-5
$\begingroup$

Galatius and Randal-Williams proved the following generalized Mumford conjecture in their joint paper, "Stable Moduli spaces of High Dimensional Manifolds". For each characteristic class of oriented $2n$-dimensional vector bundles $c\in H^{2n+k}(BSO(2n))$, one can define the associated generalized Mumford–Morita–Miller class of a smooth fiber bundle $\pi: E\to B$ with oriented $2n$-dimensional fibers as $\kappa_c(E)=\pi_!(c(T_vE))\in H^k(B),$ where $T_v(E) = Ker(D\pi)$ is the fiberwise tangent bundle of $\pi$. When the fiber is taken to be $W_g=\#^g S^n\times S^n$, the connected sum of $g$ copies of $S^n\times S^n,$ there is a corresponding universal class $\kappa_c\in H^k(BDiff(W_g, D^{2n}))$ which for $k>0$ is compatible with increasing $g$.

Theorem (Galatius-R-W) Let $2n>4$ and let $\mathcal{B}\subset H^*(BSO(2n);\mathbb{Q})$ be the set of monomials in the classes $e, p_{n−1}, p_{n−2},...,p_{\left\lceil\frac{n+1}{4}\right\rceil}$ of total degree greater than $2n$. Then the natural map $$\mathbb{Q}[\kappa_c|c\in\mathcal{B}]=\lim_{\longleftarrow g}H^*(BDiff(W_g,D^{2n});\mathbb{Q})$$ is an isomorphism.

I would like to determine the stable cohomology $$\lim_{\longleftarrow g}H^*(BDiff(W_{g},D^{\infty});\mathbb{Q}).$$ Under the the isomorphism of G-R-W, this reduces to determining the MMM classes $\kappa_c$ and the classes $e,p_{\infty}$. How do I determine such classes? Moreover, is letting $n\to\infty$ in this construction permissible?

Thanks!

Improve this question
$\endgroup$
10
  • 1
    $\begingroup$ Right, but $g$ doesn't appear in its expression. Did you mean $\varprojlim_g H^*(\mathit{BDiff}(W_g, D^{2n}); \mathbb Q)$ or $\varprojlim_g H^*(\mathit{BDiff}(W_\infty, D^\infty); \mathbb Q)$? $\endgroup$
    – Arun Debray
    Commented Aug 4, 2018 at 22:10
  • 1
    $\begingroup$ Oh, I see. Sorry, that is my mistake; I meant the former with $n=\infty$. @ArunDebray $\endgroup$
    – Sergio Charles
    Commented Aug 4, 2018 at 22:25
  • 1
    $\begingroup$ If you mean to take the limit as $n$ goes to $\infty$ you need a way to map diffeomorphisms of the $n$-dimensional manifolds $W^n_g$ to diffeomorphisms of the $(n+1)$-dimensional manifolds $W_g^{n+1}$. It seems unlikely that any interesting map like this could exist. What map do you have in mind? $\endgroup$
    – Allen Hatcher
    Commented Aug 5, 2018 at 0:32
  • 2
    $\begingroup$ In Definition 1.1 of that paper the dimension of the manifold $W_g$ (which is $2n$, not $n$ as I mistakenly said in my previous comment) is fixed and one is considering embedded submanifolds of ${\mathbb R}^N$ diffeomorphic to $W_g$. Then one lets $N$ go to infinity via the natural inclusion of ${\mathbb R}^N$ in ${\mathbb R}^{N+1}$, so submanifolds of ${\mathbb R}^N$ are regarded as submanifolds of ${\mathbb R}^{N+1}$. However the dimension $2n$ of the submanifolds is not changing. There is no natural way to let $n$ go to infinity in this construction. $\endgroup$
    – Allen Hatcher
    Commented Aug 5, 2018 at 1:50
  • 2
    $\begingroup$ I don't understand what you mean by "compute the MMM classes". The definition of $\kappa_c$ is given on page 2 of the paper you link to, and the statement of the theorem is that those classes generate and are algebraically independent, as $c$ ranges over the set $\mathcal{B}$. I don't think there's more to say. $\endgroup$
    – user80296
    Commented Aug 6, 2018 at 3:23

1 Answer 1

Reset to default
0
$\begingroup$

As suggested by Dr. Hatcher, the dimension of the manifold $W_g$, $2n$, is fixed and one considers embedded submanifolds of $\mathbb{R}^N$ diffeomorphic to $W_g$. Then one lets $N$ go to infinity via the natural inclusion of $\mathbb{R}^N$ in $\mathbb{R}^{n+1}$, so submanifolds of $\mathbb{R}^N$ are regarded as submanifolds of $\mathbb{R}^{N+1}$. However the dimension $2n$ of the submanifolds is not changing. Thus, there is no natural way to let $n$ go to infinity in this construction.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .