Editing Thom space (section)

==Thom spectrum==

=== Real cobordism ===
There are two ways to think about bordism: one as considering two <math>n</math>-manifolds <math>M,M'</math> are cobordant if there is an <math>(n+1)</math>-manifold with boundary <math>W</math> such that

:<math>\partial W = M \coprod M'</math>

Another technique to encode this kind of information is to take an embedding <math>M \hookrightarrow \R^{N + n}</math> and considering the normal bundle

:<math>\nu: N_{\R^{N+n}/M} \to M</math>

The embedded manifold together with the isomorphism class of the normal bundle actually encodes the same information as the cobordism class <math>[M]</math>. This can be shown<ref>{{Cite web|last=|first=|date=|title=Thom's theorem| url=https://sites.math.northwestern.edu/~jnkf/classes/mflds/3thom.pdf| url-status=live|archive-url=https://web.archive.org/web/20210117195051/https://sites.math.northwestern.edu/~jnkf/classes/mflds/3thom.pdf|archive-date=17 Jan 2021|access-date=|website=}}</ref> by using a cobordism <math>W</math> and finding an embedding to some <math>\R^{N_W + n}\times [0,1]</math> which gives a homotopy class of maps to the Thom space <math>MO(n)</math> defined below. Showing the isomorphism of

:<math>\pi_nMO \cong \Omega^O_n</math>

requires a little more work.<ref>{{Cite web| last=| first=| date=| title=Transversality |url= https://sites.math.northwestern.edu/~jnkf/classes/mflds/4transversality.pdf | url-status=live|archive-url= https://web.archive.org/web/20210117200636/https://sites.math.northwestern.edu/~jnkf/classes/mflds/4transversality.pdf |archive-date=17 Jan 2021|access-date=|website=}}</ref>

=== Definition of Thom spectrum ===
By definition, the '''Thom spectrum'''<ref>See pp. 8-9 in {{cite arXiv|last=Greenlees|first=J. P. C.|date=2006-09-15|title=Spectra for commutative algebraists|eprint=math/0609452}}</ref> is a sequence of Thom spaces

:<math>MO(n) = T(\gamma^n)</math>

where we wrote <math>\gamma^n\to BO(n)</math> for the [[universal vector bundle]] of rank ''n''. The sequence forms a [[spectrum (topology)|spectrum]].<ref>{{cite web|url=http://math.northwestern.edu/~jnkf/classes/mflds/2cobordism.pdf|title=Math 465, lecture 2: cobordism|first=J.|last=Francis|others=Notes by O. Gwilliam|publisher=Northwestern University}}</ref> A theorem of Thom says that <math>\pi_*(MO)</math> is the unoriented [[cobordism ring]];<ref>{{harvnb|Stong|1968|loc=p. 18}}</ref> the proof of this theorem relies crucially on [[Transversality theorem|Thom’s transversality theorem]].<ref>{{cite web|url=http://math.northwestern.edu/~jnkf/classes/mflds/4transversality.pdf|title=Math 465, lecture 4: transversality|first=J.|last=Francis|others=Notes by I. Bobovka|publisher=Northwestern University}}</ref> The lack of transversality requires that alternative methods be found to compute cobordism rings of, say, [[topological manifold]]s from Thom spectra.