It's a standard fact that, for finite CW complexes, the relative (edit: oriented) bordism group $\Omega_n(X,A)$ coincides with the homology $H_n(X,A;\Omega_0)\simeq [H_\ast(X,A;\Omega_\ast(pt))]_n\simeq H_n(X,A)$ where $\Omega_0\simeq \mathbb{Z}$ is the zero-dimensional bordism group of a point, for $n<5$. <5$ (edit: this should be $n<4$). 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. Surely this must have been known in the 1950's, probably by Thom.
What is the original reference for the above proof (and the fact itself) that relative bordism and relative homology coincide in low dimensions?
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It's a standard fact that, for finite CW complexes, the relative bordism group $\Omega_n(X,A)$ coincides with the homology $\Omega_n(X,A;\Omega_0)\simeq H_n(X,A;\Omega_0)\simeq H_n(X,A)$ where $\Omega_0\simeq \mathbb{Z}$ is the zero-dimensional bordism group of a point, for $n<5$. 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. Surely this must have been known in the 1950's, probably by Thom. |
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