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:
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For $E$ a homotopy commutative ring spectrum, there is a bijection between complex orientations on $E$ and homotopy ring spectrum homomorphism $MU \longrightarrow E$ from MU.
Hence $MU$ is the universal complex oriented cohomology theory.
(e.g Lurie 10, lect. 6, theorem 8, Ravenel, chapter 4, lemma 4.1.13)
We discuss that for $E$ a complex oriented cohomology theory, the $n$th universal Conner-Floyd-Chern class $c^E_n$ is in fact a universal Thom class for rank $n$ complex vector bundles. On the one hand this says that the choice of a complex orientation on $E$ indeed universally orients all complex vector bundles. On the other hand, we interpret this fact below as the unitality condition on a homomorphism of homotopy commutative ring spectra $M U \to E$ which represent that universal orienation.
For $n \in \mathbb{N}$, the fiber sequence (prop.)
exhibits $B U(n-1)$ as the sphere bundle of the universal complex vector bundle over $B U(n)$.
When exhibited by a fibration, here the vertical morphism is equivalently the quotient map
(by the proof of this prop.).
Now the universal principal bundle $E U(n)$ is (def.) equivalently the colimit
Here each Stiefel manifold/coset spaces $U(k)/U(k-n)$ is equivalently the space of (complex) $n$-dimensional subspaces of $\mathbb{C}^k$ that are equipped with an orthonormal (hermitian) linear basis. The universal vector bundle
has as fiber precisely the linear span of any such choice of basis.
While the quotient $U(k)/(U(n-k)\times U(n))$ (the Grassmannian) divides out the entire choice of basis, the quotient $U(k)/(U(n-k) \times U(n-1))$ leaves the choice of precisly one unit vector. This is parameterized by the sphere $S^{2n-1}$ which is thereby identified as the unit sphere in the respective fiber of $E U(n) \underset{U(n)}{\times} \mathbb{C}^n$.
In particular:
(zero-section into Thom space of universal line bundle is weak equivalence)
The zero-section map from the classifying space $B U(1) \simeq \mathbb{C}P^\infty$ (the infinite complex projective space) to the Thom space of the universal complex line bundle (the tautological line bundle on infinite complex projective space) is a weak homotopy equivalence
(e.g. Adams 74, Part I, Example 2.1)
Observe that the circle group $U(1)$ is naturally identified with the unit sphere in $\mathbb{C}$: $U (1) \simeq S(\mathbb{S})$. Therefore the sphere bundle of the universal complex line bundle is equivalently the $U(1)$-universal principal bundle
But the universal principal bundle is contractible
(Alternatively this is the special case of lemma for $n = 1$.)
Therefore the Thom space of the universal complex line bundle is:
For $E$ a generalized cohomology theory, the $E$-reduced cohomology of the Thom space of the complex universal vector bundle is equivalently the $E$-relative cohomology of $B U(n)$ relative $B U(n-1)$:
If $E$ is equipped with the structure of a complex oriented cohomology theory then
where the $c_i$ are the universal $E$-Conner-Floyd-Chern classes.
Regarding the first statement:
In view of lemma and using that the disk bundle is homotopy equivalent to the base space we have
Regarding the second statement: the Conner-Floyd-Chern classes freely generate the $E$-cohomology of $B U(n)$ for all $n$ (prop.):
and the restriction morphism
projects out $c_n^E$. Since this is in particular a surjective map, the relative cohomology $E^\bullet( B U(n), B U(n-1) )$ is just the kernel of this map.
Let $E$ be a complex oriented cohomology theory. Then the $n$th $E$-Conner-Floyd-Chern class
(using the identification of lemma ) is a Thom class in that its restriction to the Thom space of any fiber is a suspension of a unit in $\pi_0(E)$.
(e.g. Tamaki-Kono 06, p. 61, Lurie 10, lecture 5, prop. 6)
Since $B U(n)$ is connected, it is sufficient to check the statement over the base point. Since that fixed fiber is canonically isomorphic to the direct sum of $n$ complex lines, we may equivalently check that the restriction of $c^E_n$ to the pullback of the universal rank $n$ bundle along
satisfies the required condition. By the splitting principle, that restriction is the product of the $n$-copies of the first $E$-Conner-Floyd-Chern class
Hence it is now sufficient to see that each factor restricts to a unit on the fiber, but that is precisely the condition that $c_1^E$ is a complex orientation of $E$. In fact by def. the restriction is even equal to $1 \in \pi_0(E)$.
For the present purpose:
For $E$ a generalized cohomology theory, a complex orientation on $E$ is a choice of element
in the cohomology of the classifying space $B U(1)$ (given by the infinite complex projective space) such that its image under the restriction map
is the unit
Often one just requires that $\phi(c_1^E)$ is a unit, i.e. an invertible element. However we are after identifying $c_1^E$ with the degree-2 component $M U(1) \to E_2$ of homtopy ring spectrum morphisms $M U \to E$, and by unitality these necessarily send $S^2 \to M U(1)$ to the unit $\iota_2 \;\colon\; S^2 \to E$ (up to homotopy).
Let $E$ be a homotopy commutative ring spectrum (def.) equipped with a complex orientation (def. ) represented by a map
Write $\{c^E_k\}_{k \in \mathbb{N}}$ for the induced Conner-Floyd-Chern classes. Then there exists a morphism of $S^2$-sequential spectra (def.)
whose component map $M U_{2n} \longrightarrow E_{2n}$ represents $c_n^E$ (under the identification of lemma ), for all $n \in \mathbb{N}$.
Consider the standard model of MU as a sequential $S^2$-spectrum with component spaces the Thom spaces of the complex universal vector bundle
Notice that this is a CW-spectrum (def., lemma).
In order to get a homomorphism of $S^2$-sequential spectra, we need to find representatives $f _{2n} \;\colon\; M U_{2n} \longrightarrow E_{2n}$ of $c^E_n$ (under the identification of lemma ) such that all the squares
commute strictly (not just up to homotopy).
To begin with, pick a map
that represents $c_0 = 1$.
Assume then by induction that maps $f_{2k}$ have been found for $k \leq n$. Observe that we have a homotopy-commuting diagram of the form
where the maps denoted $c_k$ are any representatives of the Chern classes of the same name, under the identification of lemma . Here the homotopy in the top square exhibits the fact that $c_1^E$ is a complex orientation, while the homotopy in the bottom square exhibits the Whitney sum formula for Chern classes (prop.).
Now since $M U$ is a CW-spectrum, the total left vertical morphism here is a Serre-cofibration, hence a Hurewicz cofibration, hence satisfies the homotopy extension property. This means precisely that we may find a map $f_{2n+1} \colon M U_{2(n+1)} \longrightarrow E_{2(n+1)}$ homotopic to the given representative $c_{n+1}$ such that the required square commutes strictly.
For $E$ a complex oriented homotopy commutative ring spectrum, the morphism of spectra
that represents the complex orientation by lemma is a homomorphism of homotopy commutative ring spectra.
(Lurie 10, lecture 6, prop. 6)
The unitality condition demands that the diagram
commutes in the stable homotopy category $Ho(Spectra)$. In components this means that
commutes up to homotopy, hence that the restriction of $c_n$ to a fiber is the $2n$-fold suspension of the unit of $E_{2n}$. But this is the statement of prop. : the Chern classes are universal Thom classes.
Hence componentwise all these triangles commute up to some homotopy. Now we invoke the Milnor sequence for generalized cohomology of spectra (prop.). Observe that the tower of abelian groups $n \mapsto E^{n_1}(S^n)$ is actually constant (suspension isomorphism) hence trivially satisfies the Mittag-Leffler condition and so a homotopy of morphisms of spectra $\mathbb{S} \to E$ exists as soon as there are componentwise homotopies (cor.).
Next, the respect for the product demands that the square
commutes in the stable homotopy category $Ho(Spectra)$. In order to rephrase this as a condition on the components of the ring spectra, regard this as happening in the homotopy category $Ho(OrthSpec(Top_{cg}))_{stable}$ of the model structure on orthogonal spectra, which is equivalent to the stable homotopy category (thm.).
Here the derived symmetric monoidal smash product of spectra is given by Day convolution (def.) and maps out of such a product are equivalently as in the above diagram is equivalent (cor.) to a suitably equivariant collection diagrams of the form
where on the left we have the standard pairing operations for $M U$ (def.) and on the right we have the given pairing on $E$.
That this indeed commutes up to homotopy is the Whitney sum formula for Chern classes (prop.).
Hence again we have componentwise homotopies. And again the relevant Mittag-Leffler condition on $n \mapsto E^{n-1}((MU \wedge MU)_n)$-holds, by the nature of the universal Conner-Floyd classes (prop.). Therefore these componentwise homotopies imply the required homotopy of morphisms of spectra (cor.).
Let $E$ be a homotopy commutative ring spectrum (def.). Then the map
which sends a homomorphism $c$ of homotopy commutative ring spectra to its component map in degree 2, interpreted as a class on $B U(1)$ via lemma , constitutes a bijection from homotopy classes of homomorphisms of homotopy commutative ring spectra to complex orientations (def. ) on $E$.
(Lurie 10, lecture 6, theorem 8)
By lemma and lemma the map is surjective, hence it only remains to show that it is injective.
So let $c, c' \colon M U \to E$ be two morphisms of homotopy commutative ring spectra that have the same restriction, up to homotopy, to $c_1 \simeq c_1'\colon M U_2 \simeq B U(1)$. Since both are homotopy ring spectrum homomophisms, the restriction of their components $c_n, c'_n \colon M U_{2n} \to E_{2 n}$ to $B U(1)^{\wedge^n}$ is a product of $c_1 \simeq c'_1$, hence $c_n$ becomes homotopic to $c_n'$ after this restriction. But by the splitting principle this restriction is injective on cohomology classes, hence $c_n$ itself ist already homotopic to $c'_n$.
It remains to see that these homotopies may be chosen compatibly such as to form a single homotopy of maps of spectra
This follows due to the existence of the Milnor short exact sequence of the form
(prop.).
Here the Mittag-Leffler condition is clearly satisfied (by lemma all relevant maps are epimorphisms). Hence the lim^1-term vanishes, and so by exactness the canonical morphism
is an isomorphism. This says that two homotopy classes of morphisms $M U \to E$ are equal precisely already if all their component morphisms are homotopic (represent the same cohomology class).
The analogous universal finite-rank complex orientation on MΩΩSU(n): Hopkins 84, Prop. 1.2.1.
Frank Adams, part II, lemma 4.6, example 4.7 in: Stable homotopy and generalised homology, 1974
Doug Ravenel, chapter 4 of: Complex cobordism and stable homotopy groups of spheres, 1986
Stanley Kochman, section 4.4 of: Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Dai Tamaki, Akira Kono, Section 3.7 of: Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (ISBN: 978-0-8218-3514-2)
Jacob Lurie, MU and complex orientations (pdf), Lecture 6 in: Chromatic Homotopy Theory, Lecture series 2010 (web),
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