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. | ||
+ | |||
+ | In introducing notation and definitions we follow [https://www.mccme.ru/circles/oim/eliminat_talk.pdf slides by A. Skopenkov] | ||
The ''deleted product'' | The ''deleted product'' | ||
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This is the configuration space of ordered pairs of distinct points of $K$. | This is the configuration space of ordered pairs of distinct points of $K$. | ||
− | Suppose that $f:K\to\R^ | + | 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^{ | + | Then the map $\widetilde f:\widetilde K\to S^{m-1}$ is well-defined by the Gauss formula |
$$\widetilde f(x,y)=\frac{f(x)-f(y)}{|f(x)-f(y)|}.$$ | $$\widetilde f(x,y)=\frac{f(x)-f(y)}{|f(x)-f(y)|}.$$ | ||
We have $\widetilde f(y,x)=-\widetilde f(x,y)$, i.e. this map is equivariant with respect to the `exchanging factors' involution | 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^{ | + | $(x,y)\mapsto(y,x)$ on $\widetilde K$ and the antipodal involution on $S^{m-1}$. |
− | Thus the existence of an equivariant map $\widetilde K\to S^{ | + | 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 $\pi_{\mathrm{eq}}^{m}(K) = [K;S^{m}]_{\mathrm{eq}}$ be the set of equivariant maps ̃$K\to S^m$ up to equivariant homotopy. | |
− | + | ||
− | + | <!--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 induced by the Gauss map. | ||
+ | I.e. $\alpha([f]) = [\widetilde f]$. | ||
+ | <!-- 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}} | {{beginthm|Theorem}} | ||
The Haefliger-Wu invariant $\alpha:\mathrm{Emb}^m K\to\pi^{m-1}_{\mathrm{eq}}( \widetilde K)$ is one-to-one for $2m\ge 3n+4$. | 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}} | ||
+ | |||
+ | 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> | </wikitex> | ||
Revision as of 11:44, 9 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.
In introducing notation and definitions we follow slides by A. Skopenkov
The deleted product
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 .
Denote by the set embeddings of into up to isotopy. Let be the set of equivariant maps ̃ up to equivariant homotopy.
The Haefliger-Wu invariant
is induced by the Gauss map.
I.e. .
Theorem 1.1. The Haefliger-Wu invariant is one-to-one for .
See [Skopenkov2006, 5] and [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1] for the DIFF case and [Skopenkov2002, Theorem 1.3] for the PL case.
2 Uniqueness theorems
Lemma 2.1. Assume that is a compact -manifold and either
(a) or
(b) is connected and .
Then every two every two equivariant maps are equivariantly homotopic.
Proof. 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 . QED
3 Invariants
...
4 Classification/Characterization
...
5 Further discussion
...
6 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.