coun terexample to the unstable Gromov- La wson-Rosenberg conjecture giv en in [5]. 1. The Resul t W e give a counterexample to the following conjecture stated by Cha ng as [1, Conjecture 1], and attributed there to Rosenberg and W einberger. Conjecture 1. 1. Supp ose that M is a c omp act oriente d manifo ld such that i ts universal c overing do es not admit a spin struct ur e, with fundamental gr oup Γ and of dimensio n n ≥ 5 . L et f : M → B Γ b e the c omp osition of the classifying map c : M → B Γ of the universa l c overing of M , and the natur al map B Γ → B Γ . Denote by [ M ] the fundamental class of M in H n ( M ) . Then M admits a metric of p ositive sc alar curvatur e if and only if f ∗ [ M ] vanishes in H n ( B Γ) . Here B Γ is the classifying space for the gro up Γ and B Γ is the quotient of the universal space for prop er actions, i.e. the quotient E Γ / Γ, where E Γ is a prope r Γ-space such that for every finite s ubg roup F ≤ Γ the fixed p oint set E Γ F is con- tractible (in particular, no n-empty), but such tha t E Γ H = ∅ for all other subgro ups H ≤ Γ, compar e [1 , p. 1623]. Our co unt erex ample is based on the c ounterexample to the Gromov-Lawson- Rosenberg conjecture given in [5]. There, a 5-dimensional connected closed spin manifold M with fundamental gr oup Γ = Z 4 ⊕ Z / 3 is constr uc ted, whose Rosen- ber g index v anishes but which nevertheless does no t admit a metric of po sitive scalar cur v ature. By taking the co nnected sum of this manifold M with a simply- connected non-spin manifold N , we obtain a totally non-spin ma nifo ld X whic h has the same fundamen tal gro up a s M . One ha s B Γ = T 4 × B Z / 3 a nd analogously Receiv ed by the editors F eb 1, 2011. 2010 Mathematics Subje ct Classific ation. 57R65. Daniel Pape wa s supp orted by the Germ an Research F oundation (DF G) through the Researc h T raining Group 1493 ‘M athematical structu res i n mo dern quantu m physics’. www.uni- math.gwdg.de/ pape . Thomas Schic k was partially funded by the C ourant Research Center ‘Higher order s tructures in M athematics’ within the German ini tiativ e of excellence . www.uni-math.gw dg.de/schick . c  XXXX American Mathematical So ciety 1 2 DANIEL P APE AND THOMAS SCHICK B Γ = T 4 by [1, (1) and (4), p. 1624]. Esp ecia lly , H n ( B Γ) = 0 for n ≥ 5 , so that the condition on f ∗ [ X ] from C o njecture 1.1 is satisfied in the ca se at hand. The argument in [5] relies o n the following obs e rv a tion b y Stolz and we will also make significant use of this re sult. Lemma 1 .2. L et X b e a top olo gic al sp ac e and s et for n ∈ N ≥ 2 H + n ( X ) := { f ∗ [ M ] ∈ H n ( X ) ; f : M n → X and M admits a metric with sca l > 0 } Then for any class u ∈ H 1 ( X ) t he m ap u ∩ : H n ( X ) → H n − 1 ( X ) , x 7→ u ∩ x maps H + n ( X ) into H + n − 1 ( X ) if 3 ≤ n ≤ 8 . Pr o of. See [5, Co rollar y 1.5] for 3 ≤ n ≤ 7 a nd [3, Theorem 4.4] for n = 8 .  Our result reads now as follows. Prop ositio n 1. 3. L et M b e the manifold c onstructe d in [5] (we r e c al l its c onstru c- tion in Se ction 2) and N a simply c onne cte d manifold of dimension 5 which admits no spin structu r e. The n t he manifold X := M # N has non- spin universal c overing and admits n o metric with p ositive sc alar curvature . This result is pa rt of the first named author ’s forthcoming thesis [4]. 2. The Proof Pr o of of Pr op osition 1.3 . First o f all, if X is constructed a s ab ov e, w e have al- ready noted that it has non- spin universal cov ering . T o obtain an explicit simply- connected non- s pin 5-manifold N , one ca n start with C P 2 × S 1 , which is non-spin as C P 2 is, and then do sur gery on the embedded S 1 to obtain the simply-co nnected N . Because this surg ery do es not touc h the embedded C P 1 with its no n- spin no rmal bundle, the resulting N r emains a non-spin manifold. In order to see that X admits no metric of p ositive scalar curv ature, w e use the same ar gument a s in [5]. T o b egin with, we choose the mo del B Γ = T 4 × B Z / 3 . Recall, H n ( T d ) ∼ = Z d ( n ) , d ( n ) =  d n  and H n ( B Z /k Z ) ∼ =      Z , n = 0; Z /k Z , n o dd; 0 , n even. T ogether with the K ¨ unneth formula this g ives H k ( B Γ) = M p 1 + ··· + p 5 = k H p 1 ( X 1 ) ⊗ · · · ⊗ H p 5 ( X 5 ) . Here we have written T 4 = X 1 × · · · × X 4 as pro duct of four copies o f S 1 , and X 5 for B Z / 3. Fix a basep o int x = ( x 1 , . . . , x 5 ) ∈ B Γ and let p : T → B Z / 3 b e a map whic h induces an epimorphism o n π 1 as in [5], as well as f j : X j → B Γ the ma p which includes X j ident ically and basep o in t-pres erving. W e denote by [ ∗ ] ∈ H 0 ( B Γ) the canonical generator . Next, choo se for each 1 ≤ j ≤ 4 g enerators g j ∈ H 1 ( X j ) and elements g ∗ j ∈ H 1 ( X j ) with h g ∗ j , g j i = 1, and let g 5 ∈ H 1 ( X 5 ) b e p ∗ [ S 1 ] where [ S 1 ] A COUNTEREXAMPLE TO A CONJECTURE ABOUT POSITIVE SCALAR CUR V A TURE 3 is the standard generator for H 1 ( S 1 ). Introduce the elements v j := ( f j ) ∗ ( g j ) ∈ H 1 ( B Γ) fo r j = 1 , . . . , 5 as well a s a 1 , . . . , a 4 ∈ H 1 ( B Γ) with a 1 := (pr 1 ) ∗ ( g ∗ 1 ) × 1 × 1 × 1 × 1 , a 2 := 1 × (pr 2 ) ∗ ( g ∗ 2 ) × 1 × 1 × 1 , a 3 := 1 × 1 × (pr 3 ) ∗ ( g ∗ 3 ) × 1 × 1 , a 4 := 1 × 1 × 1 × (pr 4 ) ∗ ( g ∗ 4 ) × 1 . Finally , s et w := v 1 × · · · × v 4 × v 5 ∈ H 5 ( B Γ) and z := [ ∗ ] × [ ∗ ] × [ ∗ ] × v 4 × v 5 ∈ H 2 ( B Γ) . By the K ¨ unneth formula, w 6 = 0 and z 6 = 0. F urthermore, ( ∗ ) z = a 1 ∩ ( a 2 ∩ ( a 3 ∩ w )) ∈ H 2 ( B Γ) . F or example one has a 3 ∩ w =  1 × 1 × (pr 3 ) ∗ ( g ∗ 3 )  ×  1 × 1  ∩  v 1 × v 2 × v 3  ×  v 4 × v 5  =  1 × 1 × (pr 3 ) ∗ ( g ∗ 3 )  ∩  v 1 × v 2 × v 3  ×  (1 × 1) ∩ ( v 4 × v 5 )  =  1 ∩ v 1  ×  1 ∩ v 2  ×  (pr 3 ) ∗ ( g 3 ) ∗ ∩ v 3  ×  1 ∩ v 4  ×  1 ∩ v 5  = v 1 × v 2 ×  (pr 3 ) ∗ ( g ∗ 3 ) ∩ v 3  × v 4 × v 5 = v 1 × v 2 × [ ∗ ] × v 4 × v 5 , bec ause of (pr 3 ) ∗ ( g ∗ 3 ) ∩ ( i 3 ) ∗ ( g 3 ) = h g ∗ 3 , g 3 i [ ∗ ]. Let f : T 5 → T 4 × B Z / 3 b e given b y f = ( f 1 × f 2 × f 3 × f 4 ) × ( f 5 ◦ p ) and choo s e ( g 1 × · · · × g 4 ) × [ S 1 ] = : [ T 5 ] as fundamental class for T 5 . Then f ∗ [ T 5 ] = w . As in [5] one can construct a b ordism in Ω spin 5 ( B Γ) from f to a map g : M → B Γ which induces a n is omorphism of fundamental g roups. This defines the manifold M . Now let N b e any simply- connected closed non-spin manifold of dimension 5 and se t X := M # N . Finally , ass ume that X admits a metric of p os itive sca lar curv ature. Then con- sider the map h : M ⊔ N → B Γ on the disjo int union of M and N , which equals g on M a nd sends N to a p oint. One has h ∗ [ M ⊔ N ] = g ∗ [ M ] = w a nd since M ⊔ N is b ordant to M # N , it follows that w ∈ H + 5 ( X ). But then it follows from ( ∗ ) as well as Lemma 1.2 that w is mappe d to z under the following comp ositio n H + 5 ( B Γ) a 3 ∩ · − − − → H + 4 ( B Γ) a 2 ∩ · − − − → H + 3 ( B Γ) a 1 ∩ · − − − → H + 2 ( B Γ) . Hence z = k ∗ [ S 2 ] for some k : S 2 → B Γ since S 2 is the o nly orientable s ur face which admits a metric of p os itive scalar curv ature. On the other ha nd, π 2 ( B Γ) = 0 so that k is null homotopic. This implies z = 0, which is a contradiction.  R emark 2 .1 . The metho d described in this note pro duces a co un terexa mple to Con- jecture 1.1 with fundament al gr oup Γ whenever Γ s atisfies the following homologic al conditions: • for 5 ≤ m ≤ 8 there is a homolo gy cla s s [ M ] ∈ H m ( B Γ; Z ) represented by an m -dimensiona l clo sed oriented manifold M (with surgerie s one can then arrang e that π 1 ( M ) = Γ) • there are classes α 1 , α m − 2 ∈ H 1 ( B Γ; Z ) such that α 1 ∩ ( · · · ∩ ( α m − 2 ∩ [ M ])) 6 = 0 ∈ H 2 ( B Γ; Z ) 4 DANIEL P APE AND THOMAS SCHICK • under the map H m ( B Γ) → H m ( B Γ) the class [ M ] is send to 0. Note that this condition is similar , indeed m uch easier than the general homolog ic al condition for coun terexa mples to the Gromov-Lawson-Rosenberg co ndition derived in [2]. Unfor tunately , its str ucture requir es the g r oup Γ to con tain non-trivia l torsion, to allow for a kernel of the map H ∗ ( B Γ) → H ∗ ( B Γ) (in contrast to [2]). The assumption o n H 1 ( B Γ; Z ) we hav e to ma ke is very strong, it ha s to hav e rank at least m − 2. In particula r, the metho d tells us nothing ab out finite groups. Indeed, the ques tion of existence for k o r lower dimensional manifolds with finite fundamen tal g roup ( Z /p Z ) k for p o dd is co mpletely op en (in the totally non-spin case a s well as in the spin case) and seems the first o bstacle for a full understanding of this pr o blem. Progress in this direc tio n will require a co mpletely ne w se t of ideas. References 1. Stan ley Chang, Positive sc alar curvatur e of total ly nonspin manifolds , Pr oc. Amer . Math. Soc. 138 (2010), no. 5, 1621– 1632. MR 2587446 2. William Dwyer, Thomas Sc hick, and Stephan Stolz, R emarks on a c onje ctur e of Gr omov and Lawson , High-dimensional manifold topology , W orld Sci. Publ., River Edge , NJ, 2003, pp. 159– 176. M R 2048721 (2005f:53043) 3. Mic hael Joachim and Thomas Schic k, Positive and ne gative r esults c onc erning the Gr omov- Lawson-Rosenb erg c onje ctur e , Geometry and top ology: Aarhus (1998 ), Contemp. Math., vol. 258, Amer. Math. Soc., Providence, RI, 2000, pp. 213–226. MR 1778 107 (2002g:53079) 4. Daniel P ap e, Index the ory and p ositive sc alar curvatur e , Ph.D. thesis, Georg-August- Unive rsi t¨ at of G¨ ottingen, 2011. 5. Thomas Sc hic k, A c ounter ex ample to the (unstable) Gr omov-Lawson-Rosenb er g c onje ct ur e , T opology 37 (1998), no. 6, 1165–1168. MR 16329 71 (99j:53049) Georg-A ugust-Universit ¨ at G ¨ ottingen, Bunsenstr. 3, 37073 G ¨ ottingen, Germany E-mail addr ess : pape@uni-math.gw dg.de Georg-A ugust-Universit ¨ at G ¨ ottingen, Bunsenstr. 3, 37073 G ¨ ottingen, Germany E-mail addr ess : schick@uni-math. gwdg.de