Some calculations involving configuration spaces of distinct points

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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.

For a subset $X\subset \mathbb R^{m}$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}X\subset \mathbb R^{m}$, $\widetilde X$$\widetilde X$ denotes the deleted product of $X$$X$, i.e. $\widetilde X:= \{(x, y)\in X\times X, x\neq y\} = X^2$$\widetilde X:= \{(x, y)\in X\times X, x\neq y\} = X^2$ minus the diagonal. This is the configuration space of ordered pairs of distinct points of K.

Definition 1.1.[of the Haefliger-Wu invariant $\alpha$$\alpha$]

The Haefliger-Wu invariant $\alpha:\mathrm{Emb}^{k}N\to \pi_{\mathrm{eq}}^{k-1}(\widetilde{N})$$\alpha:\mathrm{Emb}^{k}N\to \pi_{\mathrm{eq}}^{k-1}(\widetilde{N})$ is induced by the Gauss map, also denoted by $\alpha$$\alpha$. The Gauss map assigns to an individual embedding $f:N\to\R^{k}$$f:N\to\R^{k}$ an equivariant map $\widetilde{N}\to S^{k-1}$$\widetilde{N}\to S^{k-1}$ defined by the formula

$\displaystyle (x,y)\mapsto \frac{f(x)-f(y)} {\|f(x)-f(y)\|}, \quad (x,y)\in\widetilde{N}\subset N\times N.$ $$\displaystyle (x,y)\mapsto \frac{f(x)-f(y)} {\|f(x)-f(y)\|}, \quad (x,y)\in\widetilde{N}\subset N\times N.$$

Theorem 1.2. The Haefliger-Wu invariant $\alpha:\mathrm{Emb}^m N\to\pi^{m-1}_{\mathrm{eq}}( \widetilde N)$$\alpha:\mathrm{Emb}^m N\to\pi^{m-1}_{\mathrm{eq}}( \widetilde N)$ is one-to-one for $2m\ge 3n+4$$2m\ge 3n+4$.

2 Uniqueness theorems

Theorem 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 any two embeddings of $N$$N$ into $\R^m$$\R^m$ are isotopic.

In cases (a-d) inequality $2m\ge3n+4$$2m\ge3n+4$ is clearly satisfied. Hence by Haefliger-Weber theorem \ref{thm_Haef-Web} we have $\mathrm{Emb}_m(N) \cong \pi^{m-1}_{\mathrm{eq}}(\widetilde N)$$\mathrm{Emb}_m(N) \cong \pi^{m-1}_{\mathrm{eq}}(\widetilde N)$. So it is sufficient to show that $\pi^{m-1}\_{\mathrm{eq}}(\widetilde N)$$\pi^{m-1}\_{\mathrm{eq}}(\widetilde N)$ is trivial, i.e. every two every two equivariant maps $f, g:\widetilde N\to S^{m-1}$$f, g:\widetilde N\to S^{m-1}$ are equivariantly homotopic.

Take an arbitrary equivariant triangulation $T$$T$ of $\widetilde N$$\widetilde N$.

(a) One can easily construct an equivariant homotopy between restrictions of $f$$f$ and $g$$g$ on vertices of $T$$T$. By general position a homotopy of $f, g$$f, g$ on the boundary of a $k$$k$-simplex can be extended to a homotopy on the whole $k$$k$-simplex since $k<2n+1$$k<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 on the subcomplex can by constructed similarly to case~(a). This homotopy can be extended to a homotopy on $\widetilde{N}$$\widetilde{N}$.

3 Invariants

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4 Classification/Characterization

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5 Further discussion

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