# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### $d^3$ 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 ...
### 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 ...