10
$\begingroup$

It is often said [e.g. Atiyah, "Bordism and Cobordism" (1961)] that the Thom spectrum $MSO(i)$ represents oriented cobordism, in the following sense: \begin{eqnarray} MSO^n(X,Y) &:=& \lim_{i \rightarrow \infty} \langle \Sigma^{i-n}(X/Y), MSO(i) \rangle\\ &=& \lim_{i \rightarrow \infty} \langle X/Y, \Omega^{i-n} MSO(i) \rangle\\ &=& \langle X/Y, \Omega^{i-n} MSO(i) \rangle, ~~\text{large}~i, \end{eqnarray} for finite CW pairs $(X,Y)$. where $\Sigma$ is the reduced suspension, $\Omega$ is the usual loop space functor, and $\langle-, -\rangle$ is the homotopy classes of pointed maps. The direct limit was taken with respect to the maps \begin{equation} \langle \Sigma^{i-n}(X/Y), MSO(i) \rangle \rightarrow \langle \Sigma^{i+1-n}(X/Y), \Sigma MSO(i) \rangle \xrightarrow{f_{i*}} \langle \Sigma^{i+1-n}(X/Y), MSO(i+1) \rangle. ~ (1) \end{equation} where $f_{i}:\Sigma MSO(i) \rightarrow MSO(i+1)$ is the natural map mentioned in Atiyah.

By the Brown representability theorem, one should be able to represent oriented cobordism in the usual sense that \begin{equation} MSO^n(X,Y) \stackrel{?}{\cong} \langle X/Y, K_n \rangle ~~~ (2) \end{equation} for some $\Omega$-spectrum $\{K_n\}$. So this is something like moving the direct limit inside $\langle -,- \rangle$.

My question is: If $K_n$ exists, then what is it? Or is it because the Brown representability theorem hypothesized a generalized cohomology theory on all CW pairs, that there isn't an $\Omega$-spectrum $\{K_n\}$ representing oriented cobordism, which is defined only for finite CW pairs?

I was able to show that (1) is actually the same, via adjunction, as the maps \begin{equation} \langle X/Y, \Omega^{i-n} MSO(i) \rangle \rightarrow \langle X/Y, \Omega^{i+1-n}\Sigma MSO(i) \rangle \rightarrow \langle X/Y, \Omega^{i+1-n} MSO(i+1) \rangle ~~~ (3) \end{equation} induced by \begin{equation} MSO(i) \xrightarrow{\eta_{MSO(i)}} \Omega \Sigma MSO(i) \xrightarrow{\Omega(f_i)} \Omega MSO(i+1), ~~~(4) \end{equation} where $\eta_Y:Y \rightarrow \Omega \Sigma Y$ is the unit of the adjunction $\Sigma \dashv \Omega$. Can we go from here to contruct $K_n$ out of $MSO(i)$?

Sorry for this potentially elementary question.

$\endgroup$
  • 2
    $\begingroup$ You pretty much wrote the answer. $K_n = \mathrm{colim}\, \Omega^{i-n}MSO(i)$. If $X/Y$ is a finite CW-space, then it is a compact object in the category of spaces, and its stabilization $\Sigma^{\infty}(X/Y)$ is a compact object in the category of spectra. $D$ is comact if $Hom(D,\cdot)$ preserves filtered colimits, in particular colimits of towers. If $X/Y$ is infinite then the formula you wrote doesn't define a cohomology theory and the only correct way to define it is via the representing object given above. $\endgroup$ – Anton Fetisov Apr 8 '15 at 23:00
  • $\begingroup$ @Anton Fetisov Thanks a lot! I'm yet to understand your comment, but I figured we could give a pedestrian proof as follows, which is probably a translation of yours. Observe that the image of $f:X/Y\rightarrow \lim_{i\rightarrow \infty}\Omega^{i-n}MSO(i)$ is compact (recall $X$, $Y$ are finite CW complexes), and that the $\Omega^{i-n}MSO(i)$'s are Hausdorff. $\endgroup$ – user46652 Apr 26 '15 at 5:06
  • $\begingroup$ (cont'd) From the fact that a compact subspace of the colimit of a sequence of Hausdorff spaces is contained in one of the spaces, the surjectivity of the obvious map \begin{equation} \lim_{i\rightarrow \infty}\langle X/Y, \Omega^{i-n} MSO(i) \rangle \rightarrow \langle X/Y, \lim_{i\rightarrow \infty} \Omega^{i-n} MSO(i) \rangle \end{equation} follows. The same argument but for a homotopy $f_t:X/Y\rightarrow \lim_{i\rightarrow \infty}\Omega^{i-n}MSO(i)$ shows the injectivity. $\endgroup$ – user46652 Apr 26 '15 at 5:07
14
$\begingroup$

A concrete model for Ω^∞ applied to Thom spectra (which is all what we need because Thom spectra are connective) was given by Quinn in his thesis.

Very roughly, Ω^∞MG is a simplicial set whose n-simplices are n-dimensional G-manifolds with corners, with corners of codimension i assigned to simplicial faces of Δ^n of codimension i.

For precise definitions and statements see Quinn's paper “Assembly maps in bordism-type theories”, especially Sections 3 and 6.

Laures and McClure explain how to promote these simplicial sets to strictly commutative symmetric ring spectra. See their papers “Multiplicative properties of Quinn spectra” and “Commutativity properties of Quinn spectra”.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.