All Questions
Tagged with kt.k-theory-and-homology vector-bundles
58 questions
82
votes
3
answers
13k
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 ...
- 22.1k
30
votes
4
answers
2k
views
When can we cancel vector bundles from tensor products?
Let $E,F,G$ be algebraic vector bundles over $\mathbb P_{\mathbb C}^n$. My general question is:
Assume $E\otimes G \cong F\otimes G$, under what conditions can one conclude that $E\cong F$?
Some ...
- 30.5k
25
votes
1
answer
839
views
Vector bundles on $\mathbb{A}^n / G$
Let $G$ be a finite group acting linearly on $\mathbb{A}^n$. Do we expect algebraic vector bundles on $X := \mathbb{A}^n/G$ to be trivial? Here by the quotient I mean the singular scheme, not the ...
- 2,300
18
votes
2
answers
2k
views
K-Theory and the Stack of Vector Bundles
I have some understanding that vector bundles provide a basic, familiar example of what I should call a stack. Namely, consider the functor $Vect$ that assigns to a space $X$ the set of isomorphism ...
- 2,806
17
votes
3
answers
2k
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 always the best possible:...
- 5,530
12
votes
2
answers
2k
views
Is every vector bundle over a noncompact finite-dimensional manifold a summand of a trivial bundle?
In the notes of Vector Bundles and K-theory by Prof Allen Hatcher, on page 12 he proved a Proposition that for each vector bundle $E\to B$ with $B$ compact hausdorff there exists a vector bundle $E'\...
- 3,751
10
votes
1
answer
374
views
Computing K-theory for cellular vector bundles
One of the most computationally convenient properties of singular cohomology $X \mapsto H^\bullet(X;\mathbb{Z})$ is the fact that one can extract it from a good cover $\{U_i\}$ of $X$ via Cech ...
- 15.5k
9
votes
1
answer
327
views
Closed formulas for topological K-theory?
Let $X$ be a compact manifold. I'm interested in whether any of the following cases admits a general closed formula for (complex)-$K$-theory. Let $E$ be a complex vector bundle with a given line ...
- 7,789
9
votes
1
answer
336
views
Positive cones in K-groups
Let $X$ be a topological space or a scheme, and let $K^0(X)$ be $K$-group of vector bundles of $X$. One may ask when an element $x$ of $K^0(X)$ is represented by an actual vector bundle, and not just ...
- 1,274
9
votes
1
answer
514
views
K-theory on finite-dimensional (possibly not finite) CW complexes
I am trying to understand why (at least my most elementary understanding of) topological K-theory breaks down for non-compact things (which I have seen asserted in various places). In particular, as ...
- 1,225
8
votes
1
answer
544
views
Question regarding the paper by Atiyah, Bott and Shapiro: alternative description of K-theory
In Atiyah, Bott, and Shapiro - Clifford modules (journal, MSN), the authors discuss the alternative description of K-theory in terms of sequences of vector bundles. I would like to understand the ...
- 9,330
8
votes
1
answer
612
views
Triviality of direct multiples of flat complex vector bundles
Atiyah Patodi and Singer [Spectral asymmetry and Riemannian geometry III] write that if $E$ is a complex flat bundle (non holomorphic, just smooth and complex) on a compact manifold $X$ (more ...
- 221
7
votes
1
answer
434
views
classifying maps of Whitney sums of vector bundles
For an $n$-dimensional vector bundle $\xi$ with structure group $G\leq O(n)$ over a $CW$-complex $B$, we have a classifying map up to homotopy
$$
f(\xi): B\longrightarrow BG,
$$
$f(\xi)\in [B;BG]$, ...
- 519
7
votes
2
answers
805
views
What are some applications of virtual vector bundles?
K-theory gives a nice way to define vector bundles that don't actually exist. For example, given a singular variety $Y$ embedded into a smooth variety $X$ we can define the virtual normal bundle as
$$
...
- 1,716
7
votes
1
answer
485
views
Non trivial rank 2 holomorphic vector bundles in complex dimensions greater than or equal 2
Does every compact complex manifold of complex dimension greater than or equal two
possess a nontrivial rank 2 holomorphic vector bundle?
- 1,236
7
votes
2
answers
1k
views
Cancellation and splitting theorems for vector bundles etc over schemes
It is not too hard, in the theory of vector bundles over manifolds (or nice topological spaces, say locally contractible with finite covering dimension), to arrive at a splitting theorem. This ...
- 35.5k
7
votes
0
answers
270
views
The Todd class and Weyl's character formula
Let $\mathfrak{g}$ be a finite-dimensional complex semi-simple Lie algebra. Fix a Cartan sub algebra $\mathfrak{h} \subset \mathfrak{g}$ and let $R \subset \mathfrak{h}^{\ast}$ denote the root system. ...
- 1,379
7
votes
0
answers
369
views
Does the suspension isomorphism $K_1(A) \to K_0(SA)$ descend from a more refined invariant?
If $A$ is a C*-algebra, denote its minimal unitization by $\tilde A$ and its suspension by $SA$, thought of as all continuous $a:[0,1] \to A$ with $a(0)=a(1)=0$. The unitized suspension $\widetilde{SA}...
- 662
6
votes
2
answers
981
views
Relative Characteristic classes
A pair of vector bundles over a base space $X$ is a pair $(E,F)$ where $E$ is a vector bundle over $X$ and $F$ is a sub-bundle of $E$. Two pairs $(E_{1},F_{1})$ and $(E_{2}, F_{2})$ are ...
- 356
6
votes
1
answer
509
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-1)-skeleton of X. (...
- 8,803
6
votes
1
answer
752
views
Locally trivializing a G vector bundle?
In §1.6 of Atiyah's K-theory, he defines the notion of a $G$-(vector)-bundle, which is a sort of "equivariant vector bundle" with respect to a finite group action. More specifically, let $G$ ...
- 1,225
5
votes
2
answers
624
views
Topological K-theory for commutative C*-algebras
It is in some sense folklore that given two arbitrary abelian groups $G,H$ one can find a $C^*$ algebra $A$ such that $K_0(A)=G$ and $K_1(A)=H$. My question is the following: what is known in the case ...
- 9,330
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
293
views
Reference for $E_{\infty}$-ness of the Chern Character
I would like a reference/proof for the fact that the Chern character map: $$KU_{\mathbb{Q}} \rightarrow H\mathbb{Q}[u, u^{-1}]$$
is an $E_{\infty}$-ring map. Thank you in advance!
- 2,327
5
votes
1
answer
251
views
Equivalence of families indexes of Fredholm operators
Let $F=F(H,H)$ be the space of bounded Fredholm operators in a Hilbert space $H$ with topology inherited from the norm operator topology, and let $X$ be a compact topological space.
For a continuous ...
- 153
5
votes
1
answer
788
views
K-Theory as a special $\lambda$-ring
I wonder if there is a nice and short proof that the $K$-theory of a topological space is a special $\lambda$-ring without invoking the splitting principle and alike. Is it possible to show directly ...
- 63.1k
5
votes
0
answers
132
views
Riemannian version of topological $K$-theory
Let $X$ be a compact Hausdorff space.Put $Vec(X)$, the space of all real (or complex) vector bundles over $X$.We put also $Vec_g(X)$, the space of all Riemannian vector bundles over $X$, that is the ...
- 356
5
votes
0
answers
130
views
Alternative description of $K$-theory of locally compact spaces using sequences of bundles
In this paper Aityah, Bott and Shapiro give an alternative definition of (relative) $K$-theory groups $K(X,Y)$ using sequences of bundles (this group is denoted by $L_n(X,Y)$ where $n$ is the length ...
- 9,330
4
votes
1
answer
457
views
Pullback of complex vector bundles along a retraction of compact Hausdorff spaces: a direct proof instead?
Consider a pointed compact Hausdorff space $(X,x_0)$ and a closed pointed subspace $i:A\subset X$ such that there exists a continuous map $r:X\rightarrow A$ such that $r|_A=\text{Id}_A$. Set
$$q:(X,...
user490645
4
votes
1
answer
447
views
Totally non parallelizable manifold
Does there exist a manifold M which all iterated tangent bundles are non parallelizable manifolds? That is$ M, TM , T^2(M), \ldots ,T^n(M)\ldots$ are non parallelizable manifold?
What is ...
- 356
4
votes
1
answer
157
views
Geometric Motivation for Hilbert $C^*$-Bimodules
I'm trying to get an understanding of Hilbert $C^*$-bi-modules from a geometric point of view. As is well-known, we have that
i) Commutative unital $C^*$-algebras correspond to compact Hausdorff ...
- 167
4
votes
1
answer
353
views
Isomorphism classes of differential rank $k$ vectors bundles over $S^q$ [closed]
Could anybody provide a motivated sketch of why the isomorphism classes of the differentiable rank $k$ real vector bundles over the sphere $S^q$ are given by$$\text{Vect}_k(S^q) \simeq \pi_{q - 1}(\...
- 369
4
votes
1
answer
282
views
Is there a geometric interpretation of Johnson-Wilson E(n) analogous to vector bundles for K-theory?
I am reading Ravenel's Localization with Respect to Certain Periodic Homology Theories where he states;
For $n\ge2$, the spectra E(n)
represent periodic homology theories which at present have ...
4
votes
1
answer
507
views
A question on complex line bundle over $S^{2}$
Consider the trivial bundle $\epsilon_{2}=S^{2}\times \mathbb{C}^{2}$ with the standard Hermitian inner product $<(a,b), (c,d)>=a\bar{c}+b\bar{d}$.
Assume that $\ell$ is a sub line bundle of ...
- 356
4
votes
1
answer
354
views
Natural extension homomorphism and wrong-way maps in K-theory
Let $X \subset Y$ be two smooth manifolds. To the inclusion $I:X \to Y$ corresponds the so called wrong-way map in $K-theory$ $i_!:K(X) \to K(Y)$. It is constructed as follows: to the inclusion $X \...
- 9,330
4
votes
0
answers
110
views
category of vector bundles with connections and its K-theory
For the category of Hermitian vector bundles with unitary connections, an object is (of course) a Hermitian vector bundle with a Hermitian metric and a unitary connection $(E, g^E, \nabla^E)$. For ...
- 1,173
4
votes
0
answers
81
views
Minimum rank of inverse complex vector bundles
When considering vector bundles (real or complex) over a compact manifold, i know about the existence of inverse bundles. That is, if $\xi$ is a vector bundle over $M$, then there is a bundle $\nu$ ...
- 261
4
votes
0
answers
220
views
Splitting principle for real vector bundles with $w_i=0$, $0<i<2^r$
Is the following true/known?
Let $E$ be a (real) vector bundle over a compact CW-complex $X$.
Suppose that
$w_i(E)=0$ for $0<i<2^r$.
Then
there exist
a space $Y$ and a map $f\colon Y\to X$ ...
- 809
3
votes
1
answer
457
views
Euler characteristics and the difference bundle construction
I am reading on K theory in Lawson and Michelson (Spin Geometry). One has the "exact sequence spaces" $L(X,Y)$ and there is the theorem that there is a unique equivalence of functors $\chi$ between $L$...
- 9,853
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
3
votes
0
answers
214
views
Vector fields on quasi-spheres
In 1962, Adams proved that there do not exist $\rho(n)$ linearly independent vector fields on the sphere $S^{n-1}$, where $\rho(n)$ is the Hurwitz-Radon number. I wonder if this is still true in the ...
- 333
3
votes
0
answers
287
views
Basic Question: K-theory of a sphere bundle
Does anyone know how to compute the topological K-theory of the unit tangent bundle for a compact connected Riemann surface of genus greater than one?
Thank you very much in advance.
- 39
2
votes
2
answers
247
views
Preimage of $1 \in H^n(M^n)$ under Chern character
Let $M$ be a closed, oriented manifold of dimension $n$. We know that the Chern character induces an isomorphism $K^\ast(M) \otimes \mathbb{Q} \cong H^\ast(M; \mathbb{Q})$ and now I was wondering how ...
- 2,998
2
votes
1
answer
400
views
${\rm SL}_2(\mathbb C)$-equivariant K-theory of $\mathbb C P^1$
Consider the action of ${\rm SL}_2(\mathbb C)$ on $\mathbb C P^1$ induced by the action of 2×2 matrices on 2-vectors. Is it true that $K^{{\rm SL}_2(\mathbb C)}(\mathbb C P^1)$ is $\mathbb C[t,t^{-1}]$...
- 2,964
2
votes
2
answers
124
views
Invertible (isometric) sections of certain hom bundles over sphere
Assume that we have a vector bundle $E$ over $S^n$.
Is there a continuous family of invertible linear maps $T_x:E_x \to E_{-x}$?
Here continuity has the obvious meaning as soon as ...
- 356
2
votes
1
answer
266
views
Are these vector bundles, trivial bundle?
We identify the vector space tensor product $\mathbb{R}^{m} \otimes \mathbb{R}^{n}$ with $\mathbb{R}^{mn}$
Let $X$ be the space of all non zero simple tensors $X=\{a\otimes b \mid a\in \...
- 356
2
votes
0
answers
141
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 ...
- 1,903
1
vote
1
answer
331
views
realization map for K-theory of spheres
Let $\overset{\sim}{K}(X)$ and $\overset{\sim}{KO}$ denote the reduced stable isomorphic classes of complex and real bundles over X and $\rho$ be the realization map. We know that $\overset{\sim}{K}(\...
- 141
1
vote
1
answer
322
views
A (possible) equivalent relation on the space of vector bundles
Edit: According to the essential comment of Alex Degtyarev, we revise the question as follows;
Assume that $\alpha$ and $\beta$ are two oriention preserving automorphism of Lie groups $O(n)$ and $...
- 356
1
vote
1
answer
128
views
Integration from vector bundles
Let $(E,M,p)$ be a smooth n dimensional vector bundle. Then $(TE,TM,Dp)$ is a 2n dimensional vector bundle. We restrict this bundle to $M\subset TM$. We denote this restricted bundle by $F$, as a ...
- 356