Some calculations involving configuration spaces of distinct points
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‘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. | ‘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. | ||
− | + | The ''deleted product'' $\widetilde K=K^{\underline2}$ | |
+ | is | ||
+ | $\widetilde K:=\{(x,y)\in K\times K\ :\ x\ne y\}.$ | ||
+ | This is the configuration space of ordered pairs of distinct points of $K$. | ||
+ | |||
+ | Suppose that $f:K\to\R^d$ is an embedding of a subset $K\subset \mathbb R^m$. | ||
+ | Then the map $\widetilde f:\widetilde K\to S^{d-1}$ 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 $S^{d-1}$. | ||
+ | Thus the existence of an equivariant map $\widetilde K\to S^{d-1}$ is a necessary condition for the embeddability of $K$ in $\R^d$. | ||
+ | |||
{{beginthm|Definition}}[of the Haefliger-Wu invariant $\alpha$] | {{beginthm|Definition}}[of the Haefliger-Wu invariant $\alpha$] | ||
\label{DefHaef} | \label{DefHaef} | ||
− | The Haefliger-Wu invariant | + | '''The Haefliger-Wu invariant''' |
− | $\alpha:\mathrm{Emb}^{k} | + | $\alpha:\mathrm{Emb}^{k}K\to \pi_{\mathrm{eq}}^{k-1}(\widetilde{K})$ |
is induced by the Gauss map, also denoted by $\alpha$. | is induced by the Gauss map, also denoted by $\alpha$. | ||
− | + | <!-- The Haefliger-Wu invariant and the Gauss map are analogously defined for $N_0$; we will denote them by $\alpha_0$ in this case.--> | |
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | <!-- 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}} | {{beginthm|Theorem}} | ||
− | The Haefliger-Wu invariant $\alpha:\mathrm{Emb}^m | + | The Haefliger-Wu invariant $\alpha:\mathrm{Emb}^m K\to\pi^{m-1}_{\mathrm{eq}}( \widetilde K)$ is one-to-one for $2m\ge 3n+4$. |
{{endthm}} | {{endthm}} | ||
</wikitex> | </wikitex> |
Revision as of 14:31, 5 April 2020
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
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.
The deleted product is This is the configuration space of ordered pairs of distinct points of .
Suppose that is an embedding of a subset . Then the map is well-defined by the Gauss formula
We have , i.e. this map is equivariant with respect to the `exchanging factors' involution on and the antipodal involution on . Thus the existence of an equivariant map is a necessary condition for the embeddability of in .
Definition 1.1.[of the Haefliger-Wu invariant ]
The Haefliger-Wu invariant is induced by the Gauss map, also denoted by .
Theorem 1.2. The Haefliger-Wu invariant is one-to-one for .
2 Uniqueness theorems
Theorem 2.1. Assume that is a compact -manifold and either
(a) or
(b) is connected and .
Then any two embeddings of into are isotopic.
In cases (a-d) inequality is clearly satisfied. Hence by Haefliger-Weber theorem \ref{thm_Haef-Web} we have . So it is sufficient to show that is trivial, i.e. every two every two equivariant maps are equivariantly homotopic.
Take an arbitrary equivariant triangulation of .
(a) One can easily construct an equivariant homotopy between restrictions of and on vertices of . By general position a homotopy of on the boundary of a -simplex can be extended to a homotopy on the whole -simplex since . We extend equivariant homotopy on symmetric simplices in the symmetric way, so we obtain an equivariant homotopy.
(b) Since has non-empty boundary, there exists an equivariant deformational retraction of to an equivariant -subcomplex of . A homotopy on the subcomplex can by constructed similarly to case~(a). This homotopy can be extended to a homotopy on .
3 Invariants
...
4 Classification/Characterization
...
5 Further discussion
...