Some calculations involving configuration spaces of distinct points
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== Introduction == | == Introduction == | ||
<wikitex>; | <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. | + | ‘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 | + | In introducing notation and definitions we follow \cite{Skopenkov2020a}. |
The ''deleted product'' | The ''deleted product'' | ||
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Denote by $\mathrm{Emb}^{m}K$ the set embeddings of $K$ into $\mathbb R^{m}$ up to isotopy. | 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 | + | 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 homotopy class of the map $f$. |
<!--Definition of the Haefliger-Wu invariant $\alpha$--> | <!--Definition of the Haefliger-Wu invariant $\alpha$--> | ||
'''The Haefliger-Wu invariant''' | '''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 |
− | is induced by the Gauss map. | + | <!--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.--> | <!-- 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 | + | 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 $2m\ge 3n+4$ or | ||
+ | |||
+ | (b) $K$ is a compact connected $n$-manifold with nonempty boundary and $2m\ge 3n+4$ or | ||
+ | |||
+ | (c) $K$ 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 $2m\ge 3n+1-k$. | ||
{{endthm}} | {{endthm}} | ||
Revision as of 13:27, 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. See [Vassiliev1992].
In introducing notation and definitions we follow [Skopenkov2020a].
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. Denote by the homotopy class of the map .
The Haefliger-Wu invariant
is defined by formula
.
Theorem 1.1. The Haefliger-Wu invariant is one-to-one either
(a) is a compact connected -complex and or
(b) is a compact connected -manifold with nonempty boundary and or
(c) is a compact -manifold with nonempty boundary, is -connected, , , and .
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 each two equivariant maps from to are equivariantly homotopic.
Proof. Given two equivariant maps 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 between and 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 . 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 between and 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.
- [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).