All Questions
Tagged with cohomology kt.k-theory-and-homology
37
questions
6
votes
1
answer
413
views
Nilpotency of generalized cohomology
$\newcommand\pt{\mathrm{pt}}$Let $(X,\pt)$ be a connected, pointed, finite CW complex and let $h$ be a generalized cohomology theory. Let $\smash{\tilde{h}}^*(X)$ denote the kernel of restriction $h^*(...
- 757
3
votes
2
answers
145
views
Is there a $ H_* H^* $-theory which is naturally a common generalization both of singular homology and de Rham (or singular) cohomology?
It is known that $K_* K^* $-theory is a common generalization both of $K$-homology and $K$-theory as an additive bivariant functor on separable C*-algebras.
Is it possible to construct a $ H_* H^* $-...
- 595
3
votes
0
answers
82
views
When does homology preserve inverse limits of Eilenberg-MacLane spaces?
Let $... \to G_3 \to G_2 \to G_1$ be an inverse system of abelian groups and $G$ the limit of the system. By a theorem of Goerss the integral homology of the Eilenberg-MacLane space $K(G,n)$ for $n &...
- 499
3
votes
0
answers
80
views
Cohomological counterpart of the K-theory of the Roe C*-algebra in non-periodic systems
In crystalline insulating quantum systems the dynamics of electrons is governed by a Schrödinger operator which is periodic with respect to a Bravais lattice $\Gamma\cong \mathbb{Z}^d$ and whose ...
- 31
3
votes
1
answer
210
views
"High-dimensional" classes in topological $K$-theory
I am looking for a sequence of topological spaces $X_n$, $n\in\mathbb N$, with the following property. Let $\tilde{K}^0(X_n)$ be the complex reduced $K$-theory group of $X_n$ (with respect to some ...
- 1,901
5
votes
2
answers
300
views
Computation of cohomology of Morava $K$-Theory
Let $G$ be an elementary abelian group, so that $G = (\mathbb{Z}/p)^k$ for some $k$. We can then compute the Morava $K$-theory of $BG = (BZ/p)^k$ pretty easily: $K(n)^*(BG) = K(n)^*[[x]]/[p](x) (x)_{K(...
- 448
8
votes
0
answers
179
views
Conner-Floyd Chern classes and $E$-(co)homology of $BU$
In his book, Stable homotopy and generalised homology, Adams computes the $E$-(co)homology of $BU$ for a complex oriented cohomology theory $E$. In II.4, he first describes the $E$-homology of $BU$ as ...
user340793
6
votes
1
answer
475
views
Stable Adams operations
I have come across a paper by Adams, Harris and Switzer on the Hopf algebra of cooperations of real and complex K-theory. The Adams operations are stable in the $p$-local setting, however I have not ...
2
votes
0
answers
138
views
Definition of odd topological K-theory using circles
I wanted to check whether the following characterization of odd complex topological $K$-theory is correct (reposted from Math.SE).
Let $X$ be a compact Hausdorff space. Then $K^{-1}(X)$ can be defined ...
- 1,901
0
votes
1
answer
207
views
Computation of the groups $K(BU \times \mathbb{Z})$ and $H^*(BU \times \mathbb{Z})$
Let $U$ denote the limiting group of the chain $U(1) \to U(2) \to U(3) \to \cdots$
I wish to compute the group $K^{-1}\mathbb{C}/\mathbb{Z}(BU \times \mathbb{Z})$. For this, we have the long exact ...
- 709
8
votes
0
answers
343
views
Proving faithful flatness of a K-theoretic map without the moduli stack of formal groups
I'm in the process of writing an expository paper on complex K-theory and Snaith's theorem; the proof of Snaith's theorem that I'm following along (located at http://math.uchicago.edu/~amathew/snaith....
3
votes
0
answers
106
views
Reference request: $K$-theoretic wrong-way map for a boundary inclusion
Let $W$ be a compact manifold with boundary. Let $i:\partial W\hookrightarrow W$ be the natural inclusion. We have a long exact sequence in complex $K$-theory:
$$\ldots\to K^*(\partial W)\xrightarrow{...
- 1,901
3
votes
2
answers
479
views
Does the Gysin map in $K$-theory respect bordism?
Let $X_1$ and $X_2$ be two closed spin$^c$ manifolds that are bordant via a spin$^c$ manifold-with-boundary $W$.
Let $Z$ be a closed spin$^c$ manifold with $\dim Z=\dim X_1$ mod $2$. Let
$$f_1:X_1\to ...
- 1,901
5
votes
0
answers
288
views
Chern-Weil theory in the cohomological Atiyah-Singer theorem
I am interested in the following piece of data appearing in the cohomological Atiyah-Singer theorem. My reference is "The index of elliptic operators. III" by Atiyah and Singer.
Let $D:\...
- 2,237
11
votes
0
answers
509
views
Chromatic Homotopy Theory and Physics
Chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work relating ...
- 10.4k