Questions tagged [topological-k-theory]
Topological K-theory is a generalized cohomology theory, for which $K_0(X)$ is the Grothendieck group of isomorphism classes of vector bundles over topological space $X$. See also (algebraic-k-theory).
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Why $f^*$ must be the identity map in Q.3.2.3(a) in AT? And notation question. [duplicate]
Here is the question I am trying to understand its solution:
Using the cup product structure, show there is no map $\mathbb R P^n \to \mathbb R P^m$ inducing a nontrivial map $H^1(\mathbb R P^m; \...
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Question on Segal's definition of $K$-theory
Segal, in the papers "Fredholm complexes" and "Equivariant K-theory", gives the following equivalent definitions of $K$-theory.
For $X$ a compact top. space, let $\mathcal{L}(X) $ ...
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Representation of topological K-theory via Brown representability
We know that topological K-theory is a generalized cohomology theory, and reduced K-theory is a reduced cohomology theory. Thus, both are representable with a sequence of pointed homotopy functors, ...
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Reference for the following cancellation theorems
I found these two cancellation theorems in the K-book, in section 1.4, page 39:
Real Cancellation Theorem: Suppose $X$ is a $d$-dimensional CW complex and $\eta:E\to X$ is an $n$-dimensional real ...
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Reduced real connective K theory of a point
I’ve just started reading about reduced real connective k-theory, denoted $\widetilde{ko}_*$. I’m familiar with real k-theory and complex k-theory, they’re reduced counterparts, and the definition of $...
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complexification of real vector bundles on $S^4$ and $S^8$
Prove there is a surjection $\tilde{KO}(S^8) \to \tilde{K}(S^8)$.(similarly for $S^4$)
My idea:
It is equivalent to show in every class of complex vector bundles on $S^8(S^4)$, there is a ...
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K Group Determined by Chern Classes
Why is it that the complex reduced K group of $\mathbb{CP}^2$ is determined by Chern classes $c_1$ and $c_2$?
I am aware of the fact that the cohomology ring of complex Grassmannians is generated by ...
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K functor of riemann surface
I have read about K-theory, but i didnt find any computational ideas there. Suppose i have a riemann surface of genus g - $S_g$, how can i compute $K_0(S_g)$?
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Bundle homomorphisms and sections of Hom-bundle
I reading "Complex Topological K-theory" by Efton Park and came across exercise 1.4.
Let $V,W$ be vector bundles over a compact Hausdorff space $X$.
a) Show that the collection $Hom(V,W)$ of ...
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Lemma 3.6.9 from Complex Topological K-Theory by Efton Park
The lemma goes like this (in author's notation, $\pi: V\to X$ is the projection map; if $f: Y\to X$, and $W$ is a vector bundle over $X$, then $f^* W$ is the pullback of $W$ over $Y$):
Let $X$ be a ...
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What is the $\langle \cdot, \cdot \rangle$ operator when calculating Stiefel-Whitney numbers?
I've been trying to understand Stiefel-Whitney numbers and I'm reading this paper. On page 9, the author defines the Stiefel-Whitney number using an operator $\langle \cdot, \cdot \rangle$. I've tried ...
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Homotopy Invariance of K-theory.
For a locally compact Hausdorff space $X$, the $K$ ring is defined to be the $K$ ring of its one-point compactification, i.e. $K(X)\colon =K(X^+)$. Therefore, $K(\mathbb{R})\colon =K(S^1)=\mathbb Z$ ...
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Calculating the equivariant K-group $K_G^*(S^1)$ - where's the mistake?
Let a finite group $G$ act on the circle $S^1$ via a group homomorphism $\varphi \colon G \to S^1$. Let $K = \ker \varphi$. I wish to calculate the equivariant K-theory group $K_G^*(S^1)$.
One method ...
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Associated fibre bundle to a $(\Gamma,\alpha)$-equivariant $G$-principal bundle
Let $\Gamma$ and $G$ be compact Lie groups and $\alpha:\Gamma\to Aut(G)$ group homomorphism with the condition that $(\gamma,g)\mapsto \alpha(\gamma)(g)$ is continuous.
$(\Gamma,\alpha,G)$-bundles are ...
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Collapsing of Kunneth formula for equivariant K-theory of homogeneous spaces.
Minami in "K-groups of symmetric spaces" (equations 1.1, 1.2) states the following, originally due to Hodgkins:
Suppose that $G$ is a compact connected Lie group such that $\pi_1(G)$ is ...
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Hatcher K-Theory continuity of $\alpha \to \inf_{(x,z)\in X \times S^1} \det|\alpha(x,z)|$
Does Hatcher makes an error at page $45$ afirming that $f:\alpha \mapsto \inf_{(x,z)\in X \times S^1} \det|\alpha(x,z)|$ is continuous? I didn't find any reference of continuity of $\inf$ of ...
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Showing symmetric and antisymmetric tensor series are inverses in $K(M)[[t]]$?
Let $M$ be a smooth manifold, and let $K(M)$ denote its real $K$-theory ring, following Remark 2.7 at the nLab page.
Given a vector bundle $V$ we define the quantities
$$
S_t(V) = \sum_{j=0}^{\infty} (...
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isomorphic pullback through projections determine isomorphic vector bundles
The context is the following: working with $\xi = (E,p_E,X),\eta = (F,p_F,X)$ two vector bundles, denote $\pi_0:X \times D_0 \to X,\pi_{\infty} : X \times D_{\infty} \to X$ the projections over the ...
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Definition of vector bundle and K-theory
The definition of vector bundles seems to be split in the mathematical community: some sources insist that the rank of each fibre is the same, whereas some don't ask for this requirement.
I was ...
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Grothendieck group of noetherian scheme - why $\textbf{Coh}(X)$ is a set?
According to Hartshorne Exercise II.6.10, the Grothendieck group of a noetherian scheme is defined to be the quotient of the free abelian group generated by all the coherent sheaves on $X$ by the ...
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Is the first Stiefel-Whitney class an isomorphism if there is a unique orientable class?
Suppose that $X$ is a nice compact manifold such that its reduced real $K$-group $\tilde{K}\mathcal{O}(X)$ has a unique stably equivalent class of orientable bundles, i.e, $\ker(\omega_{1})$ is the ...
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$X$ is a connected compact metric space. Is every $C^*$-embedding $M_n(\mathbb C)\to M_n(C(X))$ unitarily equivalent to the trivial one?
$M_n=M_n(\mathbb C)$ is the $n\times n$ complex matrix algebra and $M_n(C(X))$ is the algebra of continuous functions from $X$ to $M_n(\mathbb C)$.
There is a trivial homomorphism $M_n\to M_n(C(X))$ ...
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Is real topological $K$-theory subsumed by complex topological $K$-theory?
The title pretty much captures my question. I understand that there are both real and complex $K$-theories of Hausdorff topological spaces depending on whether we look at real or complex bundles over ...
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Stably isomorphic complex vector bundles which are not isomorphic
Is there an example of complex vector bundles $E_1$ and $E_2$ which are not isomorphic but for which $E_1 \oplus \mathbb{C}^{n}$ isomorphic to $E_2 \oplus \mathbb{C}^{n}$ for some $n$?
I'm a beginner ...
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Thom class in K-theory for real vector bundles of odd rank and pushforwards in K-theory with degree shift
If $E \to X$ is a complex vector bundle, its Thom class is defined using the exterior algebra of $E$, giving the Thom isomorphism $K^*(X) \cong \widetilde{K^*}(X^E)$. Atiyah-Bott-Shapiro use Clifford ...
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Relationship between Chern characters in the sense of Hatcher and in the sense of Roe
Let $\pi: E \to X$ be a vector bundle over some manifold $X=\mathcal M$.
Definition 1:
Hatcher defines Chern classes
$$c_i(E)\in H^{2i}_\text{singular}(X)$$
as elements of the singular cohomology ring ...
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Different coefficient rings for the Chern classes and the Chern character
Following Hatcher, we define Chern classes $c_i$ taking vector bundles $E \to X$ to some equivalence class $c_i(E) \in H^{2i}(X; \mathbb Z)$
But when we define the Chern character, $$\operatorname{ch}(...
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Why is universal coefficient theorem for $C^*$-algebras natural in both variables?
This theorem 4.4 from The Kunneth Theorem and the Universal Coefficient Theorem for Kasparov’s Generalized K-functor, Jonathan Rosenberg and Claude Schochet.
My question is, how does naturality of $\...
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What does degree mean here? Question about universal coefficient theorem and $KK$ theory.
Universal Coefficient Theorem 1.17. Let $A\in\mathcal N$. Then there is a short exact sequence
$$
0\to \text{Ext}(K_*(A),K_*(B))\stackrel{\delta}\to KK_*(A,B)\stackrel{\gamma}\to \text{Hom}(K_*(A),K_*(...
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What is the product in KO-theory in terms of concrete homogeneous spaces?
It is well-known that the spaces $KO_i$ of the $\Omega$-spectrum representing KO-theory a.k.a. real K-theory are $KO_0,KO_{-1}, \ldots, KO_{-7} = BO \times \mathbb{Z}, O, O/U, U/Sp, BSp \times \mathbb{...
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Spaces with only even dimensional cells has zero $K^{-1}$ group?
This is from D. Quillen's "On the Cohomology and K-Theory of the General Linear Groups Over a Finite Field". Here $BU= \bigcup _1^\infty X_m$ is the union of finite subcomplexes with only ...
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Is there a paper that explains Kasparov's KK theory in English?
I need to learn Kasparov's KK theory but the original paper is written in Russian.
G.G. KASPAROV, The operator K-functor and extensions of C*-algebras, Izv. Akad. Nauk SSSR,
Ser. Mat. 44 (1980), 571-...
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Milnor's construction
In the Milnor's construction,
$$
\mathcal{J}(G):=\underrightarrow{\lim}G^{*(k+1)}
$$
where $G$ is a topological group.
I know that there is a natural freely (right) $G$-action on $\mathcal{J}(G)$ and ...
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Intuition behind 4-fold periodicity of $L$-theory
The quadratic and the symmetric L-groups are 4-fold periodic.
What is the simple argument to obtain the intuition behind the 4-fold periodicity of $L$-theory?
(For example, why not have the Bott ...
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K-Theory Equivalence Classes
Let $M$ be a finite dimensional compact manifold and $(Vect(M),\oplus)$ be the abelian monoid of complex vector bundles on $M$. I just read that it is possible to construct an equivalence relation on $...
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Generalizing the clutching construction to more contractible open sets
In computing the topological $K$-theory of $\mathbb{R}P^2$ I had an idea to mimic the clutching construction: Recall that this says that for the open cover $S^k = D^k_+\cup D^k_-$ a rank $n$ vector ...
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Definition of odd topological K-theory using circles
I wanted to check whether the following characterization of odd complex topological $K$-theory is correct.
Let $X$ be a compact Hausdorff space. Then $K^{-1}(X)$ can be defined as $\tilde{K}^0((X\...
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K-theory proof of index theorem - some minor confusion
I am trying to understand the general approach to the $K$-theory proof of the Atiyah-Singer index theorem, using this https://arxiv.org/pdf/math/0504555.pdf paper. I ran into some confusion on page 29,...
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Example of a space whose complex K-theory is not easily computable from singular cohomology
I am looking for a counterexample to the formula
$$ K^n(X) \cong \prod_{i\equiv n \mod 2} H^i(X) $$
where $K^*$ denotes complex topological $K$-theory, $H^*$ singular cohomology and $X$ a compact ...
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Topological index in Atiyah Singer
I'm a beginner at Atiyah-Singer index theorem and I've reviewed some results about theorem. Here's some questions.
Ive seen the topological index is equal to $$\operatorname{ch}(D) \operatorname{Td}(X)...
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Help with the setup for Atiyah's proof of Bott Periodicity.
I'm trying to understand Atiyah's proof of Bott Periodicity from his little book on K-Theory - in particular his formulation in terms of $K(P(L \oplus 1))$ where $L$ is a line bundle on a space $X$.
...
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Complex topological K-theory - the book
I am reading the first chapter of the book "Complex topological K-theory" by Efton Park, which is in general very good. However, for some reasons, which I don't understand, when working with ...
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What is the meaning of $KO^{-1}(S^1)$?
I am interested in the KO-theory of the circle $S^1$. In particular $KO^{-1}(S^1)$. Using the suspension theorem and reduced $K$-theory I can easily show that
\begin{equation}
KO^{-1}(S^1) \simeq KO^{-...
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Equivalence of two definitions of $K^{-1}$ in complex $K$-theory
In complex $K$-theory, one way I have seen the group $K^{-1}(X)$ defined, for a compact Hausdorff space $X$, is
$$K^{-1}(X):= K^0_c(X\times\mathbb{R}),$$
where the right-hand side refers to compactly ...
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A map from a $ G_1 $ - equivariant KK-theory of Kasparov, to a $ G_2 $ - equivariant KK-theory of Kasparov
Let $ G $ be a locally compact group.
Let $ H $ and $ K $ be two normal subgroups of $ G $.
In order to construct a map, $$ \psi \ : \ \ F(G/H,G/K) \to F(G/K,G/H) $$
where, $$ F(G/H,G/K) = KK^{G/H} ( ...
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Why does this short exact sequence split?
I am currently reading K-Theory, Anderson and Atiyah, and trying to understand the index bundle (of families of Fredholm operators).
Here is some context: let $X$ be a compact topological space, $F\...
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Is there a sense in which $\tilde{K}$ is an exact functor?
I'm going through Hatcher's K-Theory script for the first time and noticed that following theorem looked quite like a statement of the form “this functor is exact”:
If $X$ is compact Hausdorff and $A\...
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Reference books on the Baum Connes conjecture
Do there exist readable reference books about Baum-Connes Conjecture for beginners ?
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Tangent bundle of the incidence variety and its Chern class
I am trying to learn about Thom polynomials and I often find arguments in the literature that just make no real sense to me. Maybe it is due to my lack of knowledge in $K$ - theory. I apologize for ...
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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 ...
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