Some calculations involving configuration spaces of distinct points
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== Uniqueness theorems == | == Uniqueness theorems == | ||
<wikitex>; | <wikitex>; | ||
− | {{beginthm| | + | {{beginthm|Lemma}}\label{th::unknotting} |
Assume that $N$ is a compact $n$-manifold and either | Assume that $N$ is a compact $n$-manifold and either | ||
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(b) $N$ is connected and $m \ge 2n+1 \ge 5$. | (b) $N$ is connected and $m \ge 2n+1 \ge 5$. | ||
− | Then | + | Then every two every two equivariant maps $f, g:\widetilde N\to S^{m-1}$ are equivariantly homotopic. |
{{endthm}} | {{endthm}} | ||
− | In cases (a- | + | ''Proof.'' |
− | + | In cases (a-b) inequality $2m\ge3n+4$ is clearly satisfied. | |
− | + | ||
Take an arbitrary equivariant triangulation $T$ of $\widetilde N$. | Take an arbitrary equivariant triangulation $T$ of $\widetilde N$. | ||
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(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 $\widetilde{N}$. | (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 $\widetilde{N}$. | ||
+ | QED | ||
</wikitex> | </wikitex> | ||
Revision as of 14:36, 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
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. In cases (a-b) inequality is clearly satisfied.
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
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4 Classification/Characterization
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5 Further discussion
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