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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13 views

Do limits of Waldhausen categories comute with ordinary limits?

Let $(A,\mathcal{W}, \mathcal{C})$ be a Waldhausen category with $A$ an additive category. On one hand, we can define the ordinary limits $lim_A$ of the underlying category $A$. On other hand, we can ...
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Is there a categorical way to see that $K_1(A)\twoheadrightarrow K_1(A/I)$ for a nilpotent ideal $I\subset A$?

Let $A$ ba a (not necessarily commutative) unital ring. The $K_1$ group of $A$ is defined as $K_1(A)=\pi_1((\mathrm{Proj}(A)^\simeq)^\mathrm{gp})$. Here $\mathrm{Proj}^\simeq$ is the core of the ...
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Show that $[u]_1$ belongs to $Im(K_1(\varphi))$ if and only if …

Let $\varphi: A \to B$ be a surjective $^*$-homomorphism between unital $C^*$-algebras A and B, and let $u$ be a unitairy in $\mathcal{U}_n(B)$. I want to show that $[u]_1$ belongs to $Im(K_1(\varphi))...
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Excision sequence for $K_0$ in algebraic $K$-theory

I am looking at Exercise I.2.3 in Weibel's $K$-book. Here is the statement: If $I$ is an ideal of a ring $R$, we form the augmented ring $R\oplus I$ and let $K_0(R,I)$ denote the kernel of $K_0(R\...
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Why shift operator is not homotopic to 1 ($K_1$-approach)?

Let us recall that via fourier transform it holds true that $C^*(S)\cong C(\mathbb{T})$, with map given by $S\mapsto e^{2\pi i x}$ (considering $\mathbb{T}=\mathbb{R}/\mathbb{Z}$). It is also true ...
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74 views

Elements in $K_0(A)$

Let A be a $C^*$-algebra, unital or not. I want to show that each element in $K_0(A)$ is of the form $$[p]_0 - \bigg[ \begin{pmatrix} 1_n & 0_n \\ 0_n & 0_n \\ \end{pmatrix} \bigg]_0$$ for ...
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Univesal Coefficient Theorem for $C^*$-algebras

The UCT theorem is shown in the sreenshot. I have a question : What is the definition of $Ext_{\Bbb Z}^1(K_{*}(A),K_{*}(B))$? Does it have a relationship with Tor functor?
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Show that $K_0(A)$ is a countable abelian group when $A$ is a separable $C^*$-algebra.

I want to show that $K_0(A)$ is a countable abelian group when $A$ is a separable $C^*$-algebra and I know that this is the case when $A$ is a unital separable $C^*$-algebra as this has been shown ...
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$C_0(X)$ isomorphic to $C_0(X_1) \oplus C_0(X_2)$ when $X=X_1 \cup X_2$ for X locally compact Hausdorff

If X is a locally compact Hausdorff space and $X=X_1 \cup X_2$ where $X_1, X_2$ are disjoint open and closed subsets X, I want to show that $C_0(X)$ isomorphic to $C_0(X_1) \oplus C_0(X_2)$ I have ...
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Show that $\mathbf{D}(\mathbb{C}\bigoplus \mathbb{C})$ is isomorphic to the additive semigroup $\mathbb{Z}^+\bigoplus \mathbb{Z}^+.$

I am reading "An introduction to $C^*$ Algebra" by Rordam. Show that $\mathbf{D}(\mathbb{C}\bigoplus \mathbb{C})$ is isomorphic to the additive semigroup $\mathbb{Z}^+\bigoplus \mathbb{Z}^+.$ I don'...
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35 views

Show that $\mathcal{D}(\mathbb{C}) \cong \mathbb{Z}^+$

Let $Tr: M_n(\mathbb{C}) \to \mathbb{C}$ be the standard trace given by $$Tr \begin{pmatrix} x_{11} & x_{12} & x_{13} & \dots & x_{1n} \\ x_{21} & x_{22} & x_{23} &...
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Generators for $K_1(A\otimes \mathbb{K})$

I've been working computing generator for several $C^*$-algebras involved in my Master's thesis, however I've got stucked with the generators of $K_1(C(\mathbb{T})\otimes\mathbb{K})$, which is in my ...
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57 views

Trace, dimension and equivalence in $M_n(\mathbb{C})$

Let $tr: M_n(\mathbb{C}) \to \mathbb{C}$ be the standard trace given by $tr \begin{pmatrix} x_{11} & x_{12} & x_{13} & \dots & x_{1n} \\ x_{21} & x_{22} & x_{23} &...
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Chern Character of the dual bundle

Let $(E,\nabla)$ be a complex vector bundle with connection. Is there a formula for the Chern character of the dual vector bundle $(E^*,\nabla^*)$ in terms of the chern character/classes of $E$?
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39 views

What are the K groups of $X_n$, the wedge sum of $n$ circles at a single point?

If $X_n$ is the bouquet of n circles, then what is $K_0(X_n)$ and $K_1(X_n)$? I am super confused on how to approach this problem, since I've only seen $K_0$, $K_1$ computed for algebras, but I don't ...
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Help with a problem in K-theory/C- algebras.

For this problem you may assume the fact that $K_1(C_0(D)) = 0$. Let $n > 1$, let $\omega = e^{\frac{2πi}{n}}$, and let $E_n$ be the space obtained from $D$ by identifying $z$ and $\omega z$ for ...
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What are the K groups of $X_n$?

If $X_n$ is the bouquet of n circles, then what is $K_0(X_n)$ and $K_1(X_n)$? I am super confused on how to approach this problem, since I've only seen $K_0$, $K_1$ computed for algebras, but I don't ...
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Problem 22.39(c) in “Modern Classical Homotopy Theory ” by Jeffery Strom on pg.511.

Here is the problem: Suppose $R$ is a field. (a) Show that $h^{n}(?) = Hom_{R}(H_{n}(?; R), R)$ is a cohomology theory defined on (at least) the category of finite CW complexes. (b) Show that $u$ ...
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63 views

Show that a retract of a cofibration is also a cofibration.

Here is the question: Suppose that $g: A \rightarrow C $ is a retract of $f: B \rightarrow D.$ Show that if $f$ is a cofibration, then so is $g.$ Could anyone help me in answering this question, ...
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Problem 22.39(b) in “ Modern classical homotopy Theory ” by Jeffery Strom on pg.511.(u is the natural transformation of cohomology theories.)

Here is the problem: Suppose $R$ is a field. (a) Show that $h^{n}(?) = Hom_{R}(H_{n}(?; R), R)$ is a cohomology theory defined on (at least) the category of finite CW complexes. (b) Show that $u$ ...
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Problem 22.39(a) in “ Modern classical homotopy Theory ” by Jeffery Strom on pg.511.

Here is the problem: Suppose $R$ is a field. (a) Show that $h^{n}(?) = Hom_{R}(H_{n}(?; R), R)$ is a cohomology theory defined on (at least) the category of finite CW complexes. I got a hint to ...
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99 views

Verify that $\langle\alpha \cup \beta, u\rangle = \langle\beta, \alpha \cap u\rangle.$

For a commutative ring $R, \alpha \in \tilde{H}^p(X;R), \beta \in \tilde{H}^q(X;R)$ and $ u \in \tilde{H}_{p+q}(X;R),$ Verify $\langle\alpha \cup \beta, u\rangle = \langle\beta, \alpha \cap u\rangle.$ ...
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Problem 22.35 in “Modern Classical Homotopy Theory ” by Jeffery Strom on pg.510.

Here is the question: Show that is natural in both variables. That is suppose $f: X \rightarrow Y, u \in \tilde{H^{*}}(Y), \alpha \in \tilde{H_{*}}(X).$ Then we can form the cap products$$ \langle u,...
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Motivation for studying Operator $K_1$

I'm struggling to find motivation for studying $K_1$ for C*-algebras (Here I am talking about $K_1$ as the abelian group built from unitaries in $M_\infty(A)$ up to homotopy). Operator $K_0$ is the ...
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Subgroup of $K_0(A)$ generated by projections in $A$, where $A$ is a unital $C^*$-algebra

Let $A$ be a unital $C^*$-algebra. In what follows, I use the standard notation of $K$-theory for $C^*$-algebras. That is, $P_\infty(A):=\bigcup_{n\in\mathbb{N}} P_n(A)$, where $P_n(A)$ is the set of ...
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Calculating the cochain complex of $S^2 \times S^4. $

Calculating the cochain complex of $S^2 \times S^4. $ My calculation: $C^6 = \mathbb{Z}$ $C^5 = 0$ $C^4 = \mathbb{Z}$ $C^3 = 0 $ $C^2 = \mathbb{Z}$ $C^1 = 0$ $C^0 = \mathbb{Z}$ Am I correct? ...
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62 views

Describe a map $f: S^2 \times S^2 \rightarrow S^4$ such that $f^{*}$ is an isomorphism.

Here is the question: Describe a map $f: S^2 \times S^2 \rightarrow S^4$ such that $f^{*}: \tilde{H^4}(S^4, \mathbb{Z}) \rightarrow \tilde{H^4}(S^2 \times S^2, \mathbb{Z})$ is an isomorphism. Does ...
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Isomorphism of Projective Modules

Good folks! Having problem sorting out the solution to a question that should be relatively simple from Weibel's book of $K$-theory. Hoping you could help me. The question in particular is Exercise 2....
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172 views

Show that the orthogonal group acts transitively on the sphere $S^n.$

Show that the orthogonal group $$ O(n + 1) = \{ A \in GL(n+1 , \mathbb{R}) \mid A^{-1} = A^{T}\}$$acts transitively on the sphere $S^n,$ with stabilizer subgroup $O(n).$ Then use this to determine, ...
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What are $\tilde{H_*}(K;\mathbb{Z})$ and $\tilde{H^*}(K;\mathbb{Z})$?

Here is the question: Let $P$ be the projective plane and $K$ be the Klein bottle. What are $\tilde{H_*}(K;\mathbb{Z})$ and $\tilde{H^*}(K;\mathbb{Z})$? What is $\tilde{H_{*}}(K \times P; \mathbb{Z})$...
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Show that the space $Y = S^3 \vee S^6$ has precisely two distinct homotopy classes of comultiplications.

Here is the question: A comultiplication for a pointed space $X$ is a map $\phi : X \rightarrow X \vee X$ so that the composite $$X \xrightarrow{\phi} X \vee X \xrightarrow{i_{X}} X \times X$$ is ...
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Determine with justification, whether $S^2 \times S^4$ is homeomorphic to $\mathbb{C}P^3$.

Determine with justification, whether $S^2 \times S^4$ is homeomorphic to $\mathbb{C}P^{3}$. I know that they are not, but I do not know how to justify it , I got a hint that squaring a generator in ...
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Identify the homotopy type of $F.$

Here is the question: Let $F$ be the homotopy fiber of the inclusion $X \rightarrow X \times X.$ (1)Show that $\pi_{i}(F) \cong \pi_{i +1}(X).$ Here is the answer of this part: Show that $\pi_{i}(...
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Show that $\pi_{i}(F) \cong \pi_{i +1}(X) $ where $F$ is the homotopy fiber of the inclusion $X \rightarrow X \times X.$

Let $F$ be the homotopy fiber of the inclusion $X \rightarrow X \times X.$ (1)Show that $\pi_{i}(F) \cong \pi_{i +1}(X).$ Could anyone explain to me how to prove this, please?
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Exercise 16.10. on pg.356 in “Modern Classical Homotopy Theory.”

The question is asking to show that it suffices to prove Theorem 16.2 for a path connected space $X \in \mathcal{T_*}.$ Here is Theorem 16.2: I believe that the answer of this question is ...
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Show that $K_2(F)$ is a direct summand of $K_2F(t)$

I have a question regarding Example 6.1.2 (page 252) from the book "The K-Book" by Charles Weibel. Here is the statement: Example 6.1.2: Let $F(t)$ be a rational function field in one variable $t$ ...
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Progress on the Bass Finiteness Conjecture?

I am interested in proving the Bass Finiteness Conjecture, and making it my main project... I was wondering what recent developments there have been in the proof of the Bass Finiteness Conjecture... ...
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78 views

$a\otimes(-a) = a \otimes (1-a) + a^{-1}\otimes(1-a^{-1})$???

J. Browkin writes in his article "K-Theory, Cyclotomic Equations, and Clausen's Function" (which appears on Chapter 11 of Lewin's book "Structural properties of polylogarithms") that for any field $F$ ...
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$KO_*$ groups of $\mathbb{R}P^\infty$, “Snaiths” theorem for $KO$

I wonder whether anyone has taken the time to compute $KO_*(\mathbb{R}P^\infty)$? The standard tools to compute these Groups in the complex case rest on the requirement for the cohomology theories $E$...
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Identification of $M_n(\tilde{SA})$,where $\tilde{SA} $ is the unitalization of suspension $C^*$-algebra

In the section 11.1 of Rordam's book, there is a remark : If $A$ is a unital $C^*$-algebra, denote $SA$ by the suspension of $A$, $\tilde{SA} $ is the unitalization of $SA$. We can identify $M_n(\...
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How to find the set of the unitization of suspension $C^*$ algebras

Suppose $A$ is a $C^*$-algebra, then the suspension $SA$ is given by $SA=\{f\in C(\Bbb T,A):f(1)=0\}$. I saw the following conclusions from Olsen's book (page 136) Denote the unitization of $SA$ ...
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51 views

isomorphism between $K_1(A)$ and $K_0(SA)$.

The above theorem is from Rordam's book. I have a question: In the proof of Thorem 10.1.3,the author mentioned that we can use the identifications.How to prove the above two identifications?
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Geometric Resolutions of C* -Algebras

I am reading "K-theory FOR OPERATOR ALGEBRAS" Bruce Blackadar, Proposition 23.5.1. Let B be a separable $C^{*}$-algebra. Then there is a separable commutative $C^{∗}$-algebra F, whose spectrum ...
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Existence of finite dimensional ample subspace

This result is from M F Atiyah , K Theory. Let $E$ be a vector bundle over $X, ~ \Gamma(E)$ be set of sections of $E$. An ample subspace is a subspace $V$ of $\Gamma(E)$ such that the map $\varphi: X \...
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Simplicial resolution in cotangent complex

I am reading Cotangent complex from Loday's book Cyclic Homology. My doubt is related to the simplicial resolution of $k$- algebra $A$ which is used in the definition of cotangent complex.. Let me ...
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Functor between derivators to commute with homotopy colimits?

I am reading the following paper by Non Connective K theory via Universal Invariants. On page 35. One defines $$Hom_{!}(D,D')$$ To be the "homotopy colimit" preservsing functors of two derivators $D$...
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If $(E, \varphi, F) \in D(A, B)$, then $(E, \varphi, F)$ is homotopic to the 0-module.

I am reading proposition 17.2.3 Blackadar: If $ \varepsilon =(E, \varphi, T) \in \mathbb{D}(A, B)$, then $(E, \varphi, T)$ is homotopic to the $0$-module. I have some questions about the proof. We ...
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22 views

Let $X$ be an element in $\mathbb{E}(A,B)$, and there is a homomorphism between B and C then we can define an element in $\mathbb{E}(A,C).$

I am self-studying KK-theory. I came up with the following lemma: Let A, B and C be graded $C^{*}$-algebras, and $\phi: B \rightarrow C $ be an even *-homomorphism, and $X=(E, \pi , T)\in \mathbb{E}(...
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$d^3$ in the Atiyah-Hirzebruch spectral sequence for (twisted) $KO$

Cross posted to MathOverflow after no response with a bounty. Let $h^n(-)$ be a generalised cohomology theory. For a space $X$ there is a spectral sequence known as the Atiyah-Hirzebruch spectral ...
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40 views

definition of $G$-$C^*$ algebra

I wonder what is the precise definition of a $G$-$C^*$ algebra.The document I read gave the definition of $G$-$C^*$ algebras as following: $C^*$-algebras with a strongly continuous action by ...

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