p -local stable cohomological rigidity of quasitoric manifolds
aa r X i v : . [ m a t h . A T ] M a y p -LOCAL STABLE COHOMOLOGICAL RIGIDITY OF QUASITORICMANIFOLDS SHO HASUI AND DAISUKE KISHIMOTO
Abstract.
It is proved that if two quasitoric manifolds of dimension ≤ p − p have isomorphic cohomology rings, then they have the same p -local stable homotopy type. Introduction
A class C of spaces is called cohomologically rigid if any spaces in C having isomorphic co-homology rings are homeomorphic with each other. It is well known that cohomology ringsdo not distinguish closed manifolds up to homeomorphism (or even homotopy equivalence), sothe class of all closed manifolds is not cohomologically rigid. But what can we say about thecohomological rigidity if we restrict to a class of manifolds with good symmetries? The mani-folds that we consider in this paper are quasitoric manifolds which were introduced by Davisand Januszkiewicz [DJ] as a topological counterpart of smooth projective toric varieties. Sincetheir introduction, quasitoric manifolds have been prominent objects which produce fruitfulinteractions of algebra, combinatorics, geometry, and topology. Formally, a quasitoric manifoldis defined by a 2 n -dimensional manifold M with a locally standard n -dimensional torus, say T n ,action such that the orbit space M/T n is identified with a simple polytope as manifolds withcorners, where a locally standard T n -action means that it is locally a coordinatewise T n -actionon C n . We refer to [BP] for details.The cohomological rigidity problem for quasitoric manifolds was originally posed by Masuda,where there is a good survey [CMS2]. For several simple quasitoric manifolds, the cohomologicalrigidity problem was affirmatively solved as in [DJ, CMS1, CPS, H1, H2], but their approachesare quite ad-hoc. So we would like to consider the cohomological rigidity probem for generalquasitoric manifolds. In general, we can approach to the cohomological rigidity in two stepswhich are quite different in nature: the first step is to show that spaces in question withisomorphic cohomology rings are homotopy equivalent, and the second step is to convert thehomotopy equivalences obtained in the first step into homeomorphisms. In this paper, we study Date : September 10, 2018.2010
Mathematics Subject Classification.
Primary 55P15; Secondary 57S25.
Key words and phrases. quasitoric manifold, cohomological rigidity, stable homotopy type, localization,Adams e -invariant. the first step for quasitoric manifolds from homotopy theoretical point of view. We will actuallyconsider the following problem. Problem 1.1.
Do quasitoric manifolds with isomorphic cohomology rings have the same p -localstable homotopy type? As a first step to attack this problem, the authors and Sato [HKS] obtained the followingresult which is a consequence of the p -local splitting of Σ M and Σ N in [HKS]; under theassumption of the theorem, the splitting shows that Σ M ( p ) and Σ N ( p ) are wedges of p -localspheres. Theorem 1.2 (Hsaui, Kishimoto, and Sato [HKS]) . If M, N are quasitoric manifolds with thesame betti numbers and dim M = dim N < p , then Σ M ( p ) ≃ Σ N ( p ) . This paper shows a much more general p -local stable cohomological rigidity of quasitoricmanifolds by considering K -theory, where we do not employ the p -local stable splitting ofquasitoric manifolds. We say that an isomorphism θ : H ∗ ( X ) ∼ = −→ H ∗ ( Y ) is p -locally realized bya stable map if there is a stable map h : Σ ∞ Y ( p ) → Σ ∞ X ( p ) which is θ ⊗ Z ( p ) in cohomologywith Z ( p ) -coefficient. Note in particular that h is a p -local stable homotopy equivalence by theJ.H.C. Whitehead theorem whenever X, Y are CW-complexes. We now state our main result.
Theorem 1.3.
Any cohomology isomorphism between quasitoric manifolds of dimension ≤ p − is p -locally realized by a stable map. Corollary 1.4.
If two quasitoric manifolds of dimension ≤ p − have isomorphic cohomologyrings, then they have the same p -local stable homotopy type. Hereafter, let p denote an odd prime unless otherwise specified. The 2-primary case will bedealt with only at the end of this paper.2. Adams e -invariant In this section, we recall the definition of the (complex) Adams e -invariant and its properties,and generalize it to maps from an odd sphere into a CW-complex without odd dimensionalcells, where we refer to [A] for details. Let π S ∗ denote the stable homotopy groups of spheres.Take f ∈ π S k − . Then it is a map f : S n +2 k − → S n for n large. We now consider the K -theory of the mapping cone of f . Since there is a homotopy cofibration S n → C f → S n +2 k , K ( C f ) is a free abelian group of rank 2, and we can choose generators ξ, η of K ( C f ) such thatch( ξ ) = u n + au n +2 k and ch( η ) = u n +2 k for a ∈ Q , where ch : K ( X ) → H ∗ ( X ) ⊗ Q and u i -LOCAL STABLE COHOMOLOGICAL RIGIDITY OF QUASITORIC MANIFOLDS 3 denote the Chern character and a generator of H i ( C f ) ∼ = Z respectively. Then the assignment e : π S k − → Q / Z , f [ a ]turns out to be a well-defined homomorphism, which is the Adams e -invariant. The propertyof the complex Adams e -invariant that we are going to use is the following. Theorem 2.1 (Adams [A, Example 12.8] and Toda [T, Theorem 4.15]) . The Adams e -invariant e : π S k − → Q / Z is injective for k ≤ p − when localized at the prime p . We call a CW-complex consisting only of even dimensional cells evenly generated. We gen-eralize the Adams e -invariant for maps from odd dimensional spheres into evenly generatedCW-complexes. Let X be a connected evenly generated finite CW-complex of dimension 2 d ,and let X ( r ) denote its r -skeleton. We choose a basis of K ( X (2 k ) ) called an admissible basis byinduction on k : • Fix a basis x i , . . . , x in i of H i ( X ) for i > • Choose a basis B := { ξ , . . . , ξ n } of e K ( X (2) ) satisfying ch( ξ i ) = x i . • Choose a basis B k := b B k − ∪ { ξ k , . . . , ξ kn k } of e K ( X (2 k ) ) such that b B k − restricts to B k − and ch( ξ ki ) = x ki , where the element of b B k − restricting to ξ ij ∈ B k − is denoted by ξ ij .The following property of admissible bases is clear from the definition. Proposition 2.2.
Let
X, Y be connected evenly generated finite CW-complexes. For a homo-topy equivalence h : X ≃ −→ Y and an admissible basis B of e K ( Y ) , h ∗ ( B ) is an admissible basisof e K ( X ) , where h ∗ ( B ) := { h ∗ ( ξ ) | ξ ∈ B} . For a map f : S r − → X , we define a basis B d ( f ) of e K ( C f ) from an admissible basis B d of e K ( X ) by B d ( f ) := b B d ∪ { η } such that b B d restricts to B d and ch( η ) = u r , where u r representsthe cell attached by f and ξ ij ∈ b B d denotes the element restricting to ξ ij ∈ B d . We now define e ( B d ( f )) ij ∈ Q by ch( ξ ij ) = e ( B d ( f )) ij u r + other terms ∈ H ∗ ( C f ) ⊗ Q which is a generalization of the Adams e -invariant that we are going to use to detect thetriviality of f . We observe basic properties of our generalization of the Adams e -invariant.Note that X/X (2 d − ≃ W n d S d such that j th sphere S d corresponds to the cohomology class x dj . Let π j be the composite X proj −−→ X/X (2 d − ≃ W n d S d → S d , where the last arrow is thepinch map onto the j th sphere. By definition, we immediately have the following. Lemma 2.3.
For r > d , e ( B d ( f )) dj ≡ e ( π j ◦ f ) mod 1 . SHO HASUI AND DAISUKE KISHIMOTO
When f deforms into the 2 k -skeleton X (2 k ) , we can construct both e ( B k ( f )) ij and e ( B d ( f )) ij for i ≤ k by regarding f as a map into X (2 k ) and X , respectively. By construction, we havethe following. Lemma 2.4. If f deforms into X (2 k ) , then e ( B k ( f )) ij = e ( B d ( f )) ij for i ≤ k . Proposition 2.5. If e ( B d ( f )) ij is an integer for all i, j and d ≤ p − , then the p -localizationof f is stably null homotopic.Proof. Localize everything at the prime p , so we abbreviate the notation − ( p ) for the p -localization. By the cellular approximation theorem, f deforms into X (2 r − , so we consider amap f : S r − → X (2 r − for which we can assume the same condition on the generalized Adams e -invariant by Lemma 2.4. Consider the composite¯ f : S r − f −→ X (2 r −
2) proj −−→ X (2 r − /X (2 r − ≃ _ n r − S r − where the j th sphere in the last space corresponds to x r − j . Let π j : W n r − S r − → S r − be thepinch map onto the j th sphere. Then by Lemma 2.3 and the assumption, we have e ( π j ◦ ¯ f ) ≡ π j ◦ ¯ f is stably null homotopic by Theorem 2.1. Thus we obtain that ¯ f itselfis stably null homotopic. Consider the exact sequence of the stable homotopy groups π S r − ( X (2 r − ) → π S r − ( X (2 r − ) → π S r − ( X (2 r − /X (2 r − ) . Then f belongs to the middle group and is mapped to ¯ f by the last arrow, so it deforms into X (2 r − stably. Hence, to continue the induction, it suffices to consider a map f : S r − → X (2 r − for which we can assume the same condition on the generalized Adams e -invariant byLemma 2.4 as well. Thus by iterating this procedure, we obtain that f deforms stably into X (2 k ) for any k , implying f is stably null homotopic. Therefore the proof is completed. (cid:3) Realization of cohomology isomorphisms and K -theory This section studies the p -local stable realizability of cohomology isomorphisms betweenevenly generated CW-complexes by using K -theory. Throughout this section, let X , X beconnected evenly generated finite CW-complexes. We say that θ : K ( X ) → K ( X ) is a lift of¯ θ : H ∗ ( X ) → H ∗ ( X ) if the equality ch ◦ θ = (¯ θ ⊗ Q ) ◦ chholds. For the rest of this section, we assume that there are isomorphisms θ : K ( X ) ∼ = −→ K ( X ) and ¯ θ : H ∗ ( X ) ∼ = −→ H ∗ ( X ) -LOCAL STABLE COHOMOLOGICAL RIGIDITY OF QUASITORIC MANIFOLDS 5 which are compatible with the Chern character. We consider the p -local realizability of theisomorphism θ by a stable homotopy equivalence between X and X . We first observe inducedmaps of θ, ¯ θ on subcomplexes and their quotinets. Proposition 3.1.
Let Y i be a subcomplex of X i for i = 1 , such that ¯ θ restricts to an isomor-phism ˆ θ | Y : H ∗ ( Y ) ∼ = −→ H ∗ ( Y ) . Then θ, ¯ θ induce (1) an isomorphism θ | Y : K ( Y ) ∼ = −→ K ( Y ) which is a lift of ¯ θ | Y , and (2) isomorphisms Θ : K ( X /Y ) ∼ = −→ K ( X /Y ) and Θ : H ∗ ( X /Y ) ∼ = −→ H ∗ ( X /Y ) such that Θ is a lift of Θ .Proof. We first show (2). Note that X i /Y i is an evenly generated CW-complex since so are X i , Y i and Y i is a subcomplex of X i . Then there is a commutative diagram of solid arrows(3.1) 0 / / H ∗ ( X /Y ) ¯Θ ∼ = (cid:15) (cid:15) ✤✤✤ / / H ∗ ( X ) ¯ θ ∼ = (cid:15) (cid:15) / / H ∗ ( Y ) ¯ θ | Y ∼ = (cid:15) (cid:15) / / / / H ∗ ( X /Y ) / / H ∗ ( X ) / / H ∗ ( Y ) / / / / K ( X i /Y i ) / / ch (cid:15) (cid:15) K ( X i ) ch (cid:15) (cid:15) / / K ( Y i ) / / ch (cid:15) (cid:15) / / H ∗ ( X i /Y i ) ⊗ Q / / H ∗ ( X i ) ⊗ Q / / H ∗ ( Y i ) ⊗ Q / / K ( X i ) → H ∗ ( X i ) ⊗ Q is injective since H ∗ ( X i ) is a free abelian group. Then it follows that K ( X i /Y i ) is the kernel of the composite f i : K ( X i ) ch −→ H ∗ ( X i ) ⊗ Q → H ∗ ( Y i ) ⊗ Q . So since there is a commutative diagram K ( X ) f / / θ ∼ = (cid:15) (cid:15) H ∗ ( Y ) ⊗ Q ¯ θ | Y ⊗ Q ∼ = (cid:15) (cid:15) K ( X ) f / / H ∗ ( Y ) ⊗ Q , we get an injection Θ : K ( X /Y ) → K ( X /Y ) which becomes an isomorphism after tensoring Q . Since K ( X i /Y i ) is a direct summand of the free abelian group K ( X i ), we conclude that Θis an isomorphism. Moreover, by a straightforward diagram chasing, we see that Θ is a lift ofΘ. Therefore the proof of (2) is done. We finally prove (1). There is a commutative diagram SHO HASUI AND DAISUKE KISHIMOTO of solid arrows 0 / / K ( X /Y ) Θ ∼ = (cid:15) (cid:15) / / K ( X ) θ ∼ = (cid:15) (cid:15) / / K ( Y ) (cid:15) (cid:15) ✤✤✤ / / / / K ( X /Y ) / / K ( X ) / / K ( Y ) / / (cid:3) The cases to which we apply Proposition 3.1 are:(1) Y i = X (2 k ) i for i = 1 ,
2, and(2) Y i is a subcomplex X (2 k ) i ∪ e i for i = 1 , θ sends the cohomology class of e to that of e .We now prove the p -local realizability of ¯ θ by a stable map. Theorem 3.2.
For dim X = dim X ≤ p − , ¯ θ is p -locally realized by a stable map.Proof. We put dim X = X = 2 d , and denote the induced maps in Proposition 3.1 by the samesymbols θ, ¯ θ . We prove the p -local realizability of ¯ θ by a stable map inductively on skeleta. Weassume all spaces and maps are stabilized and p -localized, so we omit the stabilization functorΣ ∞ and the p -localization − ( p ) .The case k = 1 is trivial since the spaces are wedges of S for which any self-maps in homologyis realizable. We now assume k > h : X (2 k − → X (2 k − such that h ∗ = ¯ θ ⊗ Z ( p ) . By arranging 2 k -cells of X , we may assume that θ, ¯ θ induce the identity mapon X (2 k ) i /X (2 k − i := W a S k . Let ϕ i : W a S k − → X (2 k − i be the attaching map of the 2 k -dimensional cells of X i , and let ι ℓ : S k − → W a S k − denote the inclusion of the ℓ th sphere.Then by Proposition 3.1 there are commutative diagrams0 / / K ( S k ) / / K ( C ϕ ◦ ι ℓ ) θ (cid:15) (cid:15) / / K ( X (2 k − ) θ (cid:15) (cid:15) / / / / K ( S k ) / / K ( C ϕ ◦ ι ℓ ) / / K ( X (2 k − ) / / / / H ∗ ( S k ) / / H ∗ ( C ϕ ◦ ι ℓ ) ¯ θ (cid:15) (cid:15) / / H ∗ ( X (2 k − ) h ∗ =¯ θ (cid:15) (cid:15) / / / / H ∗ ( S k ) / / H ∗ ( C ϕ ◦ ι ℓ ) / / H ∗ ( X (2 k − ) / / -LOCAL STABLE COHOMOLOGICAL RIGIDITY OF QUASITORIC MANIFOLDS 7 with exact rows which are compatible by the Chern character. Since the Chern characters onthese two diagrams are injective, we see that h ∗ = θ in the first diagram. Then we obtain(3.2) e ( B k − ( ϕ ◦ ι ℓ )) ij = e ( h ∗ ( B k − )( ϕ ◦ ι ℓ )) ij for any i, j , where h ∗ ( B k − ) is the admissible basis of e K ( X (2 k − ) as in Proposition 2.2. Onthe other hand, it immediately follows from the definition of the generalized Adams e -invariantthat(3.3) e ( h ∗ ( B k − )( ϕ ◦ ι ℓ )) ij = e ( B k − ( h ◦ ϕ ◦ ι ℓ )) ij for any i, j . We now consider a map f := ϕ − h ◦ ϕ : _ a S k − → X (2 k − . By definition of the generalized Adams e -invariant, we have e ( B k − ( f ◦ ι ℓ )) ij = e ( B k − ( ϕ ◦ ι ℓ )) ij − e ( B k − ( h ◦ ϕ ◦ ι ℓ )) ij for any i, j , so by (3.2) and (3.3) we obtain e ( B k − ( f ◦ ι ℓ )) ij = 0 for any i, j . Then by Proposition2.5, ϕ and h ◦ ϕ are stably homotopic, implying that there is a stable map ˜ h : X (2 k )2 → X (2 k )1 satisfying a homotopy commutative diagram W a S k − ϕ / / X (2 k − / / X (2 k )1 W a S k − ϕ / / X (2 k − h O O / / X (2 k )2 . ˜ h O O Therefore by the Puppe exact sequence, we see that ˜ h realizes ¯ θ , completing the proof. (cid:3) Proof of Theorem 1.3
This section applies Theorem 3.2 to quasitoric manifolds, and then proves Theorem 1.3. Werecall from [DJ] properties of quasitoric manifolds that we are going to use.
Proposition 4.1 (Davis and Januszkiewicz [DJ]) . For a quasitoric manifold M , the followinghold: (1) M is a connected evenly generated finite CW-complex; (2) H ∗ ( M ) is generated by 2-dimensional elements. Theorem 4.2.
For quasitoric manifolds M , M , any isomorphism ¯ θ : H ∗ ( M ) ∼ = −→ H ∗ ( M ) liftsto an isomorphism θ : K ( M ) ∼ = −→ K ( M ) . SHO HASUI AND DAISUKE KISHIMOTO
Proof.
Let x , . . . , x ℓ be a basis of H ( M ). Then ¯ θ ( x ) , . . . , ¯ θ ( x ℓ ) is a basis of H ( M ). Put ρ := x × · · · × x ℓ : M → ( C P ∞ ) ℓ and ρ := ¯ θ ( x ) × · · · × ¯ θ ( x ℓ ) : M → ( C P ∞ ) ℓ . By definition,we have ρ ∗ = ¯ θ ◦ ρ ∗ in cohomology. By considering the induced map between the Atiyah-Hirzebruch spectral se-quences, we see that ρ ∗ i : K (( C P ℓ )) → K ( M i ) is surjective for i = 1 ,
2. Then in order to get amap θ : K ( M ) → K ( M ), it is sufficient to show that Ker ρ ∗ ⊂ Ker ρ ∗ . For x ∈ K (( C P ∞ ) ℓ ),we suppose ρ ∗ ( x ) = 0. Then we have0 = ( θ ⊗ Q ) ◦ ch( ρ ∗ ( x )) = ( θ ⊗ Q ) ◦ ρ ∗ (ch( x )) = ρ ∗ (ch( x )) = ch( ρ ∗ ( x )) , implying ρ ∗ ( x ) = 0 since ch : K ( M ) → H ∗ ( M ) ⊗ Q is injective by Proposition 4.1. Then weget a map θ : K ( M ) → K ( M ) such that θ ( ρ ∗ ( y )) = ρ ∗ ( y ) for any y ∈ K (( C P ∞ ) ℓ ). We havethat θ is a lift of ¯ θ . Indeed, for any y ∈ K (( C P ∞ ) ℓ ),ch( θ ( ρ ∗ ( y ))) = ch( ρ ∗ ( y )) = ρ ∗ (ch( y )) = (¯ θ ⊗ Q ) ◦ ρ ∗ (ch( y )) = (¯ θ ⊗ Q )(ch( ρ ∗ ( y )))where ρ ∗ : K (( C P ∞ ) ℓ ) → K ( M ) is surjective. It remains to show that θ is an isomorphism.Since ρ ∗ : K (( C P ∞ ) ℓ ) → K ( M ) is surjective, so is θ . If θ ( x ) = 0 for x ∈ K ( M ), we have0 = ch( θ ( x )) = (¯ θ ⊗ Q )(ch( x )) , implying x = 0 since ¯ θ ⊗ Q is an isomorphism and ch : K ( M ) → H ∗ ( M ) ⊗ Q is injective. Thus θ is injective, completing the proof. (cid:3) Proof of Theorem 1.3.
Combine Theorem 3.2 and 4.2 when p is odd. For p = 2 we only need toconsider the case dim M = dim M = 4 since dim M = dim M implies M = M = S . Thecase dim M i = 4 is proved in [DJ]. Here is an alternative proof: M i has the stable homotopytype of a wedge of S and S or C P which is distinguished by mod 2 cohomology together withthe action of the Steenrod operation Sq . By Proposition, 4.1, ¯ θ respects Sq , and thereforethe proof is completed. (cid:3) References [A] J.F. Adams,
On the groups J ( X ) IV , Topology (1996), 21-71.[BP] V. M. Buchstaber and T. E. Panov, Torus actions and their applications in topology and combinatorics ,University Lecture Series , American Mathematical Society, Providence, RI, 2002.[CMS1] S. Choi, M. Masuda, and D.Y. Suh, Quasitoric manifolds over a product of simplices , Osaka J. Math. (2010), 109-129.[CMS2] S. Choi, M. Masuda, and D.Y. Suh, Rigidity problems in toric topology, a survey , Proc. Steklov Inst.Math. (2011), 177-190.[CPS] S. Choi, S. Park, and D.Y. Suh,
Topological classification of quasitoric manifolds with the second Bettinumber 2 , Pacific J. Math. (1) (2012), 19-49. -LOCAL STABLE COHOMOLOGICAL RIGIDITY OF QUASITORIC MANIFOLDS 9 [DJ] M. W. Davis and T. Januszkiewicz,
Convex polytopes, Coxeter orbifolds and torus actions , Duke Math. J. (1991), 417-452.[H1] S. Hasui, On the classification of quasitoric manifolds over dual cyclic polytopes , Algebr. Geom. Topology (2015), 1387-1437.[H2] S. Hasui, On the cohomology equivalences between bundle-type quasitoric manifolds over a cube , arXiv:1409.7980 .[HKS] S. Hasui, D. Kishimoto, and T. Sato, p -local stable splitting of quasitoric manifolds , arXiv:1412.6886 .[T] H. Toda, p -primary components of homotopy groups IV. Compositions and toric constructions , Mem. Coll.Sci. Kyoto, Ser. A (1959), 297-332. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan
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