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 AtiyahHirzebruch 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?



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 AtiyahHirzebruch 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 MayerVietoris. 


Yuli Rudyak gave a perfect answer to this question in May, but it was by email. 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. I subsequently bought Rudyak's book. The Amazon page had a mixup 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. 

