Tagged Questions

2
votes
1answer
223 views

K-theory as a generalized cohomology theory

Which of the statements is wrong: a generalized cohomology theory (on well behaved topological spaces) is determined by its values on a point reduced complex $K$-theory $\tilde K …
16
votes
3answers
332 views

What are surprising examples of Model Categories?

Background Model categories are an axiomization of the machinery underlying the study of topological spaces up to homotopy equivalence. They consist of a category $C$, together w …
6
votes
2answers
161 views

Relation between $KO$ and $K$

What can be said about the relation between the complex and the real K-theory of a CW complex? An $n$-dimensional complex vector bundle is an $2n$-dimensional real vector bundle bu …
10
votes
3answers
346 views

How do you relate the number of independent vector fields on spheres and Bott Periodicity for real K-Theory?

The theory of Clifford algebras gives us an explicit lower bound for the number of linearly independent vector fields on the $n$-sphere, and Adams proved that this is actually alwa …
7
votes
1answer
132 views

$K_0$ of a non-separated scheme

This question is on "computing" the Grothendieck group of the projective $n$-space with $m$ origins ($m\geq 1$). For any (noetherian) scheme $X$, let $K_0(X)$ be the Grothendieck g …
2
votes
2answers
138 views

Family of Enriques surfaces and GRR, Part 2

As I mentioned in my previous post, I am studying the article Moduli of Enriques surfaces and Grothendieck-Riemann-Roch. The Grothendieck-Riemann-Roch theorem is applied there to …
1
vote
0answers
52 views

The space of spectral sections and connections to K-theory

We look at a familiy $D_\alpha$ of Dirac operators over a (compact) base space B. The projection $\Pi^+_\alpha$ onto the positive eigenspaces of $D_\alpha$ is usually not continuou …
9
votes
4answers
211 views

Why does one invert $G_m$ in the construction of the motivic stable homotopy category?

Morel and Voevodsky construct the motivic stable homotopy category, a category through which all cohomology theories factor and where they are representable, by starting with a cat …
5
votes
1answer
108 views

Products and the skeletal filtration in K-theory

Given a finite CW complex X, there is a filtration of the topological K-theory of X given by setting $K_n(X) = \ker \left(K(X) \to K(X^{(n-1)})\right)$, where $X^{(n-1)}$ is the (n …
5
votes
1answer
113 views

BU with tensor product H-space structure

Hi, I came across the space $BU_\otimes$ when struggling with twisted K-theory. Segal proved that this is an H-space, right? I have read a dozen times by now that the group $[X, …
5
votes
1answer
199 views

Freedman’s work on non-simply-connected 4-manifolds

In the late 1970's and in the 1980's, Michael Freedman showed a relationship between the topological surgery problem in 4-dimensions, the slice problem for links, and the classific …
57
votes
3answers
1k views

Proofs of Bott periodicity

K-theory sits in an intersection of a whole bunch of different fields, which has resulted in a huge variety of proof techniques for its basic results. For instance, here's a scatt …
4
votes
2answers
180 views

Understanding the product in topological K-theory

I apologize that this is perhaps not adequate for mathoverflow but I have struggled with this for days now and become desperate... The reduced K-group $\tilde{K}(S^0)$ of the zero …
3
votes
0answers
63 views

Q-construction and Gabriel-Zisman Localization

It might be a stupid question. When I took a look at the definition of Q-construction. It makes for an exact category $P$, one defines a new category $QP$ whose objects are the s …
6
votes
1answer
186 views

Quantum equivariant $K$-theory and DAHA.

Theorem 3.2 of the paper "Quantum cohomology of the Springer resolution" by Braverman, Maulik and Okounkov relates equivariant quantum cohomology of the cotangent bundle of $G/B$ …

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