Editing Thom space

In [[mathematics]], the '''Thom space,''' '''Thom complex,''' or '''Pontryagin–Thom construction''' (named after [[René Thom]] and [[Lev Pontryagin]]) of [[algebraic topology]] and [[differential topology]] is a [[topological space]] associated to a [[vector bundle]], over any [[paracompact]] space.

==Construction of the Thom space==

One way to construct this space is as follows. Let

:<math>p\colon E \to B</math>

be a rank ''n'' [[real number|real]] [[vector bundle]] over the [[paracompact space]] ''B''. Then for each point ''b'' in  ''B'', the [[Fiber (mathematics)#Fiber in naive set theory|fiber]] <math>E_b</math> is an ''n''-dimensional real [[vector space]]. We can form an ''n''-[[sphere bundle]] <math>\operatorname{Sph}(E) \to B</math> by taking the [[one-point compactification]] of each fiber and gluing them together to get the total space.{{Elucidate|What are the open sets of the new total space?|date=November 2014}} Finally, from the total space <math>\operatorname{Sph}(E)</math> we obtain the '''Thom space''' <math>T(E)</math> as the quotient of <math>\operatorname{Sph}(E)</math> by ''B''; that is, by identifying all the new points to a single point <math>\infty</math>, which we take as the [[Pointed space|basepoint]] of <math>T(E)</math>. If ''B'' is compact, then <math>T(E)</math> is the one-point compactification of ''E''.

For example, if ''E'' is the trivial bundle <math>B\times \R^n</math>, then  <math>\operatorname{Sph}(E)</math> is <math>B\times S^n</math> and, writing <math>B_+</math> for ''B'' with a disjoint basepoint, <math>T(E)</math> is the [[smash product]] of <math>B_+</math> and <math>S^n</math>; that is, the ''n''-th reduced [[suspension (topology)|suspension]] of <math>B_+</math>.

Alternatively,{{fact|date=July 2024}} since ''B'' is paracompact, ''E'' can be given a Euclidean metric and then <math>T(E)</math> can be defined as the quotient of the unit disk bundle of ''E'' by the unit <math>(n-1)</math>-sphere bundle of ''E''.

==The Thom isomorphism==

The significance of this construction begins with the following result, which belongs to the subject of [[cohomology]] of [[fiber bundle]]s. (We have stated the result in terms of <math>\Z_2</math> [[coefficients]] to avoid complications arising from [[orientability]]; see also [[Orientation of a vector bundle#Thom space]].)

Let <math>p: E\to B</math> be a real vector bundle of rank ''n''. Then there is an isomorphism called a '''Thom isomorphism'''
:<math>\Phi : H^k(B; \Z_2) \to \widetilde{H}^{k+n}(T(E); \Z_2),</math>
for all ''k'' greater than or equal to 0, where the [[Left-hand side and right-hand side of an equation|right hand side]] is [[reduced cohomology]].

This theorem was formulated and proved by [[René Thom]] in his famous 1952 thesis.

We can interpret the theorem as a global generalization of the suspension isomorphism on local trivializations, because the Thom space of a trivial bundle on ''B'' of rank ''k'' is isomorphic to the ''k''th suspension of <math>B_+</math>, ''B'' with a disjoint point added (cf. [[#Construction of the Thom space]].) This can be more easily seen in the formulation of the theorem that does not make reference to Thom space:

{{math_theorem|name=Thom isomorphism|
Let <math>\Lambda</math> be a ring and <math>p: E\to B</math> be an [[oriented bundle|oriented]] real vector bundle of rank ''n''. Then there exists a class
:<math>u \in H^n(E, E \setminus B; \Lambda),</math>
where ''B'' is embedded into ''E'' as a zero section, such that for any fiber ''F'' the restriction of ''u''
:<math>u|_{(F, F \setminus 0)} \in H^n(F, F \setminus 0; \Lambda)</math>
is the class induced by the orientation of ''F''. Moreover,
:<math>\begin{cases} H^k(E; \Lambda) \to H^{k+n}(E, E \setminus B; \Lambda) \\ x \longmapsto x \smile u \end{cases}</math>
is an isomorphism.
}}

In concise terms, the last part of the theorem says that ''u'' freely generates <math>H^*(E, E \setminus B; \Lambda)</math> as a right <math>H^*(E; \Lambda)</math>-module. The class ''u'' is usually called the '''Thom class''' of ''E''. Since the pullback <math>p^*: H^*(B; \Lambda) \to H^*(E; \Lambda)</math> is a [[ring isomorphism]], <math>\Phi</math> is given by the equation:

:<math>\Phi(b) = p^*(b) \smile u.</math>

In particular, the Thom isomorphism sends the [[identity (mathematics)|identity]] element of <math>H^*(B)</math> to ''u''. Note: for this formula to make sense, ''u'' is treated as an element of (we drop the ring <math>\Lambda</math>)

:<math>\tilde{H}^n(T(E)) = H^n(\operatorname{Sph}(E), B) \simeq H^n(E, E \setminus B).</math><ref>Proof of the isomorphism. We can embed ''B'' into <math>\operatorname{Sph}(E)</math> either as the zero section; i.e., a section at zero vector or as the infinity section; i.e., a section at infinity vector (topologically the difference is immaterial.) Using two ways of embedding we have the triple:
:<math>(\operatorname{Sph}(E), \operatorname{Sph}(E) \setminus B, B)</math>.
Clearly, <math>\operatorname{Sph}(E) \setminus B</math> deformation-retracts to ''B''. Taking the long exact sequence of this triple, we then see:

:<math>H^n(Sph(E), B) \simeq H^n(\operatorname{Sph}(E), \operatorname{Sph}(E) \setminus B),</math>

the latter of which is isomorphic to:

:<math>H^n(E, E \setminus B)</math>

by excision.</ref><!-- I can't understand this:
Note: for this formula to make sense, ''u'' is treated as an element of <math>H^k(D(E),\operatorname{Sph}(E);\Z_2)\cong \tilde H^k(T(E);\Z_2)</math>, where <math>D(E)</math> is the associated disk bundle, so we have a cup product

:<math>H^i(D(E);\Z_2)\otimes H^k(D(E),\operatorname{Sph}(E);\Z_2)\to H^{i+k}(D(E),\operatorname{Sph}(E);\Z_2)\cong \tilde H^k(T(E);\Z_2)</math>.-->

The standard reference for the Thom isomorphism is the book by Bott and Tu.

==Significance of Thom's work==

In his 1952 paper, Thom showed that the Thom class, the [[Stiefel–Whitney class]]es, and the [[Steenrod operation]]s were all related. He used these ideas to prove in the 1954 paper ''Quelques propriétés globales des variétés differentiables'' that the [[cobordism]] groups could be computed as the [[homotopy groups]] of certain Thom spaces ''MG''(''n''). The proof depends on and is intimately related to the [[transversality (mathematics)|transversality]] properties of [[smooth manifolds]]—see [[Thom transversality theorem]]. By reversing this construction, [[John Milnor]] and [[Sergei Novikov (mathematician)|Sergei Novikov]] (among many others) were able to answer questions about the existence and uniqueness of high-dimensional manifolds: this is now known as [[surgery theory]]. In addition, the spaces ''MG(n)'' fit together to form [[spectrum (homotopy theory)|spectra]] ''MG'' now known as '''Thom spectra''', and the cobordism groups are in fact [[stable homotopy theory|stable]]. Thom's construction thus also unifies [[differential topology]] and stable homotopy theory, and is in particular integral to our knowledge of the [[stable homotopy groups of spheres]].

If the Steenrod operations are available, we can use them and the isomorphism of the theorem to construct the Stiefel–Whitney classes. Recall that the Steenrod operations (mod 2) are [[natural transformation]]s

:<math>Sq^i : H^m(-; \Z_2) \to H^{m+i}(-; \Z_2),</math>

defined for all nonnegative integers ''m''. If <math>i=m</math>, then <math>Sq^i</math> coincides with the cup square. We can define the ''i''th Stiefel–Whitney class <math>w_i(p)</math> of the vector bundle <math>p: E\to B</math> by:

:<math>w_i(p) = \Phi^{-1}(Sq^i(\Phi(1))) = \Phi^{-1}(Sq^i(u)).</math>

==Consequences for differentiable manifolds==

If we take the bundle in the above to be the [[tangent bundle]] of a smooth manifold, the conclusion of the above is called the [[Wu formula]], and has the following strong consequence: since the Steenrod operations are invariant under homotopy equivalence, we conclude that the Stiefel–Whitney classes of a manifold are as well. This is an extraordinary result that does not generalize to other characteristic classes. There exists a similar famous and difficult result establishing topological invariance for rational [[Pontryagin classes]], due to [[Sergei Novikov (mathematician)|Sergei Novikov]].

==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.

==See also==
* [[Cobordism]]
* [[Cohomology operation]]
* [[Steenrod problem]]
* [[Hattori–Stong theorem]]

==Notes==
{{reflist}}

==References==
* {{cite journal |first=Dennis |last=Sullivan |author-link=Dennis Sullivan |title=René Thom's Work on Geometric Homology and Bordism |journal=[[Bulletin of the American Mathematical Society]] |volume=41 |issue=3 |year=2004 |pages=341–350 |doi=10.1090/S0273-0979-04-01026-2 |doi-access=free }}
* {{cite book |author-link=Raoul Bott |first1=Raoul |last1=Bott |first2=Loring |last2=Tu |title=Differential Forms in Algebraic Topology |location=New York |publisher=Springer |year=1982 |isbn=0-387-90613-4 }} A classic reference for [[differential topology]], treating the link to [[Poincaré duality]], [[Euler class]] of [[Sphere bundle]]s, Thom classes and Thom isomorphism, and more.
* {{cite book |author-link=John Milnor |first1=John |last1=Milnor| title=Characteristic classes }} is another standard reference for the Thom class and Thom isomorphism. See especially the paragraph 18.
* {{cite book |first=J. Peter |last=May |author-link=J. Peter May |title=A Concise Course in Algebraic Topology |publisher=[[University of Chicago Press]] |year=1999 |pages=183–198 |isbn=0-226-51182-0 }} This textbook gives a detailed construction of the Thom class for trivial vector bundles, and also formulates the theorem in case of arbitrary vector bundles. 
* {{cite book |first=Robert E.|last= Stong |author-link=Robert Evert Stong| title=Notes on cobordism theory |publisher= [[Princeton University Press]] |year=1968 }}
* {{cite journal |first=René |last=Thom |author-link=René Thom|title=[[List of important publications in mathematics#Quelques propriétés globales des variétés differentiables|Quelques propriétés globales des variétés différentiables]] |journal=[[Commentarii Mathematici Helvetici]] |volume=28 |year=1954 |pages=17–86 |doi=10.1007/BF02566923 |s2cid=120243638 }}
* {{cite journal |title=Units of ring spectra and Thom spectra |arxiv= 0810.4535 |first1 = Matthew |last1 = Ando|first2 = Andrew J.|last2 = Blumberg|first3 = David J.|last3 = Gepner|first4 = Michael J.|last4 = Hopkins|author4-link=Michael J. Hopkins| first5 = Charles|last5 = Rezk |journal=[[Journal of Topology]] |volume= 7 |year=2014|issue= 4|pages=1077–1117|doi=10.1112/jtopol/jtu009| mr=286898 |s2cid= 119613530 }}

==External links==
*http://ncatlab.org/nlab/show/Thom+spectrum
* {{springer|title=Thom space|id=p/t092680}}

[[Category:Algebraic topology]]
[[Category:Characteristic classes]]