Relation between cohomology of ordered and unordered configuration spaces? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T23:41:30Z http://mathoverflow.net/feeds/question/60794 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60794/relation-between-cohomology-of-ordered-and-unordered-configuration-spaces Relation between cohomology of ordered and unordered configuration spaces? George 2011-04-06T10:34:26Z 2011-07-12T06:11:23Z <p>For any manifold $M$, the unordered configuration space of $k$ points is obtained as a quotient of ordered configuration space of $k$ points by the group action of symmetric group on $k$ letters. Does it induce some relation between the cohomology algebras of the two spaces?</p> http://mathoverflow.net/questions/60794/relation-between-cohomology-of-ordered-and-unordered-configuration-spaces/60799#60799 Answer by Oscar Randal-Williams for Relation between cohomology of ordered and unordered configuration spaces? Oscar Randal-Williams 2011-04-06T10:58:20Z 2011-04-06T10:58:20Z <p>If $F_n(M)$ denotes the ordered configuration space and $C_n(M)$ the unordered configuration space, the quotient map gives a map $$H^*(C_n(M)) \longrightarrow H^*(F_n(M)).$$ If one takes rational coefficients, then this induces an isomorphism $$H^*(C_n(M);\mathbb{Q}) \longrightarrow H^*(F_n(M);\mathbb{Q})^{\Sigma_n}$$ onto the symmetric group invariant subspace.</p> <p>In positive characteristic I don't think there is any pleasant relation between them.</p> http://mathoverflow.net/questions/60794/relation-between-cohomology-of-ordered-and-unordered-configuration-spaces/60811#60811 Answer by Giacomo d'Antonio for Relation between cohomology of ordered and unordered configuration spaces? Giacomo d'Antonio 2011-04-06T12:41:08Z 2011-04-06T12:41:08Z <p>This is an expansion to Oscar's answer. The cited result on the cohomology with rational coefficients is an application of the Transfer Theorem:</p> <blockquote> <p>Let $G$ be a finite group, $X$ a topological manifold and $F$ a field with $\mbox{char}\,F = 0$ or $\mbox{char}\,F \nmid o(G)$, then $$H^*(X/G; F) \cong H^*(X;F)^G$$</p> </blockquote> <p>There is a nice proof of this theorem on Bredon's book "Introduction to compact transformation groups."</p> <p>For the case of $\mathbb R^k$ you can compute the cohomology (with complex coefficients) explicitly. The answer depends on the parity of $k$. For odd $k$, $H^*(F_n(\mathbb{R}^k); \mathbb{C})$ is one dimensional in degree $0$ and trivial otherwise, for even $k$ it is one-dimensional in degrees $0$ and $k-1$ and trivial otherwise.</p> http://mathoverflow.net/questions/60794/relation-between-cohomology-of-ordered-and-unordered-configuration-spaces/60830#60830 Answer by Tom Church for Relation between cohomology of ordered and unordered configuration spaces? Tom Church 2011-04-06T16:03:55Z 2011-04-06T16:03:55Z <p>Oscar Randal-Williams already mentioned the transfer isomorphism $H^*(C_n(M);\mathbb{Q}) \approx H^*(F_n(M);\mathbb{Q})^{\Sigma_n}$. At the risk of self-promotion, one place you can see this transfer in action is in my paper "Homological stability for configuration spaces of manifolds", <a href="http://arxiv.org/abs/1103.2441" rel="nofollow">arXiv:1103.2441</a>. In that paper I used an analysis of $H^*(F_n(M);\mathbb{Q})$ and the action of $\Sigma_n$ on it to conclude that the cohomology of the unordered configuration space $C_n(M)$ is eventually independent of the number of points $n$: $$H^k(C_n(M);\mathbb{Q})\approx H^k(C_{n+1}(M);\mathbb{Q})\text{ for }n\gg k.$$ The key idea is that we can relate $F_{n+1}(M)$ with $F_n(M)$, even when we cannot relate $C_{n+1}(M)$ with $C_n(M)$ directly, and then the transfer map lets us push information from $F_n(M)$ down to $C_n(M)$. The basic framework of this approach is explained in the introduction. There are also some explicit computations (some going back to Bödigheimer&ndash;Cohen&ndash;Taylor) of $H^*(C_n(M);\mathbb{Q})$ for various manifolds $M$ in Section 4.2 that might be of interest to you.</p> <p>Also, you should check out the papers on the cohomology of configuration spaces written by the other participants in this discussion, Oscar Randal-Williams and Giacomo d'Antonio, which should provide a somewhat different perspective.</p>