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

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

651
questions

**4**

votes

**1**answer

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

**0**answers

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}(\...

**7**

votes

**0**answers

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

**1**answer

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}(...

**9**

votes

**0**answers

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

**1**answer

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

**1**answer

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\...

**1**

vote

**0**answers

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

**1**answer

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

**0**answers

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

**2**answers

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**

votes

**0**answers

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**

votes

**0**answers

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

**1**answer

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**

votes

**4**answers

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{...