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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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      4
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      1answer
      88 views

      Kuenneth short exact sequence for K-homology

      Atiyah proved a Kuenneth short exact sequence for K-theory. I need one for K-homology, but can not find any reference in the literature. Do you know one? Using general spectra stuff, one gets a ...
      4
      votes
      0answers
      166 views

      Different algebra-structures on $\operatorname{THH}(\mathbb F_p)$?

      By definition, we have a ring map $\mathbb F_p\to\operatorname{THH}(\mathbb F_p)$. Post-compose with the canonical map $\mathbb Z_p\to\mathbb F_p$, we get a ring map $\mathbb Z_p\to\operatorname{THH}(\...
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      0answers
      246 views

      Reference request: complex K-theory as a commutative ring spectrum

      Does anyone know of a point-set level model for complex K-theory as a commutative ring spectrum? For real $K$-theory I know of "A symmetric ring spectrum representing KO-theory" by Michael Joachim (...
      6
      votes
      1answer
      113 views

      Coarse index of Dirac operator on $\mathbb{R}$

      Let $D=i\frac{d}{dx}$ be the Dirac operator on $\mathbb{R}$, acting on the spinor bundle $\mathbb{R}\times\mathbb{C}$. The bounded operator $F=\frac{D}{\sqrt{D^2+1}}$ has a coarse index $$\text{Ind}(...
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      0answers
      152 views

      Geometric motivation behind the Fredholm module definition

      If $A$ is an involutive algebra over the complex numbers $\mathbb{C}$, then a Fredholm module over $A$ consists of an involutive representation of $A$ on a Hilbert space $H$, together with a self-...
      3
      votes
      1answer
      172 views

      A question about the group $[HZ,KU]$

      I don't know if the following question is obvious, but can't figure it out. I want to ask if it is known what $[HZ,KU]$ is? Here $KU$ is the complex $K$-theory.
      4
      votes
      1answer
      116 views

      Mapping cone and derived tensor product

      This question is in some sense a continuation to this question: Derived Nakayama for complete modules For the setting: Let $A$ be a ring and let $I$ be some finitely generated ideal in $A$. Let $f\...
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      0answers
      453 views

      Algebraic K-theory of schemes and cohomology

      Are there examples of: two smooth projective schemes over a field having homotopy equivalent algebraic K-theory spectra and having different rational Voevodsky motives; two smooth projective schemes ...
      10
      votes
      1answer
      267 views

      Where does the $\hat A$ class get its name?

      In K-theory we have the Todd class and the $\hat A$ class. The Todd class is named after the Cambridge geometer John Arthur Todd. Where does the name $\hat A$ come from? Does the A stand for Atiyah?...
      3
      votes
      0answers
      41 views

      Minimum rank of inverse complex vector bundles

      When considering vector bundles (real or complex) over a compact manifold, i know about the existence of inverse bundles. That is, if $\xi$ is a vector bundle over $M$, then there is a bundle $\nu$ ...
      5
      votes
      2answers
      269 views

      Is every dg-coalgebra the colimit of its finite dimensional dg-subcoalgebras?

      I saw this result in A Model Category Structure for Differential Graded Coalgebras by Getzler-Goerss, but when the coalgebra is non-negatively graded, is this property also satisfied when the dg ...
      9
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      0answers
      187 views

      The term “absolute geometry”

      My question concerns the so-called absolute geometry over the "field with one element" F_1 or over the spectrum $\mathrm{Spec}(F_1)$, cf. https://ncatlab.org/nlab/show/Borger%27s+absolute+geometry. I ...
      2
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      0answers
      55 views

      Closable operators on Hilbert modules

      For $T:{\frak{Dom}}(T) \to H$ a densely defined operator, with $H$ a (separable) Hilbert space, we know that $T$ is closable if its adjoint $T^*$ has dense domain in $H$. Does this extend to the (...
      4
      votes
      1answer
      128 views

      Producing $K$-homology cycles from $KK$-cycles

      For two unital (separable) $C^*$-algebras $A$ and $B$, let $(H,\rho,F)$ be a $KK$-cycle in the sense of Kasparov, or in the sense of Wikipedia :) I wonder if there us a natural way to "forget" the ...
      10
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      4answers
      519 views

      Maps from mod-$p$ Eilenberg-MacLane spectrum to connective $K$-theory spectrum

      Let $ku$ be the connective cover of the complex $K$-theory spectrum $KU$. Consider the mod-$p$ Eilenberg-MacLane spectrum $H\mathbb{Z}/p$. I want to see that $[H\mathbb{Z}/p,ku]=0$. Since $H\mathbb{...

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