All Questions
20 questions
19
votes
1
answer
862
views
Diffeomorphism groups of h-cobordant manifolds
Do we have specific examples of h-cobordant smooth manifolds $M$ and $M'$ such that $\operatorname{BDiff}(M) \not \simeq \operatorname{BDiff}(M')$? Perhaps something can be said in terms of K-theory ...
- 5,839
18
votes
1
answer
1k
views
On the definition of A-theory
Waldhausen's A-theory is a version of algebraic K-theory of spaces. Concretely, for a (pointed) space $X$, he considers the 'Waldhausen category' $\mathcal R_f(X)$ of finite retractive CW-complexes ...
- 11.9k
14
votes
0
answers
225
views
Hauptvermutung for non-manifolds
The Hauptvermutung proposes the following: if two finite simplicial complexes are homeomorphic then they are PL-homeomorphic, meaning that they have a common refinement.
People are mostly interested ...
- 2,050
13
votes
3
answers
2k
views
Is there a good definition of (topological) K-Theory over arbitrary spaces?
Hi
(this is my very first question here, so please don't hurt me...)
for some time now i've been looking for a sufficiently aesthetical definition of (topological) K-theory of arbitrary spaces, yet ...
- 333
11
votes
1
answer
2k
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 $\tilde K$ and reduced real $K$...
- 2,782
10
votes
1
answer
838
views
Homology of infinite loop spaces $QX$
Let $X$ be a simply connected space. By $Q$ I denote $\Omega^{\infty}\Sigma^{\infty}$. Then $QX$ is an infinite loop space and the homology $H(QX)$ in $\mathbb{F}_p$ is a Hopf algebra over the Dyer-...
- 471
7
votes
2
answers
419
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 but not vice versa. ...
7
votes
1
answer
470
views
Twisted spin bordism invariants in 5 dimensions
[Note]: My question will be a bit long. So, first, thank you for your careful reading, generous comments, helps and answers, in advance!
The spin $G$-bordism invariant can be twisted in the way that ...
- 10.5k
6
votes
1
answer
142
views
Example of nonvanishing Waldhausen Nil group
In a remarkable series of papers, both anticipating development in geometric topology and algebraic K-theory, specifically what we call now the Farrell-Jones conjecture, Waldhausen ...
- 2,630
6
votes
0
answers
142
views
Pin cobordism v.s. "KO" theory in low or in any dimensions
Fact: The spin cobordism is equivalent to "KO" theory in low dimension if we only consider the 2-torsion.
This is related to a question and an answer supports the claim.
Here we denote the $p$-...
- 10.5k
6
votes
0
answers
312
views
homotopy domination that splits a non-split epimorphism and still wants to be a homotopy equivalence
Can a homotopy domination by a space supporting a free action of $G$ be promoted to a homotopy equivalence with such a space? As stated, this is not a serious question (multiply by an $EG$). But with ...
- 5,575
5
votes
1
answer
792
views
$K_0$ of integral group ring of cyclic group $\mathbb{Z}/p\mathbb{Z}$
Is there a table for the computation of $K_0(\mathbb{Z}[\mathbb{Z}/p\mathbb{Z}])$?
These groups are also known as ideal class group in number theory.In topology,they are the home of some important ...
- 593
5
votes
1
answer
403
views
covering map from spheres to projective spaces and the associated vector bundle
Let $S^n$ be the $n$-sphere and consider a $2$-sheeted covering
$$
S^n\longrightarrow\mathbb{R}P^n.
$$
We have an associated vector bundle
$$
\xi: \mathbb{R}^2\longrightarrow S^n\times_{\mathbb{Z}/2}\...
- 519
5
votes
1
answer
522
views
What is the ring structure of the complex topological K-theory of a non-singular complex quadric?
I would like to know the ring structure of $K(Q_n)$ explicitly where $Q_n \subset \mathbb{P}^{n+1}$ is the non-singular $n$-dimensional complex quadric and $K(Q_n) = K^0(Q_n)$ is the complex ...
- 1,444
4
votes
0
answers
152
views
How much vanishing of odd K-groups implies the vanishing of odd equivariant K-groups?
The main quetion is
For a compact Lie group $G$, and a $G$-space $X$ with $K^1(X)=0$.
How much can we say about the vanishing of $K_G^1(X)$? Moreover, how much $K^0_G(X)=K^0(X)\times R(G)$?
Here $...
- 719
3
votes
1
answer
470
views
Spectral sequence for H-space bundles
Let $F \rightarrow E \rightarrow B$ be a fibre bundle such that $B$ is a smooth and compact manifold and $F$ obtains an associative H-space structure. Explicitly, it is not a principal bundle.
One ...
- 31
3
votes
1
answer
323
views
self-Whitney sum of the canonical vector bundle on Grassmannians
Let $G_{k}(\mathbb{R}^N)$ be the Grassmannian manifold consisting of $k$-subspaces in $\mathbb{R}^N$. There is a canonical $k$-dimensional vector bundle
$$
\gamma_{k,N}: \mathbb{R}^k\longrightarrow E(...
- 1,990
2
votes
0
answers
175
views
Chern character of finite $CW$-complexes and rational Pontrjagin class of vector bundles
Let $K$ be a finite $CW$-complex. Could you give any references or explanations for the following two items? I do not understand. Thanks!
(1). The Chern character from $\tilde{KO}^0(K)$ to the ...
- 519
2
votes
0
answers
91
views
order of elements in a mapping space
Let $B$ be a finite CW-complex and $\xi$ be a vector bundle over $B$ with structure group $\Sigma_n$, the $n$-th symmetric group.
Then corresponding to $\xi$, we have a classifying map
$$
g\in \tilde {...
- 543
1
vote
2
answers
133
views
free complex with mod-p coefficients
How does one prove the following fact. I could not find anything in literature.
Let $\pi$ be a subgroup of the symmetric group $S_p$ and let $W$ be a free $\pi$-complex. Then for any space $X$ there ...
- 471