Some calculations involving configuration spaces of distinct points
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== Construction and examples == | == Construction and examples == | ||
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− | ... | + | For a manifold $X$, $\widetilde X$ denotes ''the deleted product'' of $X$, i.e. $X^2$ minus an open tubular neighborhood of the diagonal. It is a manifold with boundary and has the standard free involution. |
+ | {{theorem|Definition}}[of the Haefliger-Wu invariant $\alpha$] | ||
+ | \label{DefHaef} | ||
+ | The Haefliger-Wu invariant | ||
+ | $\alpha:\mathrm{Emb}^{k}N\to \pi_{\mathrm{eq}}^{k-1}(\widetilde{N})$ | ||
+ | is induced by the Gauss map, also denoted by $\alpha$. | ||
+ | The Gauss map assigns to an individual embedding $f:N\to\R^{k}$ | ||
+ | an equivariant map $\widetilde{N}\to S^{k-1}$ | ||
+ | defined by the formula | ||
+ | $$ | ||
+ | (x,y)\mapsto | ||
+ | \frac{f(x)-f(y)} | ||
+ | {\|f(x)-f(y)\|}, | ||
+ | \quad | ||
+ | (x,y)\in\widetilde{N}\subset N\times N. | ||
+ | $$ | ||
+ | The Haefliger-Wu invariant and the Gauss map are analogously defined for $N_0$; we will denote them by $\alpha_0$ in this case. | ||
+ | {{definition}} | ||
</wikitex> | </wikitex> | ||
Revision as of 13:54, 2 April 2020
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
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2 Construction and examples
For a manifold , denotes the deleted product of , i.e. minus an open tubular neighborhood of the diagonal. It is a manifold with boundary and has the standard free involution. Template:Theorem[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
The Haefliger-Wu invariant and the Gauss map are analogously defined for ; we will denote them by in this case. Template:Definition
3 Invariants
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4 Classification/Characterization
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5 Further discussion
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