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.


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…)