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Questions tagged [k-theory]

K-theory is the study of invariants of large matrices, in a suitable sense. It has many variations: (algebraic-k-theory), (topological-k-theory), or in the study of (operator-algebras).

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Equivariant Multiplicative Formula for Sphere Bundles in the Index Theorem

I am reading through the proof of the Atiyah-Singer index theorem, out of Lawson-Michelson, and I'm a bit confused about the proof of multiplicative property for indices of sphere bundles, where I ...
Elie Belkin's user avatar
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Finding a perfect complex with specified support

Let $X$ be a quasicompact quasiseparated scheme and $Z$ a closed subset of $X$ with quasicompact complement. Then there exists a perfect complex $F \in D^{\mathrm{perf}}(X)$ with $\operatorname{supp} ...
Brendan Murphy's user avatar
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Definition of Fredholm modules

I'm currently starting to learn K-homology and something bothers me about the definition of Fredholm modules. I looked in 5/6 papers and each time a different definition is given... For example in [1] ...
eomp's user avatar
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Integer times vector bundle notation in Hatcher

In Hatcher's vector bundles and K-theory, p45, there is a proposition written "If $q$ is a polynomial clutching function of degree at most $n$, then $[E,q] \oplus [nE,\mathbb{1}] \approx [(n+1)E,...
summersfreezing's user avatar
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Group actions on smash products

The complex K-theory spectrum $KU$ is equipped with a $C_2$-action coming from conjugation of vector bundles. On homotopy, we have that $\pi_*KU = \mathbb{Z}[u^{\pm 1}]$, and the action is recorded by ...
categorically_stupid'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-...
Grimp0w's user avatar
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1 answer
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Plus construction is functorial

Given a nice map (cellular map) $f : X\rightarrow Y$ between CW complexes $X$ and $Y$, how is $f^{+}$ defined between their plus constructions? Going through my old notes, I've learnt that the plus ...
May's user avatar
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2 votes
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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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Question about Proof of BDF Theorem in Analytic K-Homology book

In the proof of the BDF Theorem in the book Analytic K-Homology by Higson/Roe (chapter 7, page 187, I pasted the proof below) the authors claim that given an essentially normal operator $T$ there is a ...
craaaft's user avatar
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Table of Clifford Algebras

Thanks to Hans Lundmark, I just found the table of Clifford algebras $C_k=C_{\Bbb R}(0,k)$ and $C'_k=C_{\Bbb R}(k,0)$, $0\leq k\leq 7$. But, I couldn't understand some of the slots in the table. $\...
Bob Dobbs's user avatar
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Two Technical Details from "Categories and Cohomology Theories".

This has really bothered me for a while. I understand fully the big picture idea of what is happening in Segal's "Categories and Cohomology Theories". You take a Segal space $X$ and you ...
Johnathon Taylor's user avatar
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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 \} \...
Pierre MATSUMI's user avatar
1 vote
1 answer
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The Splitting Principle for Varieties

I am recently thinking of the following question (motivated by the splitting principle for vector bundles): Suppose $X$ is a projective smooth variety over an algebraically closed field $k$. Let \...
Dick. Y's user avatar
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K-theory and the cohomology of categories

I am thinking about K-theory. The first idea I had was about a construction which seems similar${}^{**}$. The similar idea featured some kind of cofibrant replacement of R-mod (as an infinity category)...
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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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$K_0(SS\mathbb{C})$ from the definition

For any $C^\ast$-algebra $A$ its suspension is the $C^\ast$-algebra of all continuous functions $f$ from the unit sphere $S^1$ to $A$ with $f(1)=0$. For the K-theory of $C^\ast$-algebras on has the ...
bs_math's user avatar
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Tensor powers of line bundles induces map on homology of CP^infty

The operation of taking the $n$th tensor power $\mathscr L^{\otimes n}$ of a complex line bundle $\mathscr L$ over a space induces a map $\psi^n\colon\mathbb{CP}^\infty\rightarrow\mathbb{CP}^\infty$ ...
Beckham Myers's user avatar
3 votes
1 answer
136 views

Functoriality of the one-point compactification

My question concerns the passage below: Why does continuity at infinity of $\phi$ imply that the extension $\phi^{+}$ to the one-point compactifications necessarily maps $\infty_{X}$ to $\infty_{Y}$? (...
destine's user avatar
  • 288
4 votes
0 answers
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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, ...
Nennee's user avatar
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Projective module base change

Let $M$ be a finitely generated projective $C[0,1]$-module. Without using any vector bundle theory, how does one show that the pullback of $M$ along the evaluation at $0$ and evaluation at $1$ maps ...
Anupam's user avatar
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Strictly positive compact operator commuting with a given Fredholm operator

In the following snippet from a paper by Jody Trout on the converse functional calculus, they mention the existence of a strictly positive compact operator $T$ commuting with a given self-adjoint, odd ...
Anupam's user avatar
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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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Connes' long exact sequence in cyclic homology

I'm working on the book Cyclic Homology by Loday (https://www.math.univ-paris13.fr/~vallette/GdT/Cyclic%20Homology%20-%20Loday.pdf) and in the section 2.2, he constructs the Connes' long exact ...
newuser's user avatar
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calculate $K_0(C(Z_n))$ and $K_1(C(Z_n)$)

For each natural number $n$ let $Z_n$ be the $n$-clover obtained by forming the disjoint union of $n$ circles, choosing one point in each circle, and then identifying these $n$ points, so that they ...
analysis lover's user avatar
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Reference KK-theory is stable or alternative proof verification?

I am searching for a reference that Kasparovs KK-theory is stable, that is for separable Hilbert spaces $H,H'$ there are isomorphisms $KK(A\otimes \mathbb{K}(H),B)\cong KK(A,B\otimes \mathbb{K}(H')\...
Roland's user avatar
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Show that there is a natural number n and a projection 𝑝∈𝑃𝑛(𝐶̃ ) such that 𝑔=[𝑝]0−[𝑠(𝑝)]0 and 𝜑̃ (𝑝)∼ℎ𝑠((𝜑̃ 𝑝)) .

Let $ \varphi :C\to D $ be a $*$-homomorphism of $C^*$-algebras and suppose that $g \in\ker(K_0(\varphi))$. Show that there is a natural number $n$ and a projection $ p\in P_n(\tilde C)$ such that $ g=...
analysis lover's user avatar
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Show that $δ_1([u]_1) = [e]_0−[f]_0$, where $e, f∈M_2(C_0(ℝ^2)^∼)$.

Consider the short exact sequence $$0\xrightarrow{\ \ \ }C_0(ℝ^2)\xrightarrow{\ \varphi \ }C(ⅅ)\xrightarrow{\ \psi\ }C(𝕋)\xrightarrow{\ \ \ }0,$$ where $\psi$ is the restriction mapping and $\varphi$ ...
analysis lover's user avatar
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Consider inductive sequence of $C^\ast$-algebra $A_1 \overset{φ_1}→A_2\overset{φ_2}→···$ inductive limit $(A,(μ_n)_n∈ℕ)$. How is $K_{0}$ continous?

Consider an inductive sequence of $C^\ast$-algebra $A_1 \overset{\varphi_1}\to A_2\overset{\varphi_2}\to···$with inductive limit $(A,(\mu_n)_{n\in \Bbb N})$. Explain how continuity of $K_{0}$ follows ...
analysis lover's user avatar
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1 answer
80 views

Unitary element in $M_2(A)$

Let $A$ be a unital C*-algebra and let $a∈A$ with $\|a\|≤1.$ Justify that $$af(a^* a) =f(aa^*)a$$ for every continuous function $f: [0,1]→ℂ.$ Use this to show that$$v:=\begin{pmatrix}a& (1−aa^*)...
analysis lover's user avatar
2 votes
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26 views

The Elliot invariant and the opposite algebra

Let $\mathcal{A}$ be a C$^*$ -algebra. Let $\mathcal{A}^{op}$ denote the opposite algebra, that is, the C$^*$ -algebra given as a set the same as $\mathcal{A}$, where addition is the same as before, ...
Owen Tanner's user avatar
1 vote
1 answer
106 views

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
1 vote
1 answer
220 views

Classifying Spaces of Matrix Lie Groups

I’m currently studying the classifying spaces of some of the matrix Lie groups. I’ve come across a post here that describes the classifying spaces for $SO(n)$, $SU(n)$, $GL(n)$, and $Sp(n)$. I’ve also ...
slowspider's user avatar
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3 votes
1 answer
105 views

Understanding why collapsing $CX$ and $CY$ gives us $\sum (X \times Y)$?

Here is the proposition I am trying to understand in Hatcher: But I do not understand: 1- why collapsing $CX$ and $CY$ gives us $\sum (X \times Y),$ could someone explain this to me please? as far as ...
Emptymind's user avatar
  • 2,069
1 vote
1 answer
104 views

Extension of $\{f\in C([0, 1], B)\,\vert\, f(0)=f(1)=0\}$ by $A$ with $\ast$-homomorphism $\phi:A\rightarrow B$

The following question is from An Introduction to $K$-theory for $C^{\ast}$-Algebra and an e-copy can be found here. Below is the question (since I do not know how to create a diagram in MS ...) By ...
Sanae's user avatar
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0 answers
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Classifying K-theory class in the space of Fredholm operators

I am reading Dan Freed's notes on K-theory, where he defines a vector bundle on the set of Fredholm operators transverse to a fixed finite dimensional $W$. In other words, we have $$\mathcal{O}_W=\{T\...
B. S.'s user avatar
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3 votes
1 answer
67 views

The existence of a nontrivial summand of a free module of rank $2$

I've been reading Atiyah's paper "The unity of mathematics". In section 5, he mentions "having a nontrivial summand of a free $R$-module of rank $2$ is equivalent to having a matrix $T\...
CHWang's user avatar
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1 vote
2 answers
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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 ...
user1104937's user avatar
1 vote
1 answer
45 views

The $K_0$ mapping of an automorphism induced by a derivation

Let $\mathfrak{A}$ be a unital $C^*$-Algebra and let $\delta: \mathfrak{A} \rightarrow \mathfrak{A}$ be a linear map that is not constantly zero and satisfies, for every $A, B\in\mathfrak{A}$, $\delta(...
Sanae's user avatar
  • 343
2 votes
0 answers
56 views

External Product on K-group

I am working on Prop 4.8.3 of Higson & Roe's Analytic K-Homology recently. It asks me to verify that for unital C*-Algebras $\mathcal{A}$ and $\mathcal{B}$, if $u$ is a unitary of $\mathcal{A}$ ...
Shuoyu Yan's user avatar
1 vote
0 answers
71 views

Relation between group of completion and field of fractions

I recently learned group completion in K-theory. I feel it is very similar to field of fractions of integral domain. Recall the definitions: Group completion: For an abelian monoid $M$, we define its ...
threeautumn's user avatar
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1 answer
69 views

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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0 answers
64 views

Road to Kuiper's theorem

I was scrolling through wikipedia whilst procrastinating some homework and landed on Kuiper's theorem, the statement being that GL(H) is weakly contractible, with H a infinite dimensional Hilbert ...
DevVorb's user avatar
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1 vote
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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
  • 363
2 votes
1 answer
63 views

A sufficient condition for comapct operators?

I am reading a paper about the classification of essentially normal operators right now, this paper is available in https://link.springer.com/chapter/10.1007/BFb0080022. In this paper, when he tries ...
Yanyu's user avatar
  • 390
0 votes
2 answers
58 views

Function composition on Toeplitz operator.

I am working on Prop 2.3.3 of Higson and Roe Analytic K-theory recently, which says The map $\alpha : g\mapsto \pi(T_g)$ is an injective *-homomorphism from $C(\mathbb{S}^1)$ to the Calkin algebra $\...
Shuoyu Yan's user avatar
0 votes
1 answer
94 views

Is Grothendieck group of vector bundles over M finite-degree-generated

For a topological manifold M with $KO$-the grothendieck group (actually a ring) of its vector bundles. Whether the ring $KO$ is generated by vector bundles of rank $\leq N$,where $N$ is large ...
wsh's user avatar
  • 127
1 vote
0 answers
31 views

Finiteness of K0 ring of vector bundles

It is known that,for a base space M,the equivalent classes of vector bundles over M generate a ring denoted by K0,with direct sum as + and tensor product as ×. And we can give K0 a degree:all rank-n ...
wsh's user avatar
  • 127
0 votes
1 answer
94 views

Is the Grothendieck group finite degree generated?

For a scheme $X$ with structure sheaf $\mathcal{O}_X$, is the Grothendieck ring of locally free $\mathcal{O}_X$-modules generated by the equivalence classes of rank $≤N$ modules, where $N$ is a ...
wsh's user avatar
  • 127
1 vote
0 answers
55 views

Vector bundles of K(1,G) and representation of G

What is the connection between the K0 group of vector bundles over K(1,G) and the representations of G?If G is a finite group,I guess the Grothendieck ring of its vector bundle is just the ring of its ...
wsh's user avatar
  • 127
2 votes
0 answers
81 views

How to get into K-Theory and have least interaction with Algebraic Topology

I am doing PhD in Leavitt Path Algebras (LPAs). While going through some recent works on LPAs I found this paper titled K-Theory of Leavitt path algebras by Ara etal. I tried reading it, but failed ...
cabmetric's user avatar
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