May 29, 2013
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: is a higher analog of –theory (or rather, connective -theory).
1. What is ?
The spectrum of (real) -theory is usually thought of geometrically, but it’s also possible to give a purely homotopy-theoretic description. First, one has complex -theory. As a ring spectrum, is complex orientable, and it corresponds to the formal group : the formal multiplicative group. Along with , the formal multiplicative group is one of the few “tautological” formal groups, and it is not surprising that -theory has a “tautological” formal group because the Chern classes of a line bundle (over a topological space ) in -theory are defined by
that is, one uses the class of the line bundle itself in (modulo a normalization) to define.
The formal multiplicative group has the property that it is Landweber-exact: that is, the map classifying ,
from to the moduli stack of formal groups , is a flat morphism. (more…)
July 26, 2012
The Atiyah-Segal completion theorem calculates the -theory of the classifying space of a compact Lie group . Namely, given such a , we know that there is a universal principal -bundle , with the property that is contractible. Given a -representation , we can form the vector bundle
via the “mixing” construction. In this way, we get a functor
and thus a homomorphism from the (complex) representation ring to the -theory of ,
This is not an isomorphism; one expects the cohomology of an infinite complex (at least if certain 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 induces an isomorphism from the -adic completion , where is the augmentation ideal in . Moreover, .
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…)
January 4, 2012
Let be a compact manifold, vector bundles over . Last time, I sketched the definition of what it means for a differential operator
to be elliptic: the associated symbol
was required to be an isomorphism outside the zero section. The goal of the index theorem is to use this symbol to compute the index of , which we saw last time was a well-defined number
invariant under continuous perturbations of through elliptic operators (by general facts about Fredholm operators).
The main observation is that , in virtue of its symbol, determines an element of . (Henceforth, we shall identify the tangent bundle with the cotangent bundle , by choice of a Riemannian metric; the specific metric is not really important since -theory is a homotopy invariant.) In fact, we have that is the (reduced) -theory of the Thom space, so it is equivalently for the unit ball bundle and the unit sphere bundle. But we have seen that to give an element of is the same as giving a pair of vector bundles on together with an isomorphism on , modulo certain relations.
Observation: The symbol of an elliptic operator determines an element in . (more…)
July 12, 2009
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 . Then:
Definition 1 The Grothendieck group of is the abelian group defined via generators and relations as follows: is generated by symbols for each , and by relations for each exact sequence
Note here that if are isomorphic, then in by considering the exact sequence
The Grothendieck group has an important universal property: (more…)