Some calculations involving configuration spaces of distinct points
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− | == | + | == Uniqueness theorems == |
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+ | {{beginthm|Theorem}}\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 any two embeddings of $N$ into $\R^m$ are isotopic. | ||
+ | {{endthm}} | ||
+ | |||
+ | In cases (a-d) inequality $2m\ge3n+4$ is clearly satisfied. | ||
+ | Hence by Haefliger-Weber theorem \ref{thm_Haef-Web} we have $\mathrm{Emb}_m(N) \cong \pi^{m-1}_{\mathrm{eq}}(\widetilde N)$. | ||
+ | So it is sufficient to show that $\pi^{m-1}\_{mathrm{eq}}(\widetilde N)$ is trivial, i.e. every two every two equivariant maps $f, g:\widetilde N\to S^{m-1}$ are equivariantly homotopic. | ||
+ | |||
+ | Take an arbitrary equivariant triangulation $T$ of $\widetilde N$. | ||
+ | |||
+ | (a) One can easily construct an equivariant homotopy between restrictions of $f$ and $g$ | ||
+ | on vertices of $T$. | ||
+ | By general position a homotopy of $f, g$ on the boundary of a $k$-simplex can be extended to a homotopy on the whole $k$-simplex since $k<2n+1$. | ||
+ | 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 on the subcomplex can by constructed similarly to case~(a). This homotopy can be extended to a homotopy on $\widetilde{N}$. | ||
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Revision as of 13:13, 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.
For a subset , denotes the deleted product of , i.e. minus the diagonal. This is the configuration space of ordered pairs of distinct points of K.
Definition 1.1.[of the Haefliger-Wu invariant ]
The Haefliger-Wu invariant is induced by the Gauss map, also denoted by . The Gauss map assigns to an individual embedding an equivariant map defined by the formula
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
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