Edit: When I say cobordism, I mean oriented cobordism unless stated otherwise. Also note that I accidentally flip-flopped \(\Omega^*\) and \(\Omega_*\) — \(\Omega^n\) should be cobordism classes of maps from manifolds of codimension \(n\) to \(X\), and \(\Omega_n\) is cobordism classes of maps from manifolds of dimension \(n\) to \(X\).

Let’s say \(M_1, M_2,\) and \(X\) are differentiable manifolds. We have a map \(f_1\), and a map \(f_2\).

Let’s think of a movie:
Our first frame is the map from \(M_1 \to X\)
Our last frame is the map from \(M_2 \to X\)
What is in between? The instructions for how to deform \(M_1 \to X\) to look like \(M_2 \to X\).

Each movie frame is a map from \(M \to X\), which we can stack up (like a CT-scan — such that the first frame is \(M_1 \to X\), and the last is \(M_2 \to X\)) to get another map, \(W \to X \times [0,1]\) which is “bounded” by the first and last map: \([0,1]\) is our time interval.

If such a \(W\) exists, \(M_1 \to X\) and \(M_2 \to X\) are “cobordant” (same boundary), and \(W\) is their “cobordism” (the manifold that has them as its boundary). It’s just an equivalence relation.

Note that we can also play with cobordism without \(X\) by looking at cobordism classes of \(n\)-dimensional oriented manifolds.

Note that the pair of pants is just an aesthetically pleasing example of a cobordism, and there are many other examples!

Look the collection of maps from \(n\)-dimensional oriented manifolds to \(X\) (up to cobordism of maps)

Equip the collection with connected sum.

Output: \(n\)-dimensional oriented cobordism group (wrt X).

\(\Omega^*(X)\) is the slice category \(\text{Man}/X\), up to cobordism of maps.

Why are the non-trivial groups \(\Omega^{2n}(-)\)? I’ve been told that all manifolds with odd dimension are nilbordant, but I’m not sure I believe this.

But this is just a group! Let’s make it a ring! This particular ring is called “Thom’s ring” in the literature.

Take in a manifold \(X\)

Look the collection of maps from \(n\)-dimensional oriented manifolds to \(X\) (up to cobordism of maps) for all \(n \in \mathbb{N}\)

Equip the collection with connected sum and cartesian product.

Output: oriented cobordism ring (wrt X).

The graded ring \(\Omega^*(-)\) is a cohomology theory (graded by dimension).

Notational aside: \(MU^*(-)\) and \(\Omega^*_{U}\) are alternative notations for \(\Omega^*(-)\) used in the literature.

a ring \(\mathbb{Z}[t]\) (this is just the ring of polynomials with \(t\) as a variable and coefficients in \(\mathbb{Z}\))

an arbitrary ring \(S\).

What is \(\text{hom}(\mathbb{Z}[t], S)\)?

Notice that we’re in Ring, so the members of \(\text{hom}(\mathbb{Z}[t], S)\) must be Ring homomorphisms. Let’s say we’re looking at a ring homomorphism \(\phi: \mathbb{Z}[t] \to S\).

I learned from Cris Moore that demanding examples is a useful practice, so: What is \(\phi(7t^2-4t+3)\) in simpler terms?

\(\phi\) is a ring homomorphism, so we know that \(\phi(1) = 1_s\). This implies that \(\phi(n) = n_s\).

In other words, \(\text{Spec}(\mathbb{Z}[t])\) is a forgetful functor from \(S\) as an object in \(\text{Ring}\) to its underlying set.

Exercise: show that \(\mathbb{Z}[t]\) is an initial object in an appropriate category, and that the trivial ring is the terminal object in the appropriate category.

But what is Spec?

It’s the spectrum of a ring! Intuitively, \(\text{Spec} R\) is the geometric object canonically associated to \(R\).

For example:

\(\text{Spec}(\mathbb{Z})\) is “The Point”

\(\text{Spec}(\mathbb{Z}[x,y]/ (x^2 + y^2 – 1))\) is “The Circle”

\(\text{Spec}(\mathbb{Z}[x,x^{-1}]\) is “Pairs of Invertible Elements” (e.g. a field with the origin removed)

Let’s look at \(\text{Spec}(\mathbb{Z}[x,y]/ (x^2 + y^2 – 1))(S)\) when \(S\) is \(\mathbb{R}\), it’s obviously the circle BUT IT WORKS FOR (almost) ANY S. The concept of there being a canonical geometry associated to every ring is very exciting!

Sidenote:

When people say “elliptic curve” they might mean “a family of elliptic curves”, this is commonly written \(C to \text{Spec} R\) where \(R\) is the underlying coefficient ring.

For example, \(y^2=4x^3+ax+b \mapsto a, b in R\) is the family of elliptic curves of the form \(y^2=4x^3+ax+b\) over the coefficient ring \(R\).

I’m trying to figure out what happens when \(S\) is not something nice like \(\mathbb{R}\) or \(\mathbb{C}\). What is a circle in the coefficient ring of an elliptic curve?

Thanks to Aaron Mazel-Gee for walking me through the concept of \(\text{Spec}\) (all errors in this post are mine and not his).

In singular cohomology, the first chern class of two tensored line bundles \(c_1( A \otimes B) = c_1(A) + c_1(B)\) is the additive formal group law, \(F(x,y) = x + y\).

Quillen was messing with K-theory (another cohomology theory with a notion of chern classes) trying to take the first chern class of two tensored line bundles \(c_1(A \otimes B)\), and realized that that it wasn’t \(c_1(A) + c_1(B)\). There was a more complicated formal group law there!

What do I mean by “more complicated formal group law”?

Milnor had been studying the structure of the coefficient ring of MU and showed that it was isomorphic to a polynomial ring. Quillen recalled this, and realized that complex cobordism was the ‘most general’ way to express the first chern class of the tensor product of two line bundles.

Note that the only connected algebraic groups of dimension equal to 1 over an algebraically closed field \(\mathbb{K}\) are:

The additive group over \(\mathbb{K}\)

The multiplicative group over \(\mathbb{K}\) (as a set, comprises all the non-identity elements of the field).

An elliptic curve group over \(\mathbb{K}\) (the abelian variety case)

What is an elliptic curve group?

You know, I’d really like to be able to “add” two points to get a third.

This is a nice thing, equipping our curve with an addition law gives us identity, its got inverses, and, yes, it’s associative.

How do we get a (1-dimensional) formal group law out of, say, \(y^2 = 4x^3 + ax + b\)? (for the curious, pg. 40-41 of this lecture)

First let’s homogenize \(y^2z = 4x^3 + axz^2 + bz^3\), and check that the point at infinity is smooth (i.e., its Jacobian \(\frac{\partial F_i}{\partial x_i}\) is full rank).

How do we get the elliptic formal group law’s coefficients to be in one dimension? We take a Taylor series expansion of our elliptic curve about the origin. This is commonly denoted \(\hat{C}\).

A Pedagogical Crime

Here’s an intuitive way to define K-theory:

We can think of \(K^0(X)\) as a generalization of the dimension of vector spaces. The key property of dimension is additivity for short exact sequences, so consequently one forms the universal group with that property.

Another way to define K-theory is the following:

A generalized multiplicative cohomology theory \(h^*(-)\) which is complex orientable. That is, there is an \(h\)-theoretic notion of chern class. \(h^*(-)\) is even (\(h^n(*) = 0\) for all odd \(n\)), and weakly periodic (\(h^n(*) \otimes_{h^*(pt)} h^2(*) \simeq h^{n+2}(*)\) for all \(n\)).

MISSING IMAGE
Our cohomology theory \(h^*(-)\) should behave according to the multiplicative formal group law. Let \(\hat{\mathbb{G}}_m\) be the formal completion of the multiplicative formal group law over a coefficient ring \(R\). We require that the coefficient ring of our formal group law and the coefficient ring of our cohomology theory be isomorphic, \(R \simeq h^*(*)\), and the formal group laws over these coefficient rings must also be isomorphic, \(\hat{\mathbb{G}}_m \cong \text{Spf}\) \(h^*(*)[[x]]\), (where \(x\) is the first chern class of a universal line bundle).

What’s this Spf thing? How is \(h^*(pt)[[x]]\) a formal group law? Well, the formal spectrum of a ring R[c] is something that *looks* like localization.

\(Spf R[c]\) :=

In other words, condition 2 that defines K-theory is of the form:
MISSING IMAGE

With this definition, the intimate relationship between K-theory and vector bundles is not immediately apparent. Unfortunately, this is the manner in which we currently define elliptic cohomology…

What is elliptic cohomology?

We’re using the data of an elliptic curve to construct a new way to associate a sequence of abelian groups to spaces, and this new way should behave according to the formal group law of an elliptic curve.

This is how we currently define elliptic spectra:

A family of elliptic curves \(C\) over a coefficient ring \(R\)

A generalized, complex orientable cohomology theory \(h^*(-)\).

Our cohomology theory \(h^*(-)\) should behave according to the elliptic formal group law.Let \(\hat{C}\) be the formal completion of the elliptic formal group law (over a coefficient ring \(R\)). We require that the coefficient ring of our formal group law and the coefficient ring of our cohomology theory be isomorphic, \(R \simeq h^*(*)\), and the formal group laws over these coefficient rings must also be isomorphic, \(\hat{C} \cong \text{Spf}\) \(h^*(*)[[x]]\), (where \(x\) is the first chern class of a universal line bundle).

Why do we define it this way? We don’t know any other way! Well, that’s not quite true…

K-theory is to 1-dimensional field theory (i.e., to each point \(x in X\), associate a vector space \(E_x\), and to each path in \(X\) the connection on \(E\) associates a linear map between these vector spaces) like elliptic cohomology is to 2-dimensional conformal field theory (associating Hilbert spaces to loops in \(X\) and some operators to conformal surfaces with boundary in \(X\)).

(There is a theorem that 1|1 Euclidean field theories are isomorphic to K-theory spectra. )

Relating equivariant versions of elliptic cohomology to loop groups, tmf is proposed to be closely related to supersymmetric conformal field theories.

Elliptic cohomology is a “categorification of K-theory.”

If we think the natural analogy of vector bundles for K-theory is 2-vector bundles for elliptic cohomology, there is the \(K(ku)\) interpretation. This is “like” an elliptic cohomology theory in the sense of detecting \(v_2\)-periodic phenomena, but is not complex orientable (then again, tmf isn’t complex orientable either!).

Some sources/references for the adventurous:

I recommend starting with Landweber’s introduction, which describes how elliptic genera led to elliptic cohomology, then reading Ravenel’s introduction, which leads to chromatic homotopy theory.

Another perspective on a geometric construction is exposited by Baez, which evolved into this construction. These are the product of taking seriously: ‘the key to elliptic cohomology is to study things like vector 2-bundles where the fiber lives in the 2-category not of 2-vector spaces but of bimodules, because the string 2-group has a natural representation in there (Urs Schreiber)’.

For those interested in the field theoretic developments, here are more recent notes on the work of Stolz-Teichner.

If you’re interesting in the physics-y pieces of this, the paper that introduces the Witten genus (which has values in the ring of modular forms over manifolds with rational string structure). A very different physics-y perspective on the categorical side is Loop Space Mechanics and Nonabelian Strings, which contrary to the title is quite beginner friendly.

If you’re into concrete 19th century mathematics, don’t mind reading in French, and REALLY want to get how formal group laws are useful for classification, I recommend reading Lazard’s work on formal group laws Groupes de Lie formels à un paramètre, and his concept of “analyseurs” (the beginning of operads) Groupes analytiques en caractéristique 0.

I noticed an informal “recipe” for taking a type of object and constructing invariants (of the object). It’s been useful for removing the feeling of “what, why? where did that come from?” when learning new constructions that fit this recipe. Hopefully it will help you!

Take in an object

Look at a collection of structures defined over the object (up to isomorphism)

Define a binary operation closed over this collection

Optional: Formally append inverses

Output: a useful algebraic invariant (used to study the object)

Some examples of this, and comments you are welcome to skip over (you can get a good sense of what I’m saying by just looking at the pictures).

This Burnside ring is the analogue of the representation ring in the category of finite sets, as opposed to the category of finite-dimensional vector spaces (over a field \(F\)).

The Segal theorem(proved by Gunnar Carlsson, who is a wonderful human being and patient teacher) is something that I wish to understand, which relates the Burnside ring of a finite group \(G\) to the stable cohomotopy of the classifying space \(BG\).

Let’s fill in the recipe for K-theory:

Take in a space \(M\) (compact Hausdorff to avoid pathologies)

Look at (complex/real) vector bundles over \(M\) (up to iso)

Equip the collection with the fiberwise direct sum, and the fiberwise tensor product of bundles.

Formally append rank \(-n\) vector bundles (a formal entity, defined as an object which, when directly summed with a rank \(n\) vector bundle, reduces to a point)

Output: Topological (complex/real) K-theory

Let’s do an example: What’s the K-theory of a point? Well \(\text{Vect}(pt) \simeq \mathbb{N}\), formally appending inverses gives us \(K^0(X) \simeq \mathbb{Z}\).

It’s worth stating explicitly the relationship between algebraic and topological K-theory. The algebraic K-theory of (a ring of complex valued \(C^\infty\)-maps on a space) = the topological \(K\)-theory of (the space).

Josh Grochow kindly pointed out that the representation ring and topological K-theory are similar for a good reason! Intuitively, one can think of both as “bundles of representations.”

More specifically, assuming \(G\) is finite, one can define the representation ring to be the (\(G\)-equivariant) K-theory of a point.

\(K_G(pt) \simeq R_F(G)\)

In other words, finite dimensional linear virtual \(F\)-representations of \(G\) in \(R_F(G)\) correspond to virtual equivariant bundles over a point. Note that the term “virtual” is short for “isomorphism classes of formal differences of” and that \(R_F(G)\) is sometimes written as \(\text{Rep}(G)\).

(where \(K(BG)\) is the topological \(K\)-theory of the classifying space \(BG\) of \(G\)-principal bundles) is almost an isomorphism, and becomes an isomorphism under p-adic completion.
This fits into the picture of comparing the following two processes:

completing a \(G\)-space by making the action free (a geometrical process)

completing with respect to an ideal (an algebraic process)

Edit: I just began reading Lurie’s Higher Algebra (excerpt below), and it seems that the derived category \(\mathcal{D}(R)\) of a ring \(R\) fits into ‘the recipe’.

Take in the ring \(R\)

Look at the collection of chain complexes of modules over \(R\)

Equip the collection with chain complex homomorphisms (a.k.a chain maps)

Formally make quasi-isomorphisms into isomorphisms

Output: The derived category \(\mathcal{D}(R)\)

Thank you to Semon Rezchikov (for explaining to me what a virtual vector bundle was a few weeks ago, it was a lovely and intuitive description that led me to draw the \(K\)-theory picture above), and to Qiaochu Yuan (for pointing out and correcting the errors in the recipe).

When I define a polynomial, I am simply handing you an indexed collection of coefficients.

A polynomial with two variables, \(x, y\) and coefficients \(c\), is of the form:

\(F(x, y) = \sum\limits_{ij} c_{ij} x^i y^j\)

The coefficients of a polynomial form a ring. In other words, the coefficients \(c_{ij}\) are members of a coefficient ring \(R\). When we say \(F\) is over \(R\), we mean that \(F\) has coefficients in \(R\).

Example: The polynomial
\(F(x,y) = 7 + 5xy^2 + 2x^3\) can be written out as
\(F(x,y) =7x^0y^0 + 5x^1y^2 + 2x^3y^0\) such that
\(c_{00} = 7\), \(c_{12} = 5\), \(c_{30} = 2\), and the rest of \(c_{ij} = 0\).

Alright, now let’s change the coefficients; reassign \(c_{00} = 4\), \(c_{78} = 3\), and all other \(c_{ij} = 0\).

Out pops a very different polynomial \(P(x,y) = 4 + 3x^7y^8\).

In other words, by altering the coefficients \(c_{ij}\) of \(F(x,y)\) via a ring homomorphism \(u: R \to R’\) (from the coefficient ring \(c_{ij} \in R\) to a coefficient ring \(u(c_{ij}) \in R’\))…

… we can get from \(F(x,y)\) to any other polynomial \(F'(x,y)\).

What’s a group-y polynomial?

Intuitively, a polynomial is “group-y” if there’s a constraint on our coefficients that forces the polynomial to satisfy the laws of a commutative group.

Concretely, a group-y polynomial is an operation of the form \(F(x,y) = \sum\limits_{ij}c_{ij}x^iy^j\) such that

We can make sure that our polynomial satisfies these constraints! How? We mod out our coefficient ring \(c_{ij}\) by the ideal \(I\) — generated by the relations among \(c_{ij}\) imposed by these constraints.

If you’d like to see the explicit relations, I wrote a cry for help post on stack overflow.

The ring of coefficients that results is called the Lazard ring \(L = \mathbb{Z}[c_{ij}]/I\).

It’s important to note here that group-y polynomials are morphisms out of the Lazard ring, not elements of the Lazard ring (i.e., that an assignment of values to each of the \(c_{ij}\) describes a group-y polynomial, but the ring of the \(c_{ij}\) itself is just a polynomial ring).

In other words, group-y polynomials \(f(x,y)\) are morphisms out of the Lazard ring, not elements of the Lazard ring.

More formally: for any ring \(R\) with group-y polynomial \(f(x,y) \in R[[x,y]]\) there is a unique morphism \(L \to R\) that sends \(\ell \mapsto f\).

\(L \to R \simeq F_R\)

(where \(F_R\) denotes a group-y polynomial with coefficients in \(R\))

This makes sense. If it doesn’t, then scroll up a bit! As we saw above, a change of base ring corresponds to a new group-y polynomial.

Grading the Lazard Ring

As we’ve noted, the Lazard ring \(L = \mathbb{Z}[c_{ij}]/I\) is the quotient of a polynomial ring on the \(c_{ij}\) by some relations.

Lazard proved that it is also a polynomial ring (no relations) on a different set of generators. More specifically, \(\alpha\) is a graded ring isomorphism:

Thanks to Alex Mennen for deriving constraints the associativity condition puts on our coefficients; thanks to Qiaochu Yuan and Josh Grochow for kindly explaining some basic details and mechanics of the Lazard ring.

For your ventures ahead…

In this post, I have committed two semantic sins in the name of pedagogy. Namely, sins of oversimplification which I’ll attempt to rectify s.t. you aren’t hopelessly confused by the literature:

group-y polynomial = “1-dimensional abelian formal group law”

polynomial = “formal power series”

Conventionally, a “polynomial” is a special case of a formal power series (in which we expect that our variables evaluate to a number – useful if we care about convergence).

polynomials \(\subset\) formal power series

The polynomial ring \(R[x]\) is the ring of all polynomials (in two variables) over a given coefficient ring \(R\).
The ring of formal power series \(R[[x]]\) is the ring of all formal power series (in two variables) over a given coefficient ring \(R\).

polynomial ring \(\subset\) ring of formal power series
R[x] \(\subset\) R[[x]]