# Some calculations involving configuration spaces of distinct points

## 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

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

This is the configuration space of ordered pairs of distinct points of $K$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}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

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

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

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