13
votes
1answer
524 views

Adams' theorems on the Hopf-Whitehead J-homomorphism

The J-homomorphism is a well-known and classical map $\pi_n (O(k)) \to \pi_{n+k} (S^k)$, or after stabilizing with respect to $k$, a map $J_n:\pi_n (O) \to \pi_{n}^{st}$, from the stable homotopy of ...
2
votes
1answer
161 views

Chern Character Isomorphism for non-finite CW complexes, resp. for non-CW complexes

This is a question I asked at Math.SE but got no answers: http://math.stackexchange.com/q/397164/7110/ Atiyah and Hirzebruch showed in their paper "Vector bundles and homogeneous spaces" that ...
4
votes
4answers
512 views

Characteristic classes detecting nontrivial fiberwise homotopy of sphere bundles

I am looking for characteristic classes of vector bundles (either real or complex) with values in generalized multiplicative cohomology theories such that: i) they vanish if the bundle of unit ...
3
votes
0answers
308 views

K-theory of differential graded modules over differential graded algebras

Suppose you have a smooth vector bundle $E$ over a smooth manifold $X$. If you consider the algebra $ \Omega^\ast (E)$ of differential forms on $E$, it will be homotopy equivalent to the algebra of ...
4
votes
1answer
617 views

Chern character of the index bundle for a family of Dirac operators

Suppose we have a family of compact oriented even dimensional spin manifolds $\{Y_x\}$ parameterized by a compact even dimensional manifold $X$. The $Y_x$'s are all diffeomorphic to some $Y$, of ...
24
votes
1answer
1k views

A question about the topological proofs of Bott periodicity

There are purely topological proof of Botts periodicity theorem, the first one given by Dyer and Lashof. I am heading to discuss the proof in my lecture course on homotopy theory (as a final chord ...
46
votes
3answers
4k views

Intuitive explanation for the Atiyah-Singer index theorem

My question is related to the question Explanation for the Chern Character to this question about Todd classes, and to this question about the Atiyah-Singer index theorem. I'm trying to learn the ...
5
votes
0answers
340 views

Attaching maps for Grassmann manifolds

The Grassmannian $G_n(\mathbb{R}^k)$ of n-planes in $\mathbb{R}^k$ has a CW-complex structure coming from the Schubert cell decomposition. What is known about the attaching maps in this CW-complex ...
25
votes
3answers
4k views

Explanation for the Chern character

The Chern character is often seen as just being a convenient way to get a ring homomorphism from K-theory to (ordinary) cohomology. The most usual definition in that case seems to just be to define ...