aa r X i v : . [ m a t h . A T ] A p r SOME EXAMPLES OF NON-TIDY SPACES
TAKAHIRO MATSUSHITA
Abstract.
We construct a free Z -manifold X n for a positive integer n suchthat w ( X n ) n = 0, but there is no Z -equivariant map from S to X n . Introduction and Main theorems
First we fix terminologies and notations we use in this paper. The group actson spaces from the right unless otherwise stated. Let Γ denote a group. In thispaper, the Γ-action on a space X is said to be free if for any x ∈ X , there is aneighborhood U of x such that U γ ∩ U = ∅ for any γ ∈ Γ \ { e Γ } . For a Γ-space X , we denote the orbit space of X by X . Let x ∈ X . The image of π ( X, x ) viathe group homomorphism π ( X, x ) → π ( X, x ) induced by the quotient X → X is often written by π ( X, x ) also, for simplicity. The coefficient of the singularcohomology is considered as Z , the cyclic group with order 2.We write S na for the n -dimensional sphere with the antipodal action. For a free Z -space X , we putcoind( X ) = sup { n ≥ | There is a Z -map from S na to X . } , ind( X ) = inf { n ≥ | There is a Z -map from X to S na . } .h ( X ) = sup { n ≥ | w ( X ) n = 0 } and call the coindex, the index, and the Stiefel-Whitney height of X respectively ,where w ( X ) ∈ H ( X ) is the 1st Stiefel-Whitney class of the double cover X → X .It is obvious that coind( X ) ≤ h ( X ) ≤ ind( X )for every free Z -space X . In [1], X is said to be tidy if ind( X ) = coind( X ).In this paper, we prove the following. Theorem 1.1.
For a positive integer n , there is a free Z -space X n such that h ( X ) = n but coind( X n ) = 1 . The space X n is defined as follows. Let S nb denote the Z - Z -space where its basespace is the n -dimensional sphere S n ⊂ R n +1 , and the left and the right Z -actionsare defined by τ ( x , · · · , x n ) = ( − x , x , · · · , x n ) , ( x , · · · , x n ) τ = ( − x , · · · , − x n ) , where τ is the generator of Z . Then we define X = S a , and X k +1 = X k × Z S b .In [2], Schultz proved that h ( X × Z S nb ) ≥ h ( X ) + n for any free Z -space X .We prove the equality holds, although this is not necessary to prove Theorem 1.1. Theorem 1.2. h ( X × Z S nb ) = h ( X ) + n for any free Z -space X . These terminologies are due to [1]. Many different terminologies are used, see [3] or [4].
As far as I know, there is no explicit example published whose difference betweenthe Stiefel-Whitney height and the coindex is greater than 1. On the other hand,it is known that the difference between ind( X ) and h ( X ) can be arbitrarily large.Indeed, the odd dimensional real projective space R P n − is such example by theresult of Stolz [3], see also [4]. 2. Proofs
First we prove Theorem 1.1. As is said in Section 1, Schultz proved that h ( X × Z S nb ) ≥ h ( X )+ n . So we have that h ( X n ) ≥ n . Since X n is an n -dimensional manifold(or by Theorem 1.2), we have h ( X n ) = n . So what we must show is that there isno Z -equivariant map from S a to X n . To prove this, we establish a criterion toshow the non-existence of equivariant maps, using fundamental groups.Let Γ be a discrete group, and X a path-connected free Γ-space. Let x ∈ X bea base point. Then by the covering space theory, we have an isomorphismΓ ∼ = π ( X, x ) /π ( X, x ) . Recall that this isomorphism is given as follows. For α ∈ π ( X, x ), let ϕ ∈ α andlet ˜ ϕ denote the lift of ϕ whose initial point is x . Then the terminal point of ˜ ϕ is in the fiber over x , so there is a unique Φ X ( α ) ∈ Γ such that ϕ (1) = x Φ X ( α ).This Φ X : π ( X, x ) → Γ is a group homomorphism, which is surjective since X ispath-connected, and its kernel is π ( X, x ). Hence Φ X induces the isomorphismΦ X : π ( X, x ) /π ( X, x ) −→ Γ . Let X and Y be connected free Γ-spaces and f : X → Y a Γ-equivariant map.Let x ∈ X and put y = f ( x ). Since the diagram π ( X, x ) f ∗ −−−−→ π ( Y, y ) q X ∗ y y q Y ∗ π ( X, x ) f ∗ −−−−→ π ( Y , y )is commutative, so we have a group homomorphismˆ f ∗ : π ( X, x ) /π ( X, x ) → π ( Y , y ) /π ( Y, y ) . We write Ψ( f ) : Γ → Γ for the group homomorphism which commutes the followingdiagram. Γ Ψ( f ) −−−−→ Γ Φ X x x Φ Y π ( X, x ) /π ( X, x ) ˆ f ∗ −−−−→ π ( Y , y ) /π ( Y, y ) . Then we have the following.
Proposition 2.1.
The group homomorphism Ψ( f ) : Γ → Γ is the identity.Proof. Let α ∈ π ( X, x ) /π ( X, x ) and let ϕ be the loop of ( X, x ) which repre-sents α . Let ˜ ϕ denote the lift of ϕ whose initial point is x . Then ˜ ϕ (1) = x Φ X ( α ).Then we have y Φ X ( α ) = f ◦ ˜ ϕ (1) = y Φ Y ( ˆ f ∗ α ). Hence Φ X = Φ Y ◦ ˆ f ∗ . (cid:3) OME EXAMPLES OF NON-TIDY SPACES 3
The situation we used here is the case Γ = Z . Let X be a path-connected free Z -space and x ∈ X , we say α ∈ π ( X, x ) is said to be even if α ∈ π ( X, x ),and is said to be odd if α is not even. Then Proposition 2.1 asserts that for a Z -equivariant map f : X → Y , the group homomorphism f ∗ : π ( X, x ) → π ( Y , y )preserves the parity of π ( X, x ).Let us start to the proof of Theorem 1.1. Lemma 2.2.
The group π ( X n ) has no non-trivial torsion elements.Proof. By the definition of X n , X n is the orbit space of a free and isometrical π ( X n )-action on R n . Remark that for an affine map A : R n → R n , a , · · · , a m ∈ R with P mi =1 a i = 1, and x , · · · , x m ∈ R n , we have A ( m X i =1 a i x i ) = m X i =1 a i ( Ax i ) . Let α ∈ π ( X n ) be a non-trivial torsion element and its order is denoted by k . Let x ∈ R n . Then the point y = 1 k k X i =1 xα i ∈ R n is fixed by α . This is contradiction. (cid:3) Hence to prove Theorem 1.1, it is sufficient to prove the following.
Theorem 2.3.
Let X be a path-connected free Z -space. Then there is a Z -equivariant map from S a to X if and only if there is α ∈ π ( X ) such that α = 1 and α π ( X ) .Proof. Suppose there is a Z -map f : S a → X . Let β denote the generator of π ( S a ) ∼ = Z . Since β is odd, f ∗ ( β ) is odd. Since f ∗ ( β ) · f ∗ ( β ) = f ∗ ( β · β ) = 1, wehave completed the “only if” part.On the other hand, suppose α ∈ π ( X, x ) which is odd and α = 1. Let ϕ ∈ α ,and let ˜ ϕ denote the lift of ϕ whose initial point is x . Then ψ = ( ˜ ϕτ ) · ˜ ϕ is a loopof ( X, x ), and is null-homotopic since q X ∗ [ ψ ] = α = 1 and q X ∗ is injective, where q X : X → X is the quotient map. We can regard ψ is a Z -map from S a to X , andhence we can extend it to a Z -map S a → X . This completes the proof. (cid:3) Finally, we prove Theorem 1.2.
Proof of Theorem 1.2.
It is sufficient to prove that h ( X × Z S n +1 b ) = h ( X × Z S nb )+1for n ≥
0. The part “ ≥ ” is proved by Schultz in [2], so we give the proof of thepart “ ≤ ”. Put A = { ( x , · · · , x n +1 ) ∈ S nb | | x n +1 | ≤ } ,B = { ( x , · · · , x n +1 ) ∈ S nb | | x n +1 | ≥ } . These are Z - Z -closed subset of S n +1 b . Put X ′ = X × Z S n +1 b , A ′ = X × Z A , and B ′ = X × Z B . Then the followings hold.(1) A ′ ≃ Z X × Z S nb .(2) B ′ ≃ Z X ⊔ X with the involution exchanging each X . Hence w ( B ′ ) = 0.(3) X ′ = A ′ ∪ B ′ . TAKAHIRO MATSUSHITA
Suppose w ( X × Z S nb ) k = 0. Then w ( A ′ ) k = 0. Then there is α ∈ H k ( X ′ , A ′ )which maps to w ( X ′ ) k via H k ( X ′ , A ′ ) → H k ( X ′ ). Similarly, there is β ∈ H ( X ′ , B ′ )which maps to w ( X ′ ) via H ( X ′ , B ′ ) → H ( X ′ ). Then α ∪ β ∈ H k +1 ( X ′ , A ′ ∪ B ′ ) =0, and which maps to w ( X ′ ) k +1 . Hence we have w ( X ′ ) k +1 = 0. This completesthe proof. (cid:3) Acknowledgement.
This work was supported by the Program for Leading Grad-uate Schools, MEXT, Japan.
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