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

Andrew J. Baker a.baker at maths.gla.ac.uk
Sun May 9 14:28:58 EDT 2010


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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=20
in detail including Wall's calculations and the relation with integral =
homology.=20
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=20
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=20
the cobordism ring.
J. London Math. Soc. (2) 25 (1982), no. 3, 467--472.=20

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=3Dmaths.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
=20
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Today's Topics:

   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>
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I would like to ask the following question to this list, also asked by
myself <a href=3D"
http://mathoverflow.net/questions/23440/reference-request-for-relative-bo=
rdism-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=3DISO-8859-1
Content-Transfer-Encoding: quoted-printable

<div>I would like to ask the following question to this list, also asked =
by=3D
 myself &lt;a href=3D3D&quot;<a =
href=3D3D"http://mathoverflow.net/questions/234=3D
40/reference-request-for-relative-bordism-coinciding-with-homology-in-low=
-d=3D
imensions" =
target=3D3D"_blank">http://mathoverflow.net/questions/23440/refere=3D
nce-request-for-relative-bordism-coinciding-with-homology-in-low-dimensio=
ns=3D
</a>&quot;&gt;on MathOverflow&lt;/a&gt;.</div>

<div>=3DA0</div>
<div>It's a standard fact that, for finite CW complexes, the relative =
(=3D
oriented)=3DA0bordism group=3DA0 $<span =
alt=3D3D"\Omega_n^O(X)=3D3D\pi_{n+k}(MO(k)\=3D
wedge X)"><font face=3D3D"Times New =
Roman">\Omega_n^O(X,A)</font></span>$ is =3D
isomorphic to $<span alt=3D3D"[H_\ast(X; \Omega^O_*(pt.))]_n"><font =
face=3D3D"T=3D
imes New Roman">[H_\ast(X,A; \Omega^O_*(pt.))]_n$, which is isomorphic =
to $=3D
H_n(X,A)$ for $n&lt;4$</font></span>.=3DA0One can prove this using the =
Atiyah=3D
-Hirzebruch spectral sequence, and it can be deduced from results of =
Thom,=3D
=3DA0but all papers I've seen seem to just state=3DA0this =
isomorphism=3DA0as =3D
a fact without citation. I really want to find the original reference =
for t=3D
he above isomorphism, but have wasted much time and found nothing.</div>

<div>=3DA0</div>
<div>What is the original reference for the above proof (and the fact =
itsel=3D
f) that relative bordism and relative homology coincide in low =
dimensions?<=3D
/div>
<div>=3DA0</div>
<div>=3DA0</div>

--e0cb4e887e21d06c8704861f90c1--



------------------------------

_______________________________________________
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<TITLE>RE: ALGTOP-L Digest, Vol 411, Issue 1</TITLE>
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<P><FONT SIZE=3D2>The canonical reference for classical bordism =
calculations is still Stong's book<BR>
Notes on Cobordism Theory (Princeton UP), see Chap IX which discusses =
the subject<BR>
in detail including Wall's calculations and the relation with integral =
homology.<BR>
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.<BR>
<BR>
A good source of this sort of detail and the multiplicative structure =
can be found<BR>
in David Pengelley's work:<BR>
<BR>
MR0686134 (84b:57025b)<BR>
Pengelley, David J.<BR>
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,<BR>
CMS Conf. Proc., 2, Amer. Math. Soc., Providence, R.I., 1982.<BR>
<BR>
MR0657503 (84i:57018)<BR>
Pengelley, David J.<BR>
The mod two homology of $M{\rm SO}$ and $M{\rm SU}$ as $A$ comodule =
algebras, and<BR>
the cobordism ring.<BR>
J. London Math. Soc. (2) 25 (1982), no. 3, 467--472.<BR>
<BR>
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.<BR>
<BR>
Andy Baker<BR>
<BR>
------------------------------<BR>
<BR>
Dr A. J. Baker<BR>
Department of Mathematics<BR>
University of Glasgow<BR>
University Gardens<BR>
Glasgow<BR>
G12 8QW<BR>
Scotland<BR>
<BR>
phone (0141) 330 6140<BR>
fax (0141) 330 4111<BR>
<BR>
a.baker at maths.gla.ac.uk<BR>
<A =
HREF=3D"http://www.maths.gla.ac.uk/~ajb/">http://www.maths.gla.ac.uk/~ajb=
/</A><BR>
<BR>
-----<BR>
The University of Glasgow, charity number SC004401<BR>
<BR>
<BR>
<BR>
-----Original Message-----<BR>
From: algtop-l-bounces+a.baker=3Dmaths.gla.ac.uk at lists.lehigh.edu on =
behalf of algtop-l-request at lists.lehigh.edu<BR>
Sent: Sun 5/9/2010 17:00<BR>
To: algtop-l at lists.lehigh.edu<BR>
Subject: ALGTOP-L Digest, Vol 411, Issue 1<BR>
<BR>
Send ALGTOP-L mailing list submissions to<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; algtop-l at lists.lehigh.edu<BR>
<BR>
To subscribe or unsubscribe via the World Wide Web, visit<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <A =
HREF=3D"https://lists.lehigh.edu/mailman/listinfo/algtop-l">https://lists=
.lehigh.edu/mailman/listinfo/algtop-l</A><BR>
or, via email, send a message with subject or body 'help' to<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; =
algtop-l-request at lists.lehigh.edu<BR>
<BR>
You can reach the person managing the list at<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; =
algtop-l-owner at lists.lehigh.edu<BR>
<BR>
When replying, please edit your Subject line so it is more specific<BR>
than &quot;Re: Contents of ALGTOP-L digest...&quot;<BR>
<BR>
<BR>
Today's Topics:<BR>
<BR>
&nbsp;&nbsp; 1. Submission (Daniel Moskovich)<BR>
<BR>
<BR>
----------------------------------------------------------------------<BR=
>
<BR>
Message: 1<BR>
Date: Sun, 9 May 2010 10:53:58 +0900<BR>
From: Daniel Moskovich &lt;dmoskovich at gmail.com&gt;<BR>
Subject: [ALGTOP-L] Submission<BR>
To: algtop-l at lists.lehigh.edu<BR>
Message-ID:<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; =
&lt;z2ie238cada1005081853q102f4ef1redee6a6b3c57619e at mail.gmail.com&gt;<BR=
>
Content-Type: multipart/alternative;<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; =
boundary=3De0cb4e887e21d06c8704861f90c1<BR>
<BR>
--e0cb4e887e21d06c8704861f90c1<BR>
Content-Type: text/plain; charset=3DISO-8859-1<BR>
<BR>
I would like to ask the following question to this list, also asked =
by<BR>
myself &lt;a href=3D&quot;<BR>
<A =
HREF=3D"http://mathoverflow.net/questions/23440/reference-request-for-rel=
ative-bordism-coinciding-with-homology-in-low-dimensions">http://mathover=
flow.net/questions/23440/reference-request-for-relative-bordism-coincidin=
g-with-homology-in-low-dimensions</A>&quot;&gt;on<BR>
MathOverflow&lt;/a&gt;.<BR>
<BR>
It's a standard fact that, for finite CW complexes, the relative<BR>
(oriented) bordism group&nbsp; $\Omega_n^O(X,A)$ is isomorphic to =
$[H_\ast(X,A;<BR>
\Omega^O_*(pt.))]_n$, which is isomorphic to $H_n(X,A)$ for $n&lt;4$. =
One can<BR>
prove this using the Atiyah-Hirzebruch spectral sequence, and it can =
be<BR>
deduced from results of Thom, but all papers I've seen seem to just<BR>
state this isomorphism as a fact without citation. I really want to find =
the<BR>
original reference for the above isomorphism, but have wasted much time =
and<BR>
found nothing.<BR>
<BR>
What is the original reference for the above proof (and the fact =
itself)<BR>
that relative bordism and relative homology coincide in low =
dimensions?<BR>
<BR>
--e0cb4e887e21d06c8704861f90c1<BR>
Content-Type: text/html; charset=3DISO-8859-1<BR>
Content-Transfer-Encoding: quoted-printable<BR>
<BR>
&lt;div&gt;I would like to ask the following question to this list, also =
asked by=3D<BR>
&nbsp;myself &amp;lt;a href=3D3D&amp;quot;&lt;a href=3D3D&quot;<A =
HREF=3D"http://mathoverflow.net/questions/234=3D">http://mathoverflow.net=
/questions/234=3D</A><BR>
40/reference-request-for-relative-bordism-coinciding-with-homology-in-low=
-d=3D<BR>
imensions&quot; target=3D3D&quot;_blank&quot;&gt;<A =
HREF=3D"http://mathoverflow.net/questions/23440/refere=3D">http://mathove=
rflow.net/questions/23440/refere=3D</A><BR>
nce-request-for-relative-bordism-coinciding-with-homology-in-low-dimensio=
ns=3D<BR>
&lt;/a&gt;&amp;quot;&amp;gt;on =
MathOverflow&amp;lt;/a&amp;gt;.&lt;/div&gt;<BR>
<BR>
&lt;div&gt;=3DA0&lt;/div&gt;<BR>
&lt;div&gt;It's a standard fact that, for finite CW complexes, the =
relative (=3D<BR>
oriented)=3DA0bordism group=3DA0 $&lt;span =
alt=3D3D&quot;\Omega_n^O(X)=3D3D\pi_{n+k}(MO(k)\=3D<BR>
wedge X)&quot;&gt;&lt;font face=3D3D&quot;Times New =
Roman&quot;&gt;\Omega_n^O(X,A)&lt;/font&gt;&lt;/span&gt;$ is =3D<BR>
isomorphic to $&lt;span alt=3D3D&quot;[H_\ast(X; =
\Omega^O_*(pt.))]_n&quot;&gt;&lt;font face=3D3D&quot;T=3D<BR>
imes New Roman&quot;&gt;[H_\ast(X,A; \Omega^O_*(pt.))]_n$, which is =
isomorphic to $=3D<BR>
H_n(X,A)$ for $n&amp;lt;4$&lt;/font&gt;&lt;/span&gt;.=3DA0One can prove =
this using the Atiyah=3D<BR>
-Hirzebruch spectral sequence, and it can be deduced from results of =
Thom,=3D<BR>
=3DA0but all papers I've seen seem to just state=3DA0this =
isomorphism=3DA0as =3D<BR>
a fact without citation. I really want to find the original reference =
for t=3D<BR>
he above isomorphism, but have wasted much time and found =
nothing.&lt;/div&gt;<BR>
<BR>
&lt;div&gt;=3DA0&lt;/div&gt;<BR>
&lt;div&gt;What is the original reference for the above proof (and the =
fact itsel=3D<BR>
f) that relative bordism and relative homology coincide in low =
dimensions?&lt;=3D<BR>
/div&gt;<BR>
&lt;div&gt;=3DA0&lt;/div&gt;<BR>
&lt;div&gt;=3DA0&lt;/div&gt;<BR>
<BR>
--e0cb4e887e21d06c8704861f90c1--<BR>
<BR>
<BR>
<BR>
------------------------------<BR>
<BR>
_______________________________________________<BR>
ALGTOP-L mailing list<BR>
ALGTOP-L at lists.lehigh.edu<BR>
<A =
HREF=3D"https://lists.lehigh.edu/mailman/listinfo/algtop-l">https://lists=
.lehigh.edu/mailman/listinfo/algtop-l</A><BR>
<BR>
<BR>
End of ALGTOP-L Digest, Vol 411, Issue 1<BR>
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