Cobordism classes of maps and covers for spheres

TL;DR

This paper proves that for dimensions m>n, all cobordism classes of maps from m-spheres to n-spheres are trivial, with implications for understanding sphere covers.

Contribution

It establishes the triviality of cobordism classes of sphere maps for m>n and explores applications to sphere covers.

Findings

Cobordism classes of maps from m-sphere to n-sphere are trivial for m>n.

Results have applications in the theory of covers for spheres.

Provides a new perspective on cobordism homotopy groups of spheres.

Abstract

In this paper we show that for m>n the set of cobordism classes of maps from m-sphere to n-sphere is trivial. The determination of the cobordism homotopy groups of spheres admits applications to the covers for spheres.

Equations

$${\pi }_m^C(S^n)=0.$$

$${\pi }_m^C(S^n)={\pi }_m(S^n)/N.$$

$$\bigcap _{i=0}^{n+1}{V}_i\ne \varnothing$$

$$f_{\mathcal{U},\Phi }(x):=\sum _{i=0}^{n+1}{\phi }_i(x)v_i,$$

$$U_{\ell }(\Phi ):=F_{\Phi }^{-1}(B_{\ell }),U(\Phi ):=\{U_0(\Phi ),\dots ,U_{n+1}(\Phi )\}.$$

$$Q_{\ell }:=\bigcup _{\Phi \in \Pi }{U}_{\ell }(\Phi ),\mathcal{Q}:=\{Q_0,\dots ,Q_{n+1}\}.$$

$$\mathcal{Q}{|}_{T\times \{0\}}={\mathcal{S}}_1,\mathcal{Q}{|}_{T\times \{1\}}={\mathcal{S}}_2.$$

Full text

Keywords: cobordism, homotopy group, covers

1 Introduction

Let $M_{1}$ and $M_{2}$ be compact oriented manifolds of dimension $m$. Two continuous maps $f_{1}:M_{1}\to X$ and $f_{2}:M_{2}\to X$ are called cobordant if there are a compact oriented manifold $W$ with $\partial W=M_{1}\sqcup M_{2}$ and a continuous map $F:W\to X$ such that $F|_{M_{i}}=f_{i}$ for $i=1,2$. Note that the set of cobordism classes $f:S^{m}\to X$ form a group $\pi^{C}_{m}(X)$ that is a quotient of $\pi_{m}(X)$.

In Section 2 we consider assumptions for $X$ such that $\pi^{C}_{m}(X)$=0 (Theorem 2.1). In particular, Corollary 2.2 states that $\pi^{C}_{n}(S^{n})=\pi_{n}(S^{n})={\mathbb{Z}}$ and if $m>n$ then

[TABLE]

In Section 3 we show that for manifolds the homotopy and cobordism classes of covers are equivalent to the homotopy and cobordism classes of their associated maps. Then we can apply results of Sections 2 for covers, in particular, see Corollary 3.6.

2 Cobordism classes of maps for spheres

Consider a group of oriented cobordism classes of maps $\Omega_{*}^{SO}(X)$ [3, Chapter 1]. Let $M_{i}$, $i=1,2$, be compact oriented manifolds without boundary of dimension $m$. Let $f_{i}:M_{i}\to X$, $i=1,2$, be continuous maps to a space $X$. Then $[f_{1}]_{C}=[f_{2}]_{C}$ in $\Omega_{m}^{SO}(X)$ , i.e. maps $f_{i}$ are cobordant if there are a compact oriented manifold $W$ with $\partial W=M_{1}\sqcup M_{2}$ and a continuous map $F:W\to X$ such that $F|_{M_{i}}=f_{i}$ for $i=1,2$.

If $M_{2}=\emptyset$, then $[f_{1}]_{C}=0$. In this case $f_{1}$ is called null–cobordant.

Let $M$ be a compact oriented manifold without boundary. We denote the set of cobordism classes of $f:M\to X$ by $[M,X]_{C}$.

Theorem 2.1**.**

Let $X$ be a finite $CW$-complex whose integral homology $H_{*}(X,\mathbb{Z})$ has only 2-torsion. Let $f:S^{m}\to X$ be a map that induces zero homomorphism of $m$-dimensional cohomology with coefficients in $\mathbb{Z}$ and $\mathbb{Z}_{2}$. Then $f$ is null-cobordant. in $\Omega_{m}^{SO}(X)$ the image of $f$ is 0. In particular, $[S^{m},X]_{C}=0$ if $\dim X<m$.

Proof.

By [3, Theorem 17.6], the cobordism class of $f\colon S^{m}\to X$ is determined by the Pontrjagin numbers and the Stiefel-Whitney numbers of the map $f$. From the definition, the Pontrjagin numbers and the Stiefel-Whitney numbers of the map $f$ are determined by its induced homomorphisms on cohomology with coefficients in $\mathbb{Z}$ and $\mathbb{Z}_{2}$, respectively. The hypothesis in the statement guarantees that $f$ and the constant map induce the same homomorphism on cohomology with coefficients in $\mathbb{Z}$ and $\mathbb{Z}_{2}$, and hence the result. ∎

Let $M$ be an $m$–dimensional sphere ${S}^{m}$. In this case denote $[M,S^{n}]_{C}$ by $\pi_{m}^{C}(S^{n})$. It is easy to prove that the cobordism classes $\pi_{m}^{C}(S^{n})$ form a group. Moreover, there is a subgroup $N$ in $\pi_{m}(S^{n})$ such that

[TABLE]

Corollary 2.2**.**

If $m\neq n$, then $\pi_{m}^{C}(S^{n})=0$, otherwise $\pi_{n}^{C}(S^{m})={\mathbb{Z}}$.

Proof.

We obviously have the case $m<n$. Theorem 2.1 yields the most complicated case.

Let $m=n$. The Hopf degree theorem (see [6, Sect. 7]) states that two continuous maps $f_{1},f_{2}:S^{n}\to S^{n}$ are homotopic, i.e. $[f_{1}]=[f_{2}]$ in $\pi_{n}(S^{n})$, if and only if $\deg{f_{1}}=\deg{f_{2}}$. It is clear that $[f_{1}]=[f_{2}]$ implies $[f_{1}]_{C}=[f_{2}]_{C}$. Now we show that from $[f_{1}]_{C}=[f_{2}]_{C}$ follows $\deg{f_{1}}=\deg{f_{2}}$. Indeed, then we have $F:W\to S^{n}$ with $F|_{M_{i}=S^{n}}=f_{i}$. Note that $Z:=F^{-1}(x)$ for a regular $x\in S^{n}$, is a manifold of dimension one. It is easy to see that a cobordism $(Z,Z_{1},Z_{2})$, where $Z_{i}:=f_{i}^{-1}(x)$, implies $\deg{f_{1}}=\deg{f_{2}}$. Thus, $\pi_{n}^{C}(S^{n})=\pi_{n}(S^{n})={\mathbb{Z}}$. ∎

Corollary 2.2 states that $f:S^{m}\to S^{n}$ is null-cobordant for $m>n$. Therefore, we have the following result.

Corollary 2.3**.**

Let $m>n$. Then for any continuous map $f:S^{m}\to S^{n}$ there are a compact oriented manifold $W$ with $\partial W=S^{m}$ and a continuous map $F:W\to S^{n}$ such that $F$ on the boundary coincides with $f$.

Remark. In the earlier version of this paper, we had a proof that $\pi_{m}^{C}(S^{n})=0$, where $m>n$ only for particular cases. We formulated this statement as a conjecture and sent the preprint to several topologists. Soon, Diarmuid Crowley sent us a sketch of the proof of this conjecture. Later, Alexey Volovikov pointed out to us that Theorem 2.1 follows easily from [3, Theorem 17.6].

3 Homotopy and cobordism classes of covers

For open (or closed) covers $\mathcal{U}$ of a normal space $T$ we considered certain homotopy classes $[f_{\mathcal{U}}]$ in $[T,S^{n}]$ defined in [8]. In this section we define a homotopy equivalence for covers and prove that two covers $\mathcal{U}_{1}$ and $\mathcal{U}_{2}$ are homotopy equivalent if and only if $[f_{\mathcal{U}_{1}}]=[f_{\mathcal{U}_{2}}]$ in $[T,{S}^{n}]$ (Theorem 3.2). We also prove that two covers on manifolds of the same dimension are cobordant if and only if the corresponding cobordism classes $[f_{\mathcal{U}_{i}}]_{C}$ in $\Omega_{*}(S^{n})$ are equal (Theorem 3.4).

The homotopy invariants $[f_{\mathcal{U}}]$ can be considered as obstructions for extending covers of a subspace $A\subset X$ to a cover of all of $X$. (Note that the classical obstruction theory (see [4, 10]) considers homotopy invariants that equal zero if a map can be extended from the $k$–skeleton of $X$ to the $(k+1)$–skeleton and are non-zero otherwise.) In our papers [8, 9] using these obstructions we obtain generalizations of the classic KKM (Knaster–Kuratowski–Mazurkiewicz) and Sperner lemmas [5, 11].

Let $X$ be any compact oriented manifold of dimension $(m+1)$ and $A=\partial X$ be its boundary. Let $\mathcal{U}=\{U_{0},\ldots,U_{n+1}\}$ be a cover of $A$ such that the intersection of all subsets $U_{i}$ is empty. Then $[\mathcal{U}]\in[A,S^{n}]$, where the homotopy class $[\mathcal{U}]$ is defined in [8]. In the case $m=n$ we have $[A,S^{n}]=\mathbb{Z}$ and, if $[\mathcal{U}]\neq 0$, then for any extension of this cover to a cover $\mathcal{V}=\{V_{0},\ldots,V_{n+1}\}$ of $X$ the intersection

[TABLE]

This fact is a generalization of the Sperner–KKM lemma [8, Theorem 2.6].

Another generalization of the KKM lemma is the following (see [8, Corollary 3.1]): Let $X$ is an $(m+1)$–disc and $A=S^{m}$. If $[\mathcal{U}]\neq 0$ in $\pi_{m}(S^{n})$, then we have property $(4.1)$.

However, for $m>n$ not all pairs $(X,A)$ satisfy property $(4.1)$. For instance, $X=\mathbb{C}\mathrm{P}^{2}\smallsetminus\mathrm{Int}(D^{4})$ and $A:=\partial{X}=S^{3}$. Then the Hopf map $f:S^{3}\to S^{2}$ can be extended to a continuous map $F:X\to S^{2}$. It implies that a coresponding cover $\mathcal{U}=\{U_{0},U_{1},U_{2},U_{3}\}$ can be extened to $X$ such that the intersection of all $U_{i}$ is empty.

Let $\mathcal{U}=\{U_{0},\ldots,U_{n+1}\}$ be a collection of open sets whose union contains a normal space $T$. In other words, $\mathcal{U}$ is a cover of $T$. Let $\Phi=\{\varphi_{0},\ldots,\varphi_{n+1}\}$ be a partition of unity subordinate to $\mathcal{U}$. Let

[TABLE]

where $v_{0},\ldots,v_{n+1}$ are vertices of an $(n+1)$–simplex $\Delta^{n+1}$ in ${\mathbb{R}}^{n+1}$.

Suppose the intersection of all $U_{i}$ is empty. Then $f_{\mathcal{U},\Phi}$ is a continuous map from $T$ to ${S}^{n}$. In [8, Lemmas 2.1 and 2.2] we proved that a homotopy class $[f_{\mathcal{U},\Phi}]$ in $[T,{S}^{n}]$ does not depend on $\Phi$. We denote it by $[f_{\mathcal{U}}]$.

In fact, see [8, Lemma 2.4], the homotopy classes $[f_{\mathcal{U}}]$ of covers are also well defined for closed sets. We call a family of sets $\mathcal{S}=\{S_{0},\ldots,S_{n+1}\}$ a cover of a space $T$ if $\mathcal{S}$ is either an open or closed cover of $T$.

Homotopy invariants of covers we defined through homotopy invariants of maps. Let us define them directly for covers.

Definition 3.1**.**

Let $\mathcal{S}_{i}=\{S_{0}^{i},\ldots,S_{n+1}^{i}\}$, $i=1,2$, be covers of a normal space $T$ such that for $i=1,2$ the intersection of all subsets in $\mathcal{S}_{i}$ is empty. We say that $\mathcal{S}_{1}$ is homotopic to $\mathcal{S}_{2}$ and write $[\mathcal{S}_{1}]=[\mathcal{S}_{2}]$ if   $T\times[0,1]$ can be covered by $\mathcal{Q}=\{Q_{0},\ldots,Q_{n+1}\}$ such that $\mathcal{Q}$ is an extension of $\mathcal{S}_{1}\cup\mathcal{S}_{2}$ of $T\times\{0,1\}$ and the intersection of all $Q_{k}$ is empty.

The following theorem extends Theorem 2.2 in [8].

Theorem 3.2**.**

Let $\mathcal{S}_{i}=\{S_{0}^{i},\ldots,S^{i}_{n+1}\}$, $i=1,2$, be covers of a normal space $T$. Suppose the intersection of all the $S_{j}^{i}$ in $\mathcal{S}_{i}$ is empty. Then $[\mathcal{S}_{1}]=[\mathcal{S}_{2}]$ if and only if $[f_{\mathcal{S}_{1}}]=[f_{\mathcal{S}_{2}}]$ in $[T,{S}^{n}]$.

Proof.

From [8, Lemma 1.11] it suffices to prove the theorem for open covers. It is clear that if $[\mathcal{S}_{1}]=[\mathcal{S}_{2}]$ then $[f_{\mathcal{S}_{1}}]=[f_{\mathcal{S}_{2}}]$. Now we prove the converse statement.

Suppose $[f_{\mathcal{S}_{1}}]=[f_{\mathcal{S}_{2}}]$. Let $\Phi_{i}$, $i=1,2$, be any partitions of unity subordinate to $\mathcal{S}_{i}$. Then there is a homotopy $F_{\Phi}:T\times[0,1]\to S^{n}$ between $f_{\mathcal{S}_{1},\Phi_{1}}$ and $f_{\mathcal{S}_{2},\Phi_{2}}$, where $\Phi:=(\Phi_{1},\Phi_{2})$.

Consider $S^{n}$ as the boundary of $\Delta^{n+1}$. Let $B_{i}$ be the open star of a vertex $v_{i}$ of $\Delta^{n+1}$. Let

[TABLE]

Then $U(\Phi)$ is a cover of $T\times[0,1]$.

Denote by $\Pi$ the set of all pairs $\Phi:=(\Phi_{1},\Phi_{2})$, where $\Phi_{i}$ is a partition of unity subordinate to $\mathcal{S}_{i}$. Let

[TABLE]

Then ${\mathcal{Q}}$ is a cover of $T\times[0,1]$ and

[TABLE]

This yields $[\mathcal{S}_{1}]=[\mathcal{S}_{2}]$. ∎

Definition 3.3**.**

Let $M_{i}$, $i=1,2$, be compact oriented manifolds without boundary with $\dim{M_{1}}=\dim{M_{2}}$. Let $\mathcal{S}_{i}=\{S_{0}^{i},\ldots,S^{i}_{n+1}\}$, $i=1,2$, be covers of $M_{i}$ such that for $i=1,2$ the intersection of all subsets in $\mathcal{S}_{i}$ is empty. We say that $\mathcal{S}_{1}$ is cobordant to $\mathcal{S}_{2}$ and write $[\mathcal{S}_{1}]_{C}=[\mathcal{S}_{2}]_{C}$ if there are a compact oriented manifold $W$ with $\partial W=M_{1}\sqcup M_{2}$ and its cover $\mathcal{Q}=\{Q_{1},\ldots,Q_{n}\}$ such that $\mathcal{Q}|_{M_{i}}=\mathcal{S}_{i}$, $i=1,2$, and the intersection of all $Q_{k}$ is empty. If $M_{2}=\emptyset$, then we say that $\mathcal{S}_{1}$ is null–cobordant and write $[\mathcal{S}_{1}]_{C}=0$.

Note that if $[\mathcal{S}_{1}]_{C}=[\mathcal{S}_{2}]_{C}$, then $[f_{\mathcal{S}_{1}}]_{C}=[f_{\mathcal{S}_{2}}]_{C}$ in $\Omega_{*}^{SO}(S^{n})$, where for a continuous $f:M\to S^{n}$ by $[f]_{C}$ we denote the correspondent cobordism class.

Theorem 3.4**.**

Let $M_{1}$ and $M_{2}$ be compact oriented homotopy equivalent manifolds without boundary. Let $\mathcal{S}_{i}$, $i=1,2$, be covers of $M_{i}$ such that the intersection of all covers in $\mathcal{S}_{i}$ is empty. Then $[\mathcal{S}_{1}]_{C}=[\mathcal{S}_{2}]_{C}$ if and only if $[f_{\mathcal{S}_{1}}]_{C}=[f_{\mathcal{S}_{2}}]_{C}$.

Proof.

By definition if $[f_{\mathcal{S}_{1},\Phi_{1}}]_{C}=[f_{\mathcal{S}_{2},\Phi_{2}}]_{C}$, then there is a map $F_{\Phi}:W\to S^{n}$ such that $F|_{M_{i}}=f_{\mathcal{S}_{i},\Phi_{i}}$. Actually, the theorem can be proved by the same arguments as Theorem 3.2 if we substitute $T\times[0,1]$ by a cobordism $W.$ ∎

From this theorem it is easy prove the following corollary.

Corollary 3.5**.**

Let $\mathcal{S}$ be a cover of a compact oriented manifold $M$ such that the intersection of all subsets in $\mathcal{S}$ is empty. Suppose $[\mathcal{S}]_{C}=0$. Then there is a compact oriented manifold $W$ with $\partial W=M$ such that $\mathcal{S}$ can be extended to a cover $\mathcal{Q}$ of $W$ (i.e. $\mathcal{Q}|_{M}=\mathcal{S}$) with the empty intersection of all subsets $Q_{k}$.

Theorem 3.4 and Corollary 2.2 yield

Corollary 3.6**.**

Let $m>n$. Then for any cover $\mathcal{U}=\{U_{0},\ldots,U_{n+1}\}$ of $S^{m}$ with the empty intersection of all subsets in $\mathcal{U}$ there are a compact oriented manifold $W$ with $\partial W=S^{m}$ and a cover $\mathcal{Q}$ of $W$ such that $\mathcal{Q}$ is an extension of $\mathcal{U}$ with the empty intersection of all subsets in $\mathcal{Q}$.

Acknowledgement. We wish to thank Diarmuid Crowley, Alexander Dranishnikov, Roman Karasev, Arkadiy Skopenkov and Alexey Volovikov for helpful discussions and comments.