The topic of topological modular forms is a very broad one, and a single blog post cannot do justice to the whole theory. In this section, I’ll try to answer the question as follows: ${\mathrm{tmf}}$ is a higher analog of ${KO}$theory (or rather, connective ${KO}$-theory).

1. What is ${\mathrm{tmf}}$?

The spectrum of (real) ${KO}$-theory is usually thought of geometrically, but it’s also possible to give a purely homotopy-theoretic description. First, one has complex ${K}$-theory. As a ring spectrum, ${K}$ is complex orientable, and it corresponds to the formal group ${\hat{\mathbb{G}_m}}$: the formal multiplicative group. Along with ${\hat{\mathbb{G}_a}}$, the formal multiplicative group ${\hat{\mathbb{G}_m}}$ is one of the few “tautological” formal groups, and it is not surprising that ${K}$-theory has a “tautological” formal group because the Chern classes of a line bundle ${\mathcal{L}}$ (over a topological space ${X}$) in ${K}$-theory are defined by

$\displaystyle c_1( \mathcal{L}) = [\mathcal{L}] - [\mathbf{1}];$

that is, one uses the class of the line bundle ${\mathcal{L}}$ itself in ${K^0(X)}$ (modulo a normalization) to define.

The formal multiplicative group has the property that it is Landweber-exact: that is, the map classifying ${\hat{\mathbb{G}_m}}$,

$\displaystyle \mathrm{Spec} \mathbb{Z} \rightarrow M_{FG},$

from ${\mathrm{Spec} \mathbb{Z}}$ to the moduli stack of formal groups ${M_{FG}}$, is a flat morphism. (more…)

The Atiyah-Segal completion theorem calculates the ${K}$-theory of the classifying space ${BG}$ of a compact Lie group ${G}$. Namely, given such a ${G}$, we know that there is a universal principal ${G}$-bundle ${EG \rightarrow BG}$, with the property that ${EG}$ is contractible. Given a ${G}$-representation ${V}$, we can form the vector bundle

$\displaystyle EG \times_G V \rightarrow BG$

via the “mixing” construction. In this way, we get a functor

$\displaystyle \mathrm{Rep}(G) \rightarrow \mathrm{Vect}(BG),$

and thus a homomorphism from the (complex) representation ring${R(G)}$ to the ${K}$-theory of ${BG}$,

$\displaystyle R(G) \rightarrow K^0(BG).$

This is not an isomorphism; one expects the cohomology of an infinite complex (at least if certain ${\lim^1}$ terms vanish) to have a natural structure of a complete topological group. Modulo this, however, it turns out that:

Theorem (Atiyah-Segal) The natural map ${R(G) \rightarrow K^0(BG)}$ induces an isomorphism from the ${I}$-adic completion ${R(G)_{I}^{\wedge} \simeq K^0(BG)}$, where ${I}$ is the augmentation ideal in ${R(G)}$. Moreover, ${K^1(BG) =0 }$.

The purpose of this post is to describe a proof of the Atiyah-Segal completion theorem, due to Adams, Haeberly, Jackowski, and May. This proof uses heavily the language of pro-objects, which was discussed in the previous post (or rather, the dual notion of ind-objects was discussed). Remarkably, their approach uses this formalism to eliminate almost all the actual computations, by reducing to a special case. (more…)

Let ${f: S^{2n-1} \rightarrow S^n}$ be a map with ${n >1}$. Associated to this, one can form a CW complex ${M_f = D^{2n} \cup_f S^n}$; that is, we attach a ${2n}$-cell to ${S^n}$ via the map ${f}$. This CW complex has one cell in dimension ${n}$ and one cell in dimension ${2n}$ (and one cell in dimension ${0}$). The map ${D^{2n} \rightarrow M_f}$ determines a generator ${\iota_{2n}}$ of ${H^{2n}(M_f; \mathbb{Z})}$ and the map ${S^n \rightarrow M_f}$ determines a generator ${\iota_n}$ of ${H^{n}(M_f; \mathbb{Z})}$; there are no other elements in cohomology other than the unit. Consequently, we have

$\displaystyle \iota_n^2 = a \iota_{2n} , \quad a \in \mathbb{Z}.$

Definition 1 The number ${a}$ as above such that ${\iota_n^2 = a \iota_{2n}}$ is the Hopf invariant of ${f}$.

The homotopy type of ${M_f}$ determines only on the homotopy class of ${f}$, so the Hopf invariant is a homotopy invariant.

Example 1 The Hopf fibration ${f: S^3 \rightarrow S^2}$ is, by definition, the map such that the mapping cone ${M_f}$ is ${\mathbb{CP}^2}$; it follows that the Hopf fibration has Hopf invariant one.

The Hopf invariant is clearly identically zero for ${n}$ odd, but when ${n}$ is even the Hopf invariant is never identically zero; in fact, it defines a homomorphism

$\displaystyle \pi_{2n-1}(S^n) \rightarrow \mathbb{Z},$

which for ${n}$ even has image containing the even integers. (This is where the exceptional ${\mathbb{Z}}$ summand in the homotopy groups of spheres comes from.)

A classical problem in topology was the following:

Question: For which ${n}$ does there exist a map of Hopf invariant one? (more…)

Let ${M}$ be a compact manifold, ${E, F}$ vector bundles over ${M}$. Last time, I sketched the definition of what it means for a differential operator

$\displaystyle D: \Gamma(E) \rightarrow \Gamma(F)$

to be elliptic: the associated symbol

$\displaystyle \sigma(D): \pi^* E \rightarrow \pi^* F, \quad \pi: T^* X \rightarrow X$

was required to be an isomorphism outside the zero section. The goal of the index theorem is to use this symbol ${\sigma(D)}$ to compute the index of ${D}$, which we saw last time was a well-defined number

$\displaystyle \mathrm{index} D = \dim \ker D - \dim \mathrm{coker} D \in \mathbb{Z}$

invariant under continuous perturbations of ${D}$ through elliptic operators (by general facts about Fredholm operators).

The main observation is that ${D}$, in virtue of its symbol, determines an element of ${K(TX)}$. (Henceforth, we shall identify the tangent bundle ${TX}$ with the cotangent bundle ${T^*X}$, by choice of a Riemannian metric; the specific metric is not really important since ${K}$-theory is a homotopy invariant.) In fact, we have that ${K(TX)}$ is the (reduced) ${K}$-theory of the Thom space, so it is equivalently ${K(BX, SX)}$ for ${BX}$ the unit ball bundle and ${SX}$ the unit sphere bundle. But we have seen that to give an element of ${K(BX, SX)}$ is the same as giving a pair of vector bundles on ${BX}$ together with an isomorphism on ${SX}$, modulo certain relations.

Observation: The symbol of an elliptic operator determines an element in ${K(TX)}$. (more…)

The following topic came up in a discussion with my mentor recently. Since the material is somewhat general and well-known, but relevant to my project area, I decided to write this post partially to help myself understand it better.

Definition

Consider an abelian category ${\mathbf{A}}$. Then:

Definition 1 The Grothendieck group of ${\mathbf{A}}$ is the abelian group ${K(\mathbf{A})}$ defined via generators and relations as follows: ${K(\mathbf{A})}$ is generated by symbols ${[M]}$ for each ${M \in \mathbf{A}}$, and by relations ${[M] - [M'] - [M'']}$ for each exact sequence

$\displaystyle 0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0.\ \ \ \ \ (1)$

Note here that if ${M,N}$ are isomorphic, then ${[M] = [N]}$ in ${K(\mathbf{A})}$ by considering the exact sequence

$\displaystyle 0 \rightarrow M \rightarrow N \rightarrow 0 \rightarrow 0.$

The Grothendieck group has an important universal property: (more…)