All Questions
Tagged with kt.k-theory-and-homology vector-bundles
58 questions
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 ...
- 35.5k
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 $\mathbb C[t,t^{-1}]$...
- 4,111
1
vote
0
answers
111
views
Unique Hausdorff topology on trivial vector bundle?
Question: Is there a Hausdorff topology other than the product topology on $X\times \mathbb{C}^n$, that turns $(X\times \mathbb{C}^n, \mathrm{pr}_1)$ into a vector bundle, where $\mathrm{pr_1}$ ...
- 11
1
vote
0
answers
240
views
Smooth version of the splitting principle
Inspried by this MO question A manifold whose tangent space is a sum of line bundles and higher rank vector bundles we pose the following question as a possible smooth version of the splitting ...
- 356
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
0
votes
1
answer
203
views
Equivariant sheaves on $\mathbb P^1$
Let $K(\mathbb P^1)$ be the Grothendieck group of sheaves on $\mathbb P^1$. I want to show that the map $K^{{\rm PGL}(2)\times \{\pm 1\}}(\mathbb P^1) \to K(\mathbb P^1)$ is not onto. I read somewhere ...
- 39.3k
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
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,...
- 55.9k
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 ...
- 4,713
1
vote
0
answers
68
views
Metric and connection on virtual bundles
Let $E=[E^+]-[E^-]$ be an element of the Grothendieck group $K(X)$ of a compact Kahler manifold $X$.
Does it exist a way to define more "geometric" structures on $E\in K(X)$ such that a ...
- 789
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
1
vote
0
answers
132
views
A possible kind of $K$ theory via comparison of sphere bundles associated to given vector bundles
Let $E$ be a vector bundle on a topological space $X$.Thanks to Allen Hatcher's book "Vector Bundles and K theory", the construction of sphere bundle $S(E)$ can be done without any inner ...
- 356
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 ...
- 2,017
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$ ...
- 11.1k
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
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
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
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 ...
CommunityBot
- 1
1
vote
1
answer
142
views
A kind of isomorphicity of vector bundles
Let $X$ be a connected topological space. Let $E$ be a $k$ dimensional sub vector bundle of the trivial vector bundle $X\times \mathbb{R}^n$. Then $E$ defines an idempotent with trace $k$ in $M_n(C(X))...
- 1,150
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 ...
- 35.9k
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.
- 63.9k
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
$$
...
- 7,583
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 $\nu$ ...
- 3,212
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 ...
- 4,679
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 ...
- 63.9k
0
votes
0
answers
171
views
Surjectivity of the Albanese map of the moduli space of stable vector bundles
I have a naive question:(I saw that it is related to relative K-theory of Hodge-Deligne and also Nadel-Chern-Weil theory )
Let $\mathcal M (r, d)$ be the moduli space of stable vector bundles of rank ...
- 189
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 ...
- 2,062
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 ...
- 11.6k
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 ...
- 19.3k
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
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 \...
- 35.9k
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 ...
- 2,479
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?
CommunityBot
- 1
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 \...
- 56.9k
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!
- 37.8k
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:...
CommunityBot
- 1
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 ...
- 56.9k
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$$\text{Vect}_k(S^q) \simeq \pi_{q - 1}(\...
- 7,583
1
vote
1
answer
322
views
A (possible) equivalent relation on the space of vector bundles
Edit: According to the essential comment of Alex Degtyarev, we revise the question as follows;
Assume that $\alpha$ and $\beta$ are two oriention preserving automorphism of Lie groups $O(n)$ and $...
- 6,813
3
votes
1
answer
323
views
self-Whitney sum of the canonical vector bundle on Grassmannians
Let $G_{k}(\mathbb{R}^N)$ be the Grassmannian manifold consisting of $k$-subspaces in $\mathbb{R}^N$. There is a canonical $k$-dimensional vector bundle
$$
\gamma_{k,N}: \mathbb{R}^k\longrightarrow E(...
- 6,813
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 ...
- 6,813
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
$$
\xi: \mathbb{R}^2\longrightarrow S^n\times_{\mathbb{Z}/2}\...
- 556
7
votes
1
answer
434
views
classifying maps of Whitney sums of vector bundles
For an $n$-dimensional vector bundle $\xi$ with structure group $G\leq O(n)$ over a $CW$-complex $B$, we have a classifying map up to homotopy
$$
f(\xi): B\longrightarrow BG,
$$
$f(\xi)\in [B;BG]$, ...
- 10.4k
12
votes
2
answers
2k
views
Is every vector bundle over a noncompact finite-dimensional manifold a summand of a trivial bundle?
In the notes of Vector Bundles and K-theory by Prof Allen Hatcher, on page 12 he proved a Proposition that for each vector bundle $E\to B$ with $B$ compact hausdorff there exists a vector bundle $E'\...
- 5,489
4
votes
1
answer
507
views
A question on complex line bundle over $S^{2}$
Consider the trivial bundle $\epsilon_{2}=S^{2}\times \mathbb{C}^{2}$ with the standard Hermitian inner product $<(a,b), (c,d)>=a\bar{c}+b\bar{d}$.
Assume that $\ell$ is a sub line bundle of ...
- 6,813
1
vote
1
answer
331
views
realization map for K-theory of spheres
Let $\overset{\sim}{K}(X)$ and $\overset{\sim}{KO}$ denote the reduced stable isomorphic classes of complex and real bundles over X and $\rho$ be the realization map. We know that $\overset{\sim}{K}(\...
- 55.9k
30
votes
4
answers
2k
views
When can we cancel vector bundles from tensor products?
Let $E,F,G$ be algebraic vector bundles over $\mathbb P_{\mathbb C}^n$. My general question is:
Assume $E\otimes G \cong F\otimes G$, under what conditions can one conclude that $E\cong F$?
Some ...
- 148k
5
votes
2
answers
624
views
Topological K-theory for commutative C*-algebras
It is in some sense folklore that given two arbitrary abelian groups $G,H$ one can find a $C^*$ algebra $A$ such that $K_0(A)=G$ and $K_1(A)=H$. My question is the following: what is known in the case ...
- 356
1
vote
0
answers
261
views
Two questions on canonical line bundle over $\mathbb{C}P^{n}$
The canonical line bundle over $\mathbb{C}P^{n}$ is denoted by $\ell_{n}$. It is well known that: $$(\ell_{1}\otimes \ell_{1})\oplus1 \simeq\ell_{1}\oplus\ell_{1}$$ Both sides are isomorphic ...
- 356
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 ...
- 3,784