cobordism theory = manifolds and cobordisms + stable homotopy theory/higher category theory
Concepts of cobordism theory
Pontrjagin's theorem (equivariant, twisted):
$\phantom{\leftrightarrow}$ Cohomotopy
$\leftrightarrow$ cobordism classes of normally framed submanifolds
$\phantom{\leftrightarrow}$ homotopy classes of maps to Thom space MO
$\leftrightarrow$ cobordism classes of normally oriented submanifolds
complex cobordism cohomology theory
flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:
bordism theory$\;$M(B,f) (B-bordism):
relative bordism theories:
global equivariant bordism theory:
algebraic:
(…)
The universal real Thom spectrum MO is a connective spectrum whose associated infinite loop space is the classifying space for cobordism:
In particular, $\pi_n M O$ is naturally identified with the set of cobordism classes of closed $n$-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 $B O$ for (stable) real vector bundles.
Given this, for any topological group $G$ equipped with a homomorphism to the orthogonal group there is a corresponding Thom spectrum
This is considered particularly for the stages $G$ in the Whitehead tower of the orthogonal group, where it yields $M$Spin, $M$String group, etc.
All these Thom spectra happen to naturally have the structure of E-∞ rings and $E_\infty$-ring homomorphisms $M O\to E$ into another $E_\infty$-ring $E$ are equivalently universal orientations in E-cohomology. On homotopy groups these are genera with coefficients in the underlying ring $\pi_\bullet(E)$.
First recall the following two basic facts about the construction of Thom spaces. See at Thom space this prop..
For $V \to X$ a real vector bundle, there is a homeomorphism
between, on the one hand, the Thom space of the direct sum of vector bundles of $V$ with the trivial vector bundle of rank $n$ and, on the other, the $n$-fold reduced suspension of the Thom space of $V$.
For $V_1 \to X_1$ and $V_2 \to X_2$ two real vector bundles, let $V_1 \boxtimes V_2 \to X_1 \times X_2$ be the direct sum of vector bundles of their pullbacks to $X_1 \times X_2$ (their external tensor product). The corresponding Thom space is the smash product of the individual Thom spaces:
Prop. will give rise to universal Thom spectra in the following, while Prop. will give them the struture of ring spectra.
For $V \to X$ a real vector bundle, its Thom spectrum is the suspension spectrum $\Sigma^\infty Th(V)$ of the Thom space of $V$. By prop. this may be written as
with structure maps (as a sequential spectrum) the equivalences
For each $n \in \mathbb{N}$ the pullback of the rank-$(n+1)$ universal vector bundle to the classifying space of rank $n$ vector bundles is the direct sum of vector bundles of the rank $n$ universal vector bundle with the trivial rank-1 bundle: there is a pullback diagram in Top of the form
where the bottom morphism is the canonical one (this def.).
(e.g. Kochmann 96, p. 25)
For each $k \in \mathbb{N}$, $k \geq n$ there are such pullbacks of the canonical vector bundles over Grassmannians
where the bottom morphism is the canonical inclusion (def.). Under taking the colimit over $k$, this produces the claimed pullback.
The $n$-fold looping of the Thom spectrum, according to def. , of the rank-$n$ universal vector bundle is written
The image of the top horizontal maps in prop. under $\Sigma^{\infty - n}Th(-)$ are, via prop. , maps of the form
The homotopy colimit over these maps is the universal Thom spectrum:
More explicitly:
As a sequential spectrum, the universal Thom spectrum $M O$ is represented by the sequential prespectrum whose $n$th component space is the Thom space
of the rank-$n$ universal vector bundle, and whose structure maps are the image under the Thom space functor $Th(-)$ of the top morphisms in prop. , via the homeomorphisms of prop. :
As an orthogonal ring spectrum, the universal Thom spectrum $M O$ has
component spaces
the Thom spaces of the universal vector bundle with fiber $V$;
left $O(V)$-action induced by the remaining canonical left action of $E O(V)$;
multiplication maps
unit maps given by
induced by the fiber inclusion $O(V) \to E O(V)$.
For discussion of the refinement of the Thom spectrum $M O$ 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 $SO(n)$, or the spin groups $Spin(n)$, or the string 2-group $String(n)$, or the fivebrane 6-group $Fivebrane(n)$,…, or any level in the Whitehead tower of $O(n)$. To any of these groups there corresponds a Thom spectrum (denoted, respectively, $M SO$, $M Spin$, $M String$, $M Fivebrane$, etc.), which is in turn related to oriented cobordism, spin cobordism, string cobordism, et cetera.
Recall:
A (B,f)-structure $\mathcal{B}$, is a system of Serre fibrations
for $n \in \mathbb{N}$, equipped with maps $j_n \colon B_n \to B_{n+1}$ covering the canonical maps $i_n \colon B O(n) \to B O(n+1)$ ((def.)) in that there are commuting squares
Similarly, an $S^2$-$(B,f)$-structure is a compatible system
indexed only on the even natural numbers.
Given a (B,f)-structure $\mathcal{B}$ (def. ), write
for the pullback of the rank-$n$ universal vector bundle from $B O(n)$ to $B_n$ along $f_n$.
Observe that the analog of prop. still holds:
Given a (B,f)-structure $\mathcal{B}$ (def. ), then the pullback of its rank-$(n+1)$ vector bundle $E^{\mathcal{B}}_{n+1}$ (def. ) along the map $j_n \colon B_n \to B_{n+1}$ is the direct sum of vector bundles of the rank-$n$ bundle $E^{\mathcal{B}}_n$ with the trivial rank-1-bundle: there is a pullback square
Unwinding the definitions, the pullback in question is
Given a (B,f)-structure $\mathcal{B}$ (def. ), its Thom spectrum $M \mathcal{B}$ is, as a sequential prespectrum, given by component spaces being the Thom spaces of the $\mathcal{B}$-associated vector bundles of def.
and with structure maps given via prop. by the top maps in prop. :
Similarly for a $(B,f)$-structure indexed on the even natural numbers, there is the corresponding Thom spectrum as an $S^2$-sequential spectrum (def.).
If $B_n = B G_n$ for some natural system of groups $G_n \to O(n)$, then one usually writes $M G$ for $M \mathcal{B}$. For instance $M SO$, $M Spin$, $M U$ etc.
If the $(B,f)$-structure is multiplicative (def. ), then the Thom spectrum $M \mathcal{B}$ canonical becomes a ring spectrum: the multiplication maps $B_{n_1} \times B_{n_2}\to B_{n_1 + n_2}$ are covered by maps of vector bundles
and under forming Thom spaces this yields, by prop. , maps
The Thom spectrum of the framing structure (exmpl.) is equivalently the sphere spectrum
Because in this case $B_n \simeq \ast$ and so $E^{\mathcal{B}}_n \simeq \mathbb{R}^n$, whence $Th(E^{\mathcal{B}}_n) \simeq S^n$.
(…)
We discuss the Thom spectrum construction for general (∞,1)-module bundles.
There is pair of adjoint (∞,1)-functors
where $(\Sigma^\infty \dashv \Omega^\infty) : Spec \to Top$ is the stabilization adjunction between Top and Spec ($\Sigma^\infty$ 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).
Here $gl_1$ forms the “general linear group-of rank 1”-spectrum of an E-∞ ring: its ∞-group of units“. The adjunction is the generalization of the adjunction
between CRing and Ab, where $\mathbb{Z}[-]$ forms the group ring.
Write
for the suspension of the group of units $gl_1(R)$.
This plays the role of the classifying space for $gl_1(R)$-principal ∞-bundles.
For $f : b \to b gl_1(R)$ a morphism (a cocycle for $gl_1(R)$-bundles) in Spec, write $p \to b$ for the corresponding bundle: the homotopy fiber
Given a $R$-algebra $A$, hence an A-∞ algebra over $R$, exhibited by a morphism $\rho : R \to A$, the composite
is that for the corresponding associated ∞-bundle.
We write capital letters for the underlying spaces of these spectra:
The Thom spectrum $M f$ of $f : b \to gl_1(R)$ is the (∞,1)-pushout
hence the derived smash product
This means that a morphism $M f \to A$ is an $GL_1(R)$-equivariant map $P \to A$.
Notice that for $R = \mathbb{C}$ the complex numbers, $B \to GL_1(R)$ is the cocycle for a circle bundle $P \to B$. A $U(1)$-equivariant morphism $P \to A$ to some representation $A$ is equivalently a section of the A-associated bundle.
Therefore the Thom spectrum may be thought of as co-representing spaces of sections of associated bundles
“$Hom(M f, A) \simeq \Gamma(P \wedge_{GL_1(R)} A)$”.
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 $R = S$ be the sphere spectrum. Then for $f \colon b \to gl_1(S)$ a cocycle for an $S$-bundle,
is the classifying map for a spherical fibration over $B \in Top$.
The Thom spectrum $M f$ of def. is equivalent to the Thom spectrum of the spherical fibration, according to def. .
This is in (ABGHR, section 8).
Equivalently the Thom spectrum is characterized as follows:
For $\chi \colon X \to R Line$ a map to the ∞-group of $R$-(∞,1)-lines inside $R Mod$, 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 is the $R$-(∞,1)-module of sections of the (∞,1)-module bundle classified by $X \stackrel{\chi}{\to} Pic(R) \hookrightarrow R Mod$.
By the universal property of the (∞,1)-colimit we have for $\underline{R} \colon X \to R Mod$ the trivial $R$-bundle that
The section/Thom spectrum functor is the left (∞,1)-Kan extension of the canonical embedding $R Line \hookrightarrow R Mod$ along the (∞,1)-Yoneda embedding
(where the equivalence of (∞,1)-categories on the right is given by the (∞,1)-Grothendieck construction). In other words, it is the essentially unique (∞,1)-colimit-preserving (∞,1)-functor $\infty Grpd_{/ R Line} \to R Mod$ which restricts along this inclusion to the canonical embedding.
This observation appears as (Wilson 13, prop. 4.4).
The universal Thom spectrum, def. , 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. gives morphisms
that combine to a functor with smash products and hence give $M O$ the structure of a ring spectrum.
More abstractly, the sufficient condition for a Thom spectrum of an ∞-module bundle (as above) to have an E-∞ ring structure is that it arises as the (∞,1)-colimit of a homomorphism of E-∞ spaces $B \to A Line$ (ABG 10, prop.6.21).
The cobordism group of unoriented $n$-dimensional manifolds is naturally isomorphic to the $n$th homotopy group of the Thom spectrum $M O$. That is, there is a natural isomorphism
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 $\Theta : \Omega_n^{un} \to \pi_n M O$.
Given a class $[X] \in \Omega_n^{un}$ we can choose a representative $X \in$ SmthMfd and a closed embedding $\nu$ of $X$ into the Cartesian space $\mathbb{R}^{n+k}$ of sufficiently large dimension. By the tubular neighbourhood theorem $\nu$ factors as the embedding of the zero section into the normal bundle $N_\nu$ followed by an open embedding of $N_\nu$ into $\mathbb{R}^{n+k}$
Now use the Pontrjagin-Thom construction to produce an element of the homotopy group first in the Thom space $Th(N_\nu)$ of $N_\nu$ and then eventually in $M O$. To that end, let
be the map into the one-point compactification. Define a map
by sending points in the image of $Disk(N_\nu)$ under $i$ to their preimage, and all other points to the collapsed point $Sphere(N_\nu)$. This defines an element in the homotopy group $\pi_{n+k}(Th(N_\nu))$.
To turn this into an element in the homotopy group of $M O$, notice that since $N_\nu$ is a vector bundle of rank $k$, it is the pullback by a map $\mu$ of the universal rank $k$ vector bundle $\gamma_k \to B O(k)$
By forming Thom spaces the top map induces a map
Its composite with the map $t$ constructed above gives an element in $\pi_{n+k} M O(k)$
and by $\pi_{n+k} M O(k) \to {\lim_\to}_k \pi_{n+k} M O(k) =: \pi_n M O$ this is finally an element
We show now that this element does not depend on the choice of embedding $\nu : X \to \mathbb{R}^{n+k}$.
(…)
Finally, to show that $\Theta$ is an isomorphism we proceed by constructing an inverse.
For that, observe that the sphere $S^{n+k}$ is a compact topological space and in fact a compact object in Top. This implies that every map $f$ from $S^{n+k}$ into the filtered colimit
factors through one of the terms as
By Thom's transversality theorem we may find an embedding $j : Gr_k(\mathbb{R}^s) \to Th(\gamma^k_s)$ by a transverse map to $f$. Define then $X$ to be the pullback
We check that this construction provides an inverse to $\Theta$.
(…)
The homotopy equivalence $\Omega^\infty M O \simeq \vert Cob_\infty \vert$ is the content of the Galatius-Madsen-Tillmann-Weiss theorem, and is now seen as a part of the cobordism hypothesis theorem.
Write Spec for the category of spectra and $Ho(Spec)$ for its standard homotopy category: the stable homotopy category. By the symmetric monoidal smash product of spectra this becomes a monoidal category.
For $X$ any topological space, we may regard it as an object in $Ho(Spec)$ by forming its suspension spectrum $\Sigma^\infty_+ X$. We may ask under which conditions on $X$ this is a dualizable object with respect to the smash-product monoidal structure.
It turns out that a sufficient condition is that $X$ a closed smooth manifold or more generally a compact Euclidean neighbourhood retract. In that case $Th(N X)$ – 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 $\Sigma^\infty_+ X$.
Using this one shows that the trace of the identity on $\Sigma^\infty_+ X$ in $Ho(Spec)$ – the categorical dimension of $\Sigma^\infty_+ X$ – is the Euler characteristic of $X$.
For a brief exposition see (PontoShulman, example 3.7). For more see at Spanier-Whitehead duality.
The J-homomorphism is a canonical map $B O \to B gl_1(\mathbb{S})$ 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 $B O$ 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.
The geometric realization for the (infinity,n)-category of cobordisms for $n \to \infty$ 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.
flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:
bordism theory$\;$M(B,f) (B-bordism):
relative bordism theories:
global equivariant bordism theory:
algebraic:
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
Further original articles include
Textbook accounts include
Frank Adams, part III, section 2 of Stable homotopy and generalised homology, 1974
Stanley Kochman, section 1.5 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Lecture notes include
John Francis, Topology of manifolds course notes (2010) (web)
Lecture 3 Thom’s theorem (notes by A. Smith) (pdf)
A remark of the relation of the Thom spectrum to (∞,n)-category of cobordisms for $n = \infty$ is in:
Lecture 2 Cobordisms (notes by Owen Gwilliam) (pdf)
Johannes Ebert, A lecture course on Cobordism Theory, 2012 (pdf)
Mark Hovey, Brooke Shipley, Jeff Smith, example 6.2.3 of Symmetric spectra, J. Amer. Math. Soc. 13 (2000), 149-208 (arXiv:math/9801077)
Christian Schlichtkrull, Thom Spectra that Are Symmetric Spectra, Documenta Mathematica 14 (2009) 699-748 (arXiv:0811.0592)
Stefan Schwede, Example I.2.6 in Symmetric spectra, 2012 (pdf)
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
See also
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
which is reviewed in
and in the context of motivic quantization via pushforward in twisted generalized cohomology in section 3.1 of
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
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