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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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Equivalence between K-theory for C*-algebras and K-theory for rings.

My question is motivated for the Swan’s theorem that give us an isomorphism between the $K(X)$ and $K(C(X))$. When you think as $K(C(X))$ as the algebraic K theory everything works perfectly. Because ...
Gomífero's user avatar
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What is the real K-theory of the spheres?

In many sources one finds a computation of the complex topological K-theory of the spheres $S^n$, but the real theory $KO(S^n)$ is usually not computed. Maybe it can be read off some more advanced ...
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Completing proof of theorem 3.2 in Yiannis Loizides' The Atiyah-Hirzebruch Spectral Sequence.

Let $X$ be a CW-complex, and denote its $k$-skeleton by $X_k$. Let $h^*$ be a cohomology theory, and consider its AHSS for $X$: call $E_n^{p,q}$ the groups in the $n$-th page, and call $d_n^{p,q}$ the ...
Ezio Greggio's user avatar
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Relation between topological and operator K-theory

Given a compact Hausdorff space $X$ we can construct its topological K-theory. On the other hand the continuous functions on $X$ give rise to a C-algebra $C(X)$ and we can consider the operator K-...
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Computing the reduced $K$-theory of $\mathbb CP^n$ using AHSS

I'm studying Atiyah-Hirzebruch spectral sequence from some slides, a bit hard to follow as they are a collage of pages from various books and papers, filled with writings and deletions made by hand by ...
Ezio Greggio's user avatar
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240 views

Reference for Atiyah-Hirzebruch SS and stunted projective spaces

I would like to know if there are any textbooks on topological $K$-theory that explain the Atiyah-Hirzebruch spectral sequence and use it to compute $KO$ of (real) stunted projective spaces. I know ...
Ezio Greggio's user avatar
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Defining Atiyah-Hirzebruch spectral sequence from exact couple

For any integer $n$, let $h^n:\sf Top^{op}_*\to Ab$ be functors satisfying the axioms of a reduced generalized cohomology theory, as defined in Yiannis Loizides' paper on Atiyah-Hirzebruch spectral ...
Ezio Greggio's user avatar
2 votes
1 answer
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Ring structure in $K$-theory

I don't understand a statement in Randal-Williams Characteristic classes and $K$-theory, at the beginning of page 29. The book says: Tensor product (of vector bundles) gives a homomorphism of abelian ...
Ezio Greggio's user avatar
2 votes
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Non-invertible tangent bundle

I was wondering whether there is an example of a connected, open, smooth manifold $M$, whose tangent bundle $\tau$ is not a sub-bundle of a trivial bundle. Clearly, for closed $M$ this is impossible, ...
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On Bloch's higher chow.

I stumbled across the statement that for an integral scheme $X$ and an elements $f_1, \ldots, f_n \in {\cal O}_X^{\times}$, there is a symbol map, viz., $$ {\mathrm S} \colon \{ f_1,\ldots,f_n \} \...
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Elementary proof of Bott periodicity

The elementary proof of Bott periodicity was found by Atiyah and Bott, and published in the paper "On the periodicity theorem for complex vector bundles". In Hatcher's book p.41-51, this ...
Dasheng Wang's user avatar
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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) $ ...
Overflowian's user avatar
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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 $...
slowspider's user avatar
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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 ...
user884626's user avatar
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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 ...
user884626's user avatar
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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 ...
roundsquare's user avatar
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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, $K(\mathbb{R})\colon =K(S^1)=\mathbb Z$ ...
SUDEEP PODDER's user avatar
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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 ...
Motmot's user avatar
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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 ...
Dylan's user avatar
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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 ...
jacopoburelli's user avatar
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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 ...
jacopoburelli's user avatar
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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 ...
mathable's user avatar
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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 ...
user avatar
2 votes
1 answer
92 views

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 ...
user1036923's user avatar
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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 $M_n\to M_n(C(X))$ ...
Yuz's user avatar
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4 votes
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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 ...
AJY's user avatar
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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 ...
taiat's user avatar
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5 votes
1 answer
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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 ...
Motmot's user avatar
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0 answers
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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 ...
Markus Klyver's user avatar
4 votes
1 answer
80 views

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}(...
Markus Klyver's user avatar
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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 $\...
Yuz's user avatar
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1 answer
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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_*(...
Yuz's user avatar
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5 votes
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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{...
Matt C's user avatar
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1 vote
2 answers
70 views

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 ...
Flying pencil's user avatar
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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-...
Yuz's user avatar
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4 votes
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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 ...
Topological_CAT's user avatar
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1 answer
76 views

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 ...
wonderich's user avatar
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4 votes
1 answer
96 views

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 $...
Watanabe's user avatar
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2 votes
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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 ...
ThePuix's user avatar
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3 votes
1 answer
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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\...
geometricK's user avatar
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1 vote
1 answer
257 views

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,...
Quaere Verum's user avatar
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6 votes
2 answers
153 views

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 ...
Christian's user avatar
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0 answers
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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)...
LSY's user avatar
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209 views

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 $X$. ...
Jordan Levin's user avatar

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