Timeline for How to write the Thom spectrum representing cobordism as an $\Omega$-spectrum?
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| Jun 4, 2018 at 21:16 | history | edited | j.c. | CC BY-SA 4.0 |
add paper link
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| Apr 26, 2015 at 5:07 | comment | added | user46652 | (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. | |
| Apr 26, 2015 at 5:06 | comment | added | user46652 | @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. | |
| Apr 9, 2015 at 10:43 | answer | added | Dmitri Pavlov | timeline score: 14 | |
| Apr 8, 2015 at 23:00 | comment | added | Anton Fetisov | 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. | |
| Apr 8, 2015 at 22:02 | history | asked | user46652 | CC BY-SA 3.0 |