Some calculations involving configuration spaces of distinct points

This page has not been refereed. The information given here might be incomplete or provisional.

1 Introduction

‘The complement of the diagonal’ and ‘the Gauss map’ ideas play a great role in different branches of mathematics. The Haefliger-Wu invariant is a manifestation of these ideas in the theory of embeddings. The complement to the diagonal idea originated from two celebrated theorems: the Lefschetz Fixed Point Theorem and the Borsuk-Ulam Antipodes Theorem. See [Vassiliev1992].

In introducing notation and definitions we follow [Skopenkov2020a].

The deleted product

$\displaystyle \widetilde K=K^{\underline2}:=\{(x,y)\in K\times K\ :\ x\ne y\}.$ $$\displaystyle \widetilde K=K^{\underline2}:=\{(x,y)\in K\times K\ :\ x\ne y\}.$$

This is the configuration space of ordered pairs of distinct points of $K$${{Stub}} == Introduction == <wikitex>; ‘The complement of the diagonal’ and ‘the Gauss map’ ideas play a great role in different branches of mathematics<!-- [Gl68, Va92]-->. The Haefliger-Wu invariant is a manifestation of these ideas in the theory of embeddings. The complement to the diagonal idea originated from two celebrated theorems: the Lefschetz Fixed Point Theorem and the Borsuk-Ulam Antipodes Theorem. See \cite{Vassiliev1992}. In introducing notation and definitions we follow \cite{Skopenkov2020a}. The ''deleted product'' $$\widetilde K=K^{\underline2}:=\{(x,y)\in K\times K\ :\ x\ne y\}.$$ This is the configuration space of ordered pairs of distinct points of $K$. Suppose that $f:K\to\R^m$ is an embedding of a subset $K\subset \mathbb R^N$. Then the map $\widetilde f:\widetilde K\to S^{m-1}$ is well-defined by the Gauss formula $$\widetilde f(x,y)=\frac{f(x)-f(y)}{|f(x)-f(y)|}.$$ We have $\widetilde f(y,x)=-\widetilde f(x,y)$, i.e. this map is equivariant with respect to the `exchanging factors' involution $(x,y)\mapsto(y,x)$ on $\widetilde K$ and the antipodal involution on $S^{m-1}$. Thus the existence of an equivariant map $\widetilde K\to S^{m-1}$ is a necessary condition for the embeddability of $K$ in $\R^m$. Denote by $\mathrm{Emb}^{m}K$ the set embeddings of $K$ into $\mathbb R^{m}$ up to isotopy. Let $\pi_{\mathrm{eq}}^{m}(\widetilde K) = [\widetilde K;S^{m}]_{\mathrm{eq}}$ be the set of equivariant maps $\widetilde K \to S^m$ up to equivariant homotopy. Denote by $[f]$ the equivalence class of the map $f$. <!--Definition of the Haefliger-Wu invariant $\alpha$--> '''The Haefliger-Wu invariant''' $\alpha:\mathrm{Emb}^{m}K\to \pi_{\mathrm{eq}}^{m-1}(\widetilde{K})$ is defined by formula <!--is induced by the Gauss map.--> $\alpha([f]) = [\widetilde f]$. <!-- The Haefliger-Wu invariant and the Gauss map are analogously defined for $N_0$; we will denote them by $\alpha_0$ in this case.--> {{beginthm|Theorem}} The Haefliger-Wu invariant $\alpha:\mathrm{Emb}^m K\to\pi^{m-1}_{\mathrm{eq}}( \widetilde K)$ is one-to-one either (a) $K$ is a compact connected $n$-complex and m\ge 3n+4$ or <!--(b) $K$ is a compact connected $n$-manifold with nonempty boundary and m\ge 3n+4$ or--> (b) $K$ is a compact $n$-manifold with nonempty boundary, $(K, \partial K)$ is $k$-connected, $\pi_1(\partial K) = 0$, $k + 3 \le n$, $(n, k) \notin \{(5, 2), (4, 1)\}$ and m\ge 3n+2-k$. {{endthm}} See \cite[$\S$ 5]{Skopenkov2006} and \cite[6.4]{Haefliger1963}, \cite[Theorem 1.1$\alpha\partial$]{Skopenkov2002} for the DIFF case and \cite[Theorem 1.3$\alpha\partial$]{Skopenkov2002} for the PL case. </wikitex> == Uniqueness theorems == <wikitex>; {{beginthm|Lemma}}\label{th::unknotting} Assume that $N$ is a compact $n$-manifold and either (a) $m \ge 2n+2$ or (b) $N$ is connected and $m \ge 2n+1 \ge 5$. Then each two equivariant maps from $\widetilde N$ to $S^{m-1}$ are equivariantly homotopic. {{endthm}} Hereafter denote by $\widetilde K$ the product $K\times K$ minus tubular neighborhood of the diagonal. ''Proof.'' Given two equivariant maps $\phi, \psi\colon\widetilde N \to S^{m-1}$ take an arbitrary equivariant triangulation $T$ of $\widetilde N$. (a) One can easily construct an equivariant homotopy between restrictions of $\phi$ and $\psi$ on vertices of $T$. By general position a homotopy between $\phi$ and $\psi$ on the boundary of a simplex can be extended to a homotopy on the whole simplex since the dimension of the simplex does not exceed n+1$. We extend equivariant homotopy on symmetric simplices in the symmetric way, so we obtain an equivariant homotopy. (b) Since $\widetilde{N}$ has non-empty boundary, there exists an equivariant deformational retraction of $\widetilde{N}$ to an equivariant $(2n-1)$-subcomplex of $T$. A homotopy between $\phi$ and $\psi$ on the subcomplex can by constructed similarly to case (a). This homotopy can be extended to a homotopy on $\widetilde{N}$. QED </wikitex> <!--== Invariants == <wikitex>; ... </wikitex> == Classification/Characterization == <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex> == Acknowledgments == ... == Footnotes == <references/> --> == References == {{#RefList:}} <!-- == External links == * The Wikipedia page about [[Wikipedia:Page_name|link text]]. --> <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]]K$.

Suppose that $f:K\to\R^m$$f:K\to\R^m$ is an embedding of a subset $K\subset \mathbb R^N$$K\subset \mathbb R^N$. Then the map $\widetilde f:\widetilde K\to S^{m-1}$$\widetilde f:\widetilde K\to S^{m-1}$ is well-defined by the Gauss formula

$\displaystyle \widetilde f(x,y)=\frac{f(x)-f(y)}{|f(x)-f(y)|}.$ $$\displaystyle \widetilde f(x,y)=\frac{f(x)-f(y)}{|f(x)-f(y)|}.$$

We have $\widetilde f(y,x)=-\widetilde f(x,y)$$\widetilde f(y,x)=-\widetilde f(x,y)$, i.e. this map is equivariant with respect to the `exchanging factors' involution $(x,y)\mapsto(y,x)$$(x,y)\mapsto(y,x)$ on $\widetilde K$$\widetilde K$ and the antipodal involution on $S^{m-1}$$S^{m-1}$. Thus the existence of an equivariant map $\widetilde K\to S^{m-1}$$\widetilde K\to S^{m-1}$ is a necessary condition for the embeddability of $K$$K$ in $\R^m$$\R^m$.

Denote by $\mathrm{Emb}^{m}K$$\mathrm{Emb}^{m}K$ the set embeddings of $K$$K$ into $\mathbb R^{m}$$\mathbb R^{m}$ up to isotopy. Let $\pi_{\mathrm{eq}}^{m}(\widetilde K) = [\widetilde K;S^{m}]_{\mathrm{eq}}$$\pi_{\mathrm{eq}}^{m}(\widetilde K) = [\widetilde K;S^{m}]_{\mathrm{eq}}$ be the set of equivariant maps $\widetilde K \to S^m$$\widetilde K \to S^m$ up to equivariant homotopy. Denote by $[f]$$[f]$ the equivalence class of the map $f$$f$.

The Haefliger-Wu invariant $\alpha:\mathrm{Emb}^{m}K\to \pi_{\mathrm{eq}}^{m-1}(\widetilde{K})$$\alpha:\mathrm{Emb}^{m}K\to \pi_{\mathrm{eq}}^{m-1}(\widetilde{K})$ is defined by formula $\alpha([f]) = [\widetilde f]$$\alpha([f]) = [\widetilde f]$.

Theorem 1.1. The Haefliger-Wu invariant $\alpha:\mathrm{Emb}^m K\to\pi^{m-1}_{\mathrm{eq}}( \widetilde K)$$\alpha:\mathrm{Emb}^m K\to\pi^{m-1}_{\mathrm{eq}}( \widetilde K)$ is one-to-one either

(a) $K$$K$ is a compact connected $n$$n$-complex and $2m\ge 3n+4$$2m\ge 3n+4$ or

(b) $K$$K$ is a compact $n$$n$-manifold with nonempty boundary, $(K, \partial K)$$(K, \partial K)$ is $k$$k$-connected, $\pi_1(\partial K) = 0$$\pi_1(\partial K) = 0$, $k + 3 \le n$$k + 3 \le n$, $(n, k) \notin \{(5, 2), (4, 1)\}$$(n, k) \notin \{(5, 2), (4, 1)\}$ and $2m\ge 3n+2-k$$2m\ge 3n+2-k$.

See [Skopenkov2006, $\S$$\S$ 5] and [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1$\alpha\partial$$\alpha\partial$] for the DIFF case and [Skopenkov2002, Theorem 1.3$\alpha\partial$$\alpha\partial$] for the PL case.

2 Uniqueness theorems

Lemma 2.1. Assume that $N$$N$ is a compact $n$$n$-manifold and either

(a) $m \ge 2n+2$$m \ge 2n+2$ or

(b) $N$$N$ is connected and $m \ge 2n+1 \ge 5$$m \ge 2n+1 \ge 5$.

Then each two equivariant maps from $\widetilde N$$\widetilde N$ to $S^{m-1}$$S^{m-1}$ are equivariantly homotopic.

Hereafter denote by $\widetilde K$$\widetilde K$ the product $K\times K$$K\times K$ minus tubular neighborhood of the diagonal.

Proof. Given two equivariant maps $\phi, \psi\colon\widetilde N \to S^{m-1}$$\phi, \psi\colon\widetilde N \to S^{m-1}$ take an arbitrary equivariant triangulation $T$$T$ of $\widetilde N$$\widetilde N$.

(a) One can easily construct an equivariant homotopy between restrictions of $\phi$$\phi$ and $\psi$$\psi$ on vertices of $T$$T$. By general position a homotopy between $\phi$$\phi$ and $\psi$$\psi$ on the boundary of a simplex can be extended to a homotopy on the whole simplex since the dimension of the simplex does not exceed $2n+1$$2n+1$. We extend equivariant homotopy on symmetric simplices in the symmetric way, so we obtain an equivariant homotopy.

(b) Since $\widetilde{N}$$\widetilde{N}$ has non-empty boundary, there exists an equivariant deformational retraction of $\widetilde{N}$$\widetilde{N}$ to an equivariant $(2n-1)$$(2n-1)$-subcomplex of $T$$T$. A homotopy between $\phi$$\phi$ and $\psi$$\psi$ on the subcomplex can by constructed similarly to case (a). This homotopy can be extended to a homotopy on $\widetilde{N}$$\widetilde{N}$. QED

3 References

• [Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
• [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
• [Skopenkov2020a] https://www.mccme.ru/circles/oim/eliminat_talk.pdf
• [Vassiliev1992]

V. A. Vassiliev, Complements of discriminants of smooth maps: Topology and applications., Amer. Math. Soc., Providence, RI, (1992).