Stable Homotopy theory
The universal real Thom spectrum MO is a connective spectrum whose associated infinite loop space is the classifying space for cobordism:
In particular, is naturally identified with the set of cobordism classes of closed -manifolds (Thom's theorem).
More abstractly, MO is the homotopy colimit of the J-homomorphism in Spectra
hence the “total space” of the universal spherical fibration on the classifying space for (stable) real vector bundles.
Given this, for any topological group equipped with a homomorphism to the orthogonal group there is a corresponding Thom spectrum
This is considered particularly for the stages in the Whitehead tower of the orthogonal group, where it yields Spin, String group, etc.
All these Thom spectra happen to naturally have the structure of E-∞ rings and -ring homomorphisms into another -ring are equivalently universal orientations in E-cohomology. On homotopy groups these are genera with coefficients in the underlying ring .
For vector bundles
First recall the following two basic facts about the construction of Thom spaces. See at Thom space this prop..
For and to vector bundles, let be the direct sum of vector bundles of their pullbacks to . The corresponding Thom space is the smash product of the individual Thom spaces:
Prop. 1 will give rise to universal Thom spectra in the following, while prop. 2 will give them the struture of ring spectra.
For a vector bundle, its Thom spectrum is the suspension spectrum of the Thom space of . By prop. prop. 1 this may be written as
with structure maps are the equivalences
For each the pullback of the rank- universal vector bundle to the classifying space of rank vector bundles is the direct sum of vector bundles of the rank universal vector bundle with the trivial rank-1 bundle: there is a pullback diagram of topological spaces of the form
where the bottom morphism is the canonical one (def.).
(e.g. Kochmann 96, p. 25)
For each , there is such a pullback of the canonical vector bundles over Grassmannians
where the bottom morphism is the canonical inclusion (def.). Under taking colimit over , this produces the claimed pullback.
The -fold looping of the Thom spectrum, according to def. 1, of the rank- universal vector bundle is written
The image of the top horizontal maps in prop. 3 under are, via prop. 1, maps of the form
The homotopy colimit over these maps is the universal Thom spectrum:
As a sequential spectrum, the universal Thom spectrum is the is represented by the sequential prespectrum whose th component space is the Thom space
of the rank- universal vector bundle, and whose structure maps are the image under the Thom space functor of the top morphisms in prop. 3, via the homeomorphisms of prop. 1:
As an orthogonal ring spectrum, the universal Thom spectrum has
the Thom spaces of the universal vector bundle with fiber ;
left -action induced by the remaining canonical left action of ;
induced via prop. 2
unit maps given by
induced by the fiber inclusion .
For discussion of the refinement of the Thom spectrum to a symmetric spectrum see (Hovey-Shipley-Smith 00, example 6.2.3, Schwede 12, Example I.2.8). For the refinement to an orthogonal spectrum and globally equivariant spectrum see (Schwede 15, section V.4).
More generally, there are universal Thom spectra associated with any other tangent structure (“[[(B,f)]-structure]”), notably for the orthogonal group replaced by the special orthogonal groups , or the spin groups , or the string 2-group , or the fivebrane 6-group ,…, or any level in the Whitehead tower of . To any of these groups there corresponds a Thom spectrum (denoted, respectively, , , , , etc.), which is in turn related to oriented cobordism, spin cobordism, string cobordism, et cetera.
A (B,f)-structure , is a system of Serre fibrations
for , equipped with maps covering the canonical maps ((def.)) in that there are commuting squares
Similarly, an --structure is a compatible system
indexed only on the even natural numbers.
Given a (B,f)-structure (def. 5), write
for the pullback of the rank- universal vector bundle from to along .
Observe that the analog of prop. 3 still holds:
Given a (B,f)-structure (def. 5), then the pullback of its rank- vector bundle (def. 6) along the map is the direct sum of vector bundles of the rank- bundle with the trivial rank-1-bundle: there is a pullback square
Unwinding the definitions, the pullback in question is
where the second but last step is due to prop. 3.
Given a (B,f)-structure (def. 5), its Thom spectrum is, as a sequential prespectrum, given by component spaces being the Thom spaces of the -associated vector bundles of def. 6
and with structure maps given via prop. 1 by the top maps in prop. 4:
Similarly for a -structure indexed on the even natural numbers, there is the corresponding Thom spectrum as an -sequential spectrum (def.).
If for some natural system of groups , then one usually writes for . For instance , , etc.
If the -structure is multiplicative (def. 5), then the Thom spectrum canonical becomes a ring spectrum: the multiplication maps are covered by maps of vector bundles
and under forming Thom spaces this yields, by prop. 2, maps
The Thom spectrum of the framing structure (exmpl.) is equivalently the sphere spectrum
Because in this case and so , whence .
For spherical fibrations
For -module bundles
We discuss the Thom spectrum construction for general (∞,1)-module bundles.
There is pair of adjoint (∞,1)-functors
where is the stabilization adjunction between Top and Spec ( forms the suspension spectrum), restricted to connective spectra. The right adjoint is the ∞-group of units-(∞,1)-functor, see there for more details.
This is (ABGHR, theorem 2.1/3.2).
for the suspension of the group of units .
This plays the role of the classifying space for -principal ∞-bundles.
For a morphism (a cocycle for -bundles) in Spec, write for the corresponding bundle: the homotopy fiber
Given a -algebra , hence an A-∞ algebra over , exhibited by a morphism , the composite
is that for the corresponding associated ∞-bundle.
We write capital letters for the underlying spaces of these spectra:
The Thom spectrum of is the (∞,1)-pushout
hence the derived smash product
This is made precise by the following statement.
We have an (∞,1)-pullback diagram
This is (ABGHR, theorem 2.10).
This definition does subsume the above definition of Thom spectra for sphere bundles (hence also that for vector bundles, by removing their zero section):
Let be the sphere spectrum. Then for a cocycle for an -bundle,
is the classifying map for a spherical fibration over .
The Thom spectrum of def. 10 is equivalent to the Thom spectrum of the spherical fibration, according to def. 8.
This is in (ABGHR, section 8).
Equivalently the Thom spectrum is characterized as follows:
For a map to the ∞-group of -(∞,1)-lines inside , the corresponding Thom spectrum is the (∞,1)-colimit
This construction evidently extendes to an (∞,1)-functor
This is (Ando-Blumberg-Gepner 10, def. 4.1), reviewed also as (Wilson 13, def. 3.3).
This observation appears as (Wilson 13, prop. 4.4).
Ring spectrum structure
The universal Thom spectrum, def. 3, naturally inherits the structure of a ring spectrum as follows.
There are canonical commuting diagrams
Applying the Thom space functor to the top morphisms and using prop. 2 gives morphisms
that combine to a functor with smash products and hence give the structure of a ring spectrum.
More abstractly, sufficient condition for a Thom spectrum of an ∞-module bundle (as above) to have E-∞ ring structure is that it arises, as the (∞,1)-colimit of a homomorphism of E-∞ spaces (ABG 10, prop.6.21).
Relation to the cobordism ring
This is a seminal result due to (Thom 54), whose proof proceeds by the Pontryagin-Thom construction. The presentation of the following proof follows (Francis, lecture 3).
We first construct a map .
Given a class we can choose a representative SmthMfd and a closed embedding of into the Cartesian space of sufficiently large dimension. By the tubular neighbourhood theorem factors as the embedding of the zero section into the normal bundle followed by an open embedding of into
Now use the Pontrjagin-Thom construction to produce an element of the homotopy group first in the Thom space of and then eventually in . To that end, let
be the map into the one-point compactification. Define a map
by sending points in the image of under to their preimage, and all other points to the collapsed point . This defines an element in the homotopy group .
To turn this into an element in the homotopy group of , notice that since is a vector bundle of rank , it is the pullback by a map of the universal rank vector bundle
By forming Thom spaces the top map induces a map
Its composite with the map constructed above gives an element in
and by this is finally an element
We show now that this element does not depend on the choice of embedding .
Finally, to show that is an isomorphism by constructing an inverse.
For that, observe that the sphere is a compact topological space and in fact a compact object in Top. This implies that every map from into the filtered colimit
factors through one of the terms as
By Thom's transversality theorem we may find an embedding by a transverse map to . Define then to be the pullback
We check that this construction provides an inverse to .
As a dual in the stable homotopy category
Write Spec for the category of spectra and for its standard homotopy category: the stable homotopy category. By the symmetric monoidal smash product of spectra this becomes a monoidal category.
For any topological space, we may regard it as an object in by forming its suspension spectrum . We may ask under which conditions on this is a dualizable object with respect to the smash-product monoidal structure.
It turns out that a sufficient condition is that a closed smooth manifold or more generally a compact Euclidean neighbourhood retract. In that case – the Thom spectrum of its stable normal bundle is the corresponding dual object. (Atiyah 61, Dold-Puppe 78). This is called the Spanier-Whitehead dual of .
Using this one shows that the trace of the identity on in – the categorical dimension of – is the Euler characteristic of .
For a brief exposition see (PontoShulman, example 3.7). For more see at Spanier-Whitehead duality.
As the universal spherical fibration, from the -homomorphism
The J-homomorphism is a canonical map from the classifying space of the stable orthogonal group to the delooping of the infinity-group of units of the sphere spectrum. This classifies an “(∞,1)-vector bundle” of sphere spectrum-modules over and this is the Thom spectrum.
So in terms of the (∞,1)-colimit description above we have
See at orientation in generalized cohomology for more on this.
As the infinite cobordism category
The geometric realization for the (infinity,n)-category of cobordisms for is the Thom spectrum
This is implied by the Galatius-Madsen-Tillmann-Weiss theorem and by Jacob Lurie’s proof of the cobordism hypothesis. See also (Francis-Gwilliam, remark 0.9).
Under the Brown representability theorem the Thom spectrum represents the generalized (Eilenberg-Steenrod) cohomology theory called cobordism cohomology theory.
The following terms all refer to essentially the same concept:
The relation between the homotopy groups of the Thom spectrum and the cobordism ring is due to
- René Thom, Quelques propriétés globales des variétés différentiables Comment. Math. Helv. 28, (1954). 17-86
Further original articles include
- Michael Atiyah, Thom complexes, Proc. London Math. Soc. (3), 11:291–310, 1961. 10 (pdf)
Textbook accounts include
Lecture notes include
As symmetric spectra:
As orthogonal spectra and as equivariant spectra
Textbook discussion with an eye towards the generalized (Eilenberg-Steenrod) cohomology of topological K-theory and cobordism cohomology theory is in
- Yuli Rudyak, On Thom spectra, orientability and cobordism, Springer Monographs in Mathematics, 1998 (pdf)
A generalized notion of Thom spectra in terms of (∞,1)-module bundles is discussed in
a streamlined update of which is
Discussion of Thom spectra from the point of view of (∞,1)-module bundles is in
- Matthew Ando, Andrew Blumberg, David Gepner, Twists of K-theory and TMF, in Robert S. Doran, Greg Friedman, Jonathan Rosenberg, Superstrings, Geometry, Topology, and -algebras, Proceedings of Symposia in Pure Mathematics vol 81, American Mathematical Society (arXiv:1002.3004)
which is reviewed in
and in the context of motivic quantization via pushforward in twisted generalized cohomology in section 3.1 of
As dual objects in the stable homotopy category
The relation of Thom spectra to dualizable objects in the stable homotopy category is originally due to (Atiyah 61) and
Albrecht Dold, Dieter Puppe, Duality, trace, and transfer. In Proceedings of the International Conference on Geometric Topology (Warsaw, 1978), pages 81–102, Warsaw, 1980. PWN.
L. Gaunce Lewis, Jr., Peter May, M. Steinberger, and J. E. McClure, Equivariant stable homotopy theory, volume 1213 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1986. With contributions by J. E. McClure.
A brief exposition appears as example 3.7 in