Some calculations involving configuration spaces of distinct points
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+ | {{Stub}} | ||
+ | == Introduction == | ||
+ | <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. See \cite{Vassiliev1992}. | ||
− | + | In introducing notation and definitions we follow \cite{Skopenkov2020a}. | |
− | + | If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category. | |
− | + | The ''deleted product'' | |
− | {{ | + | $$\widetilde K=K^{\underline2}:=\{(x,y)\in K\times K\ :\ x\ne y\}.$$ |
+ | This is the configuration space of ordered pairs of distinct points of $K$. | ||
− | - | + | Suppose that $f:K\to\R^m$ is an embedding of a subset $K\subset \mathbb R^N$. |
+ | Then the map $\widetilde f:\widetilde K\to S^{m-1}$ 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 $S^{m-1}$. | ||
+ | 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 $\pi_{\mathrm{eq}}^{m}(\widetilde K) = [\widetilde K;S^{m}]_{\mathrm{eq}}$ be the set of equivariant maps $\widetilde K \to S^m$ up to equivariant homotopy. By $[·]$ we denote the isotopy class of an embedding or the homotopy class of a map. |
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− | + | <!--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 defined by formula | |
− | + | <!--is induced by the Gauss map.--> | |
− | + | $\alpha([f]) = [\widetilde f]$. | |
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<!-- 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 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--> | ||
+ | (b) $K$ 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 $2m\ge 3n+2-k$. | ||
+ | {{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> | ||
+ | |||
+ | == Uniqueness theorems == | ||
+ | <wikitex>; | ||
+ | {{beginthm|Lemma}}\label{th::unknotting} | ||
+ | Assume that $N$ is a compact $n$-manifold and either | ||
+ | |||
+ | (a) $m \ge 2n+2$ or | ||
+ | |||
+ | (b) $N$ is connected and $m \ge 2n+1 \ge 5$. | ||
+ | |||
+ | Then each two equivariant maps from $\widetilde N$ to $S^{m-1}$ are equivariantly homotopic. | ||
+ | {{endthm}} | ||
+ | |||
+ | Hereafter denote by $\widetilde K$ the product $K\times K$ minus tubular neighborhood of the diagonal. | ||
+ | |||
+ | ''Proof.'' | ||
+ | Given two equivariant maps $\phi, \psi\colon\widetilde N \to S^{m-1}$ take an arbitrary equivariant triangulation $T$ of $\widetilde N$. | ||
+ | |||
+ | (a) One can easily construct an equivariant homotopy between restrictions of $\phi$ and $\psi$ | ||
+ | on vertices of $T$. | ||
+ | By general position a homotopy between $\phi$ and $\psi$ 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 $2n+1$. | ||
+ | We extend equivariant homotopy on symmetric simplices in the symmetric way, so we obtain an equivariant homotopy. | ||
+ | |||
+ | (b) Since $\widetilde{N}$ has non-empty boundary, there exists an equivariant deformational retraction of $\widetilde{N}$ to an equivariant $(2n-1)$-subcomplex of $T$. A homotopy between $\phi$ and $\psi$ on the subcomplex can by constructed similarly to case (a). This homotopy can be extended to a homotopy on $\widetilde{N}$. | ||
+ | QED | ||
+ | |||
+ | {{beginthm|Lemma}}\label{th::unknotting} | ||
+ | Assume that $N$ is a closed $k$-connected $n$-manifold and $m-1 \ge 2n-k$. | ||
+ | |||
+ | Then each two equivariant maps from $\widetilde N$ to $S^{m-1}$ are equivariantly homotopic. | ||
{{endthm}} | {{endthm}} | ||
</wikitex> | </wikitex> | ||
− | == Invariants == | + | <!--== Invariants == |
<wikitex>; | <wikitex>; | ||
... | ... | ||
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... | ... | ||
</wikitex> | </wikitex> | ||
− | + | == Acknowledgments == | |
... | ... | ||
Latest revision as of 14:42, 8 January 2021
This page has not been refereed. The information given here might be incomplete or provisional. |
[edit] 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].
If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.
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. By we denote the isotopy class of an embedding or the homotopy class of a 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 -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.
[edit] 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.
Hereafter denote by the product minus tubular neighborhood of the diagonal.
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
Lemma 2.2. Assume that is a closed -connected -manifold and .
Then each two equivariant maps from to are equivariantly homotopic.
[edit] 3 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).