Cohomology ring of symmetric products (of manifolds)

Question

Let $S_g$ be a closed, orientable surface of genus $g$ (new notation in light of the first comment). I am looking for results to determine explicitly the (co)homology groups and/or cohomology ring structure (in integer or rational coefficients) of the $n$-fold symmetric product of $S_g$, denoted by $SP^n\left(S_g\right)$.

In this direction, I’m aware of a theorem of IG MacDonald for algebraic curves, but I don’t know a reference to a useful, general result for $S_g$. More generally, is there a good reference to computing the space structure of $SP^n(M)$ or the ring structure of cohomology ring $H^*\left(SP^n(M)\right)$ for $k-$dimensional closed, orientable topological manifolds $M$? In particular, I’m interested in the case when $M$ is a finite product of spheres.

I know that DV Gugnin has a result on the functoriality of the homology of symmetric products with integer coefficients but the result is purely theoretical and not helpful for my purpose of determining explicitly the ring structure of $H^*\left(SP^n(M)\right)$. I also know that Ozsvath-Szabo showed that $H_{1}\left(S_g \right) \approx H_{1} \left(SP^g\left(S_g\right) \right)$, but I’m looking for more information about either the space $SP^n(M)$ or the (co)homology of $SP^n(M)$ for arbitrary $n$ so that I can find explicitly the cohomology ring $H^*\left(SP^n(M)\right)$.

Any help or references would be appreciated.

Answer 1

Not really an answer, more an overly-long comment: A famous result of Segal, McDuff and others going back to the 70s is that the "configuration space" of $n$ unordered points in a manifold stabilises, in the sense that the rational homology $H_i(C_n(M),\mathbb Q) \cong H_i(C_{n+1}(M,\mathbb Q)$ once $n>i$. A more recent result of this nature is Church's theorem which refines this result by studying the ordered configuration space and defines a notion of "representation stability".

These stability results mean that the study of the homology of configuration spaces has two different aspects -- the study of stable and unstable phenomena. Of course this doesn't compute anything for you, but it is probably useful to keep in mind!

Update

As Qiaochu Yuan rightly points out, the "configuration space" is usually the space of $n$ distinct points, which an open submanifold of the symmetric product, and not the entire symmetric product. (That was part of why this was originally only a comment...) On the other hand, papers studying configuration spaces do consider collisions -- for example McDuff's '75 Topology paper, she studies "signed" points, where collisions between positive and negative points are allowed. Moreover the configuration space literature studies other versions of "marked" configuration spaces, which of course connects to the natural stratification of the symmetric product. Thus while configuration spaces and the symmetric products are different, they are related in many ways, and if you are trying to understand the cohomology of symmetric products, it is probably worth knowing that there is a lot known about the homology and cohomology of configuration spaces!

Answer 2

Let $\Sigma_g$ be an a (smooth) complex algebraic curve of genus $g$ ($\Sigma_g$ homeomorphic to your $S_g$ so same cohomology ring etc.). Then $\Sigma_g^{(n)}=\Sigma_g^n/\mathfrak{S}_n$ is well-known to be a (smooth) complex algebraic variety for all positive $n$.

Theorem (IG Macdonald) Let $a_i,b_i\in H^1(\Sigma_g;\mathbb{Z})$ be the one-dimensional generators, $1\leq i\leq g$, and $e\in H^2(\Sigma_g;\mathbb{Z})$ be the orientation class. Then the ring $H^*(\Sigma_g^{(n)};\mathbb{Z})$ is generated by $a^*_i, b^*_i,e^*$ subject to the following relations:

• Any two degree 1 generators i.e., any two from the $2g$ ($a^*_i$s and the $b^*_j$s) anti-commute.
• The degree 2 generator $e^*$ commutes with everything.
• If $i_1,\dots,i_m, j_1,\dots,j_n, k_1,\dots,k_p$ are distinct integers from $1$ to $g$ inclusive, then $$ a^*_{i_1}a^*_{i_2}\dots a^*_{i_m} b^*_{j_1}\dots b^*_{j_n} (a^*_{k_1}b^*_{k_1}-e^*)\dots (a^*_{k_p}b^*_{k_p}-e^*) (e^*)^q = 0 $$ provided $m+n+2p+q=n+1$.

In other words, $H^*(\Sigma_g^{(n)})$ is $\Lambda^*_{\mathbb{Z}}[a^*_1,\dots,a^*_g,b^*_1,\dots,b^*_g]\otimes \mathbb{Z}[e^*]$ quotient out by the above.