Reference request for relative bordism coinciding with homology in low dimensions

Question

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 $n<5$ (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.
What is the original reference for the above proof (and the fact itself) that relative bordism and relative homology coincide in low dimensions?

Answer 1

Thom's famous paper "Quelques propriétés globales des variétés différentiable" gives enough information about the bordism ring of a point that one can, if I'm not mistaken, read off statements like this.

For unoriented 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 $[H_\ast(X; \Omega^O_*(pt.))]_n$.

Presumably your question was about oriented 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!).

The case of pairs $(X,A)$ can then be treated e.g. by Mayer-Vietoris.

Answer 2

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.

Look Theorem IV.7.37 of my book "On Thom spectra, orientability, and cobordism", Corrected reprint, Springer, 2008.

There is proved explicitly that the map $E_i(X,A) \to H_i(X,A)$ is an isomorphism for $i<4$ and epimorphism for $i<7$, where $E$ denotes the ORIENTED bordism group.

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.

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.

For non-oriented bordism, $\Omega_n\ne H_n$ even in dimension 2.

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.