# [ALGTOP-L] Submission

Daniel Moskovich dmoskovich at gmail.com
Sat May 8 21:53:58 EDT 2010

--e0cb4e887e21d06c8704861f90c1
Content-Type: text/plain; charset=ISO-8859-1

I would like to ask the following question to this list, also asked by
myself <a href="
http://mathoverflow.net/questions/23440/reference-request-for-relative-bordism-coinciding-with-homology-in-low-dimensions">on
MathOverflow</a>.

It's a standard fact that, for finite CW complexes, the relative
(oriented) bordism group  $\Omega_n^O(X,A)$ is isomorphic to $[H_\ast(X,A; \Omega^O_*(pt.))]_n$, which is isomorphic to $H_n(X,A)$ for $n<4$. One can
prove this using the Atiyah-Hirzebruch spectral sequence, and it can be
deduced from results of Thom, but all papers I've seen seem to just
state this isomorphism 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?

--e0cb4e887e21d06c8704861f90c1
Content-Type: text/html; charset=ISO-8859-1
Content-Transfer-Encoding: quoted-printable

<div>I would like to ask the following question to this list, also asked by=
myself &lt;a href=3D&quot;<a href=3D"http://mathoverflow.net/questions/234=
40/reference-request-for-relative-bordism-coinciding-with-homology-in-low-d=
imensions" target=3D"_blank">http://mathoverflow.net/questions/23440/refere=
nce-request-for-relative-bordism-coinciding-with-homology-in-low-dimensions=
</a>&quot;&gt;on MathOverflow&lt;/a&gt;.</div>

<div>=A0</div>
<div>It&#39;s a standard fact that, for finite CW complexes, the relative (=
oriented)=A0bordism group=A0 $<span alt=3D"\Omega_n^O(X)=3D\pi_{n+k}(MO(k)\= wedge X)"><font face=3D"Times New Roman">\Omega_n^O(X,A)</font></span>$ is =
isomorphic to $<span alt=3D"[H_\ast(X; \Omega^O_*(pt.))]_n"><font face=3D"T= imes New Roman">[H_\ast(X,A; \Omega^O_*(pt.))]_n$, which is isomorphic to $= H_n(X,A)$ for $n&lt;4$</font></span>.=A0One can prove this using the Atiyah=
-Hirzebruch spectral sequence, and it can be deduced from results of Thom,=
=A0but all papers I&#39;ve seen seem to just state=A0this isomorphism=A0as =
a fact without citation. I really want to find the original reference for t=
he above isomorphism, but have wasted much time and found nothing.</div>

<div>=A0</div>
<div>What is the original reference for the above proof (and the fact itsel=
f) that relative bordism and relative homology coincide in low dimensions?<=
/div>
<div>=A0</div>
<div>=A0</div>

--e0cb4e887e21d06c8704861f90c1--