Reference request for relative bordism coinciding with homology in low dimensions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T03:13:34Z http://mathoverflow.net/feeds/question/23440 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23440/reference-request-for-relative-bordism-coinciding-with-homology-in-low-dimensions Reference request for relative bordism coinciding with homology in low dimensions Daniel Moskovich 2010-05-04T15:07:03Z 2010-12-08T12:39:41Z <p>It's a standard fact that, for finite CW complexes, the relative (edit: oriented) bordism group $\Omega_n(X,A)$ coincides with the homology $[H_\ast(X,A;\Omega_\ast(pt))]_n\simeq H_n(X,A)$ for $n&lt;5$ (edit: this should be $n&lt;4$). One can prove this using the Atiyah-Hirzebruch spectral sequence, and all papers I've seen seem to just state it as a fact without citation. I really want to find the original reference for the above isomorphism, but have wasted much time and found nothing. <br> What is the original reference for the above proof (and the fact itself) that relative bordism and relative homology coincide in low dimensions?</p> http://mathoverflow.net/questions/23440/reference-request-for-relative-bordism-coinciding-with-homology-in-low-dimensions/23592#23592 Answer by Tim Perutz for Reference request for relative bordism coinciding with homology in low dimensions Tim Perutz 2010-05-05T14:24:02Z 2010-05-05T21:54:50Z <p>Thom's famous paper "Quelques propri&eacute;t&eacute;s globales des vari&eacute;t&eacute;s diff&eacute;rentiable" gives enough information about the bordism ring of a point that one can, if I'm not mistaken, read off statements like this.</p> <p>For <i>unoriented</i> bordism, he proves (Thm. II.10) that the classifying space $MO(k)$ has the $2k$-type of a product of mod 2 Eilenberg MacLane spaces. Hence the bordism group $\Omega_n^O(X)=\pi_{n+k}(MO(k)\wedge X)$ ($k \gg 0$) is isomorphic to $[H_\ast(X; \Omega^O_*(pt.))]_n$.</p> <p>Presumably your question was about <i>oriented</i> bordism? In section 8 of his paper, Thom constructs the first few steps in a Postnikov tower for $MSO(k)$. But all that's relevant here is that $\Omega^{SO}_n(pt.)$ is $\mathbb{Z}$, $0$, $0$, $0$, $\mathbb{Z}$ for $n=0,1,2,3,4$, the isomorphism with $\mathbb{Z}$ in degree 4 being the signature. From the Atiyah-Hirzebruch spectral sequence it's then clear that $\Omega_n^{SO}(X) \cong H_n(X;\mathbb{Z})$ for $n=0, 1,2,3$. But $\Omega_4^{SO}(X)$ has an additional $\Omega_4^{SO}(pt.)=\mathbb{Z}$ summand which survives the spectral sequence, because it's the signature of the source manifold (a bordism invariant!).</p> <p>The case of pairs $(X,A)$ can then be treated e.g. by Mayer-Vietoris.</p> http://mathoverflow.net/questions/23440/reference-request-for-relative-bordism-coinciding-with-homology-in-low-dimensions/48635#48635 Answer by Daniel Moskovich for Reference request for relative bordism coinciding with homology in low dimensions Daniel Moskovich 2010-12-08T12:39:41Z 2010-12-08T12:39:41Z <p>Yuli Rudyak gave a perfect answer to this question in May, but it was by e-mail. Because his might be of interest to others, I reproduce it in full.</p> <p><blockquote> Look Theorem IV.7.37 of my book "On Thom spectra, orientability, and cobordism", Corrected reprint, Springer, 2008.</p> <p>There is proved explicitly that the map $E_i(X,A) \to H_i(X,A)$ is an isomorphism for $i&lt;4$ and epimorphism for $i&lt;7$, where $E$ denotes the ORIENTED bordism group.</p> <p>This is important to cite 2008 Corrected reprint: In previous 1998 edition I did not make an explicit claim (although is follows easily from what has been done), and many people asked about explicit citation. I included the reference in corrected reprint.</p> <p>By the way, you mention that $\Omega_n^O(X,A)$ is isomorphic to $[H_\ast(X,A; \Omega^O_*(pt.))]_n$. This is correct for NON-oriented bordism, but it is wrong for oriented one.</p> <p>For non-oriented bordism, $\Omega_n\ne H_n$ even in dimension 2. </blockquote><br></p> <p>I subsequently bought Rudyak's book. The Amazon page had a mix-up and they kept sending me the first edition, but it was resolved eventually by getting Springer to intervene. I hope it's sorted out- otherwise, I recommend ordering the book directly from Springer.</p>