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        Questions tagged [kt.k-theory-and-homology]

        Algebraic and topological K-theory, relations with topology, commutative algebra, and operator algebras

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        votes
        1answer
        101 views

        Morita equivalence of the invariant uniform Roe algebra and the reduced group C*-algebra

        In his paper "Comparing analytic assemby maps", J. Roe considers a proper and cocompact action of a countable group $\Gamma$ on a metric space $X$. He constructs the Hilbert $C^*_r(\Gamma)$-module $L^...
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        1answer
        217 views
        +50

        Comparing the definitions of $K$-theory and $K$-homology for $C^*$-algebras

        In Higson and Roe's Analytic K-homology, for a unital $C*$-algebra $A$, the definitions of K-theory and K-homology have quite a similar flavor. Roughly, the group $K_0(A)$ is given by the ...
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        votes
        1answer
        261 views

        The connective $k$-theory cohomology of Eilenberg-MacLane spectra

        Consider the connective $K$-theory spectrum $ku$. Let $H\mathbb{Z}$ be the Eilenberg-MacLane spectrum and $H\mathbb{F}_p$ be the mod-$p$ Eilenberg-MacLane spectrum. Is it known what $ku^{*}(H\...
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        votes
        1answer
        117 views

        A question on the ring structure of topological K-theory and Chern character

        Let $X$ and $Y$ be compact spaces (or closed manifolds if you want). I have two questions relating the ring structure of topological $K$-theory. To motivate my questions let me give some background ...
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        votes
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        96 views

        Is it possible for the Witt group of a scheme to have non-trivial odd torsion?

        Let $F$ be a field of characteristic not $2$. A well-known theorem of Pfister asserts that the torsion of $W(F)$, the Witt group of $F$, is $2$-primary. Baeza [B, V.6.3] extended this result to Witt ...
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        votes
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        55 views

        How does the $C^\ast$ algebra of an orbifold grupoid relate to the corresponding orbifold?

        My question is in nature a bit vague but let me try to make it concrete. Given a Lie grupoid $G$ that is étale and proper (called an orbifold grupoid) we have an associated orbifold $X$; this is ...
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        0answers
        135 views

        Blocksum induces a unital H-space structure on the space of Fredholm operators

        Fix a complex separable infinite-dimensional Hilbert space $H$. It is well known that the space of (bounded) Fredholm operators $Fred(H)$ with the norm topology is a classifying space for the ...
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        votes
        1answer
        140 views

        Is the tensor product of pretriangulated dg-categories a pretriangulated dg-category?

        In "Grothendieck ring of pretriangulated categories", Bondal, Larsen and Lunts define a product of perfect (pretriangulated with Karoubian homotopy category) dg-categories as $A\bullet B:=Perf(A\...
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        votes
        0answers
        358 views

        Comparing real topological K-theory and algebraic K-theory

        Let $R$ be a commutative unital ring and let $i$ be a non-negative integer such that $K^i_{alg}(R)$ is finitely generated abelian group. Is it possible that there does not exist weak homotopy type of ...
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        votes
        1answer
        200 views

        Twisted spin bordism invariants in 5 dimensions

        [Note]: My question will be a bit long. So, first, thank you for your careful reading, generous comments, helps and answers, in advance! The spin $G$-bordism invariant can be twisted in the way that ...
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        votes
        0answers
        65 views

        Existence of anti-symmetric hochschild homology representatives

        Let $A$ be an associative algebra over a field $k$. Let $A_{L}$ be the Lie algebra of $A$ with commutator bracket. Then if $M$ is a bimodule for $A$ there is an associated representation of $A_{L}$ ...
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        votes
        1answer
        175 views

        Hodge theory, conformal manifolds and Fredholm modules-understanding the proof of one Lemma

        I would like to understand the proof of Lemma 1, page 339 in this book. Very briefly, the context is as follows: we have even dimensional oriented conformal manifold with the Hodge star operator ...
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        votes
        1answer
        587 views

        Which spaces have trivial K-theory?

        What is known about spaces $X$ with the property that $K^*(\text{point})\to K^*(X)$ is an isomorphism? The same question for $K$-homology $K_*(X)\to K_*(\text{point})$; I don't even know whether ...
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        votes
        1answer
        142 views

        $*$-algebras, completions, and $K$-theory

        What is an example of a $*$-algebra $\cal{A}$, which admits two non-equivalent norms $\| \cdot \|_1$ and $\| \cdot \|_2$, with respect to which we can complete $\cal{A}$ to give two $C^*$-algebras $...
        6
        votes
        0answers
        105 views

        Pin cobordism v.s. “KO” theory in low or in any dimensions

        Fact: The spin cobordism is equivalent to "KO" theory in low dimension if we only consider the 2-torsion. This is related to a question and an answer supports the claim. Here we denote the $p$-...

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