[ALGTOP-L] ALGTOP-L Digest, Vol 411, Issue 1

Jim Davis jfdavis at indiana.edu
Sun May 9 16:26:47 EDT 2010


Another discussion of the issue is in section 15 of the book Differential Periodic Maps, by Conner and Floyd published in 1964.  They discuss the issue of Steenrod representability (whether a class in H_n(X,A) can be represented by the image of a compact oriented manifold pair (M,\partial M) \to (X,A), and state that if n=7, X = K(Z/3 + Z/3,1) and A is empty then there is a homology class which is not representable.  

Unfortunately they don't give a proof - they cite Thom's paper:  Quelques propriétés globales des variétés différentiables.  

I have always assumed that this is the smallest example; that the map from bordism to homology is onto for n < 7 . . . Sorry I don't have a solid reference for this.

By the way, this example of Thom shows that Andy Baker misspoke below, what he intended to say was that MSO_{(2)} splits as a wedge of suspensions of HZ_{(2)} and HF_2.  In fact Thom's example shows that MSO is not a wedge of Eilenberg-MacLane spectra.

Jim Davis




On May 9, 2010, at 8:28 PM, Andrew J. Baker wrote:

> The canonical reference for classical bordism calculations is still Stong's book
> Notes on Cobordism Theory (Princeton UP), see Chap IX which discusses the subject
> in detail including Wall's calculations and the relation with integral homology.
> The esssence is that MSO splits as a wedge of suspensions of HZ and HF_2, the tricky part is to see the wedge summands.
> 
> A good source of this sort of detail and the multiplicative structure can be found
> in David Pengelley's work:
> 
> MR0686134 (84b:57025b)
> Pengelley, David J.
> The $A$-algebra structure of Thom spectra: $M{\rm SO}$ as an example. Current trends in algebraic topology, Part 1 (London, Ont., 1981), pp. 511--513,
> CMS Conf. Proc., 2, Amer. Math. Soc., Providence, R.I., 1982.
> 
> MR0657503 (84i:57018)
> Pengelley, David J.
> The mod two homology of $M{\rm SO}$ and $M{\rm SU}$ as $A$ comodule algebras, and
> the cobordism ring.
> J. London Math. Soc. (2) 25 (1982), no. 3, 467--472.
> 
> The A_*-comodule algebra structure of H_*(BSO;F_2) can be explicitely seen as an extended comodule algebra and then the fact that the first two summand are HZ and \Sigma^4HZ is clear.
> 
> Andy Baker
> 
> ------------------------------
> 
> Dr A. J. Baker
> Department of Mathematics
> University of Glasgow
> University Gardens
> Glasgow
> G12 8QW
> Scotland
> 
> phone (0141) 330 6140
> fax (0141) 330 4111
> 
> a.baker at maths.gla.ac.uk
> http://www.maths.gla.ac.uk/~ajb/
> 
> -----
> The University of Glasgow, charity number SC004401
> 
> 
> 
> -----Original Message-----
> From: algtop-l-bounces+a.baker=maths.gla.ac.uk at lists.lehigh.edu on behalf of algtop-l-request at lists.lehigh.edu
> Sent: Sun 5/9/2010 17:00
> To: algtop-l at lists.lehigh.edu
> Subject: ALGTOP-L Digest, Vol 411, Issue 1
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> Today's Topics:
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>    1. Submission (Daniel Moskovich)
> 
> 
> ----------------------------------------------------------------------
> 
> Message: 1
> Date: Sun, 9 May 2010 10:53:58 +0900
> From: Daniel Moskovich <dmoskovich at gmail.com>
> Subject: [ALGTOP-L] Submission
> To: algtop-l at lists.lehigh.edu
> Message-ID:
>         <z2ie238cada1005081853q102f4ef1redee6a6b3c57619e at mail.gmail.com>
> Content-Type: multipart/alternative;
>         boundary=e0cb4e887e21d06c8704861f90c1
> 
> --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'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'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--
> 
> 
> 
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