January 23, 2013

The Adams conjecture I

Posted by Akhil Mathew under topology | Tags: , , , |
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The Sullivan conjecture on maps from classifying spaces originated in Sullivan’s work on localizations and completions in topology, which together with étale homotopy theory led him to a proof of the classical Adams conjecture. The purpose of this post is to briefly explain the conjecture; in the next post I’ll discuss Sullivan’s proof.

1. Vector bundles and spherical fibrations

Let {X} be a finite CW complex. Given a real {n}-dimensional vector bundle {V \rightarrow X}, one can form the associated spherical fibration {S(V) \rightarrow X} with fiber {S^{n-1}} by endowing {V} with a euclidean metric and taking the vectors of length one in each fiber. The Adams conjecture is a criterion on when the sphere bundles associated to vector bundles are fiber homotopy trivial. It will be stated in terms of the following definition:

Definition 1 Let {J(X)} be the quotient of the Grothendieck group {KO(X)} of vector bundles on {X} by the relation that {V \sim W} if {V , W} have fiber homotopy equivalent sphere bundles (or rather, the group generated by it).

Observe that if {V, W} and {V', W'} have fiber homotopy equivalent sphere bundles, then so do {V \oplus V', W \oplus W'}; for example, this is because the sphere bundle of {V \oplus V'} is the fiberwise join of that of {V} and {V'}. It is sometimes more convenient to work with pointed spherical fibrations instead: that is, to take the fiberwise one-point compactification {S^V} of a vector bundle {V \rightarrow X} rather than the sphere bundle. In this case, the fiberwise join is replaced with the fiberwise smash product; we have

\displaystyle S^{V \oplus W} \simeq S^V \wedge_X S^W,

where {\wedge_X} denotes a fiberwise smash product.

One reason is that this is of interest is that the group {KO(X)} of vector bundles on a space {X} is often very computable, thanks to Bott periodicity which identifies the {KO}-groups of a point. The set of spherical fibrations is much harder to describe: to describe the spherical fibrations over {S^n} essentially amounts to computing a bunch of homotopy groups of spheres.

(more…)

 

December 22, 2012

What is curvature, really?

Posted by Akhil Mathew under category theory, differential geometry | Tags: , , , , |
[2] Comments 

Over the past couple of days I have been brushing up on introductory differential geometry. I’ve blogged about this subject a fair bit in the past, but I’ve never really had a good feel for it. I’d therefore like to make this post, and the next, a “big picture” one, rather than focusing on the technical details.

1. Curvature of a connection

Let {M } be a manifold, and let {V \rightarrow M} be a vector bundle. Suppose given a connection {\nabla} on {V}. This determines, and is equivalent to, the data of parallel transport along each (smooth) curve {\gamma: [0, 1] \rightarrow M}. In other words, for each such {\gamma}, one gets an isomorphism of vector spaces

\displaystyle T_{\gamma}: V_{\gamma(0)} \simeq V_{\gamma(1)}

with certain nice properties: for example, given a concatenation of two smooth curves, the parallel transport behaves transitively. Moreover, a homotopy of curves induces a homotopy of the parallel transport operators.

In particular, if we fix a point {p \in M}, we get a map

\displaystyle \Omega_p M \rightarrow \mathrm{GL}( V_p)

that sends a loop at {p} to the induced automorphism of {V_p} given by parallel transport along it. (Here we’ll want to take {\Omega_p M} to consist of smooth loops; it is weakly homotopy equivalent to the usual loop space.) (more…)