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Observe that the $KK$-class $\sigma \in KK_1(A/J,J)$, which you mention in your edited paragraph only depends on the extension $$ 0 \to J \to A \to A/J \to 0 $$ and not on $B$. So we have $\delta_1^n …

Let $Y$ be the Fermat quintic, i.e. $Y \subset \mathbb{C}P^4$ is defined by $$ \sum_{i=1}^5 z_i^5 = 0 $$ In Section 5.3 of this paper by Volker Braun the author computes the K-groups of a quotient $X …

This is a variation of an argument that is called the Eilenberg swindle. It can be found for example in Blackadars book "K-theory for operator algebras" (see Proposition 12.2.1) and works as follows: …

Hi, I came across the space $BU_\otimes$ when struggling with twisted K-theory. Segal proved that this is an H-space, right? I have read a dozen times by now that the group $[X, BU_\otimes]$ consist …

This is just an addendum to Sebastian Goette's excellent answer: In fact, you can retrieve the integer-valued function that you mention from the $K$-theory class: Let $\iota_x \colon \{pt\} \to X$ be …

I think the key idea is that Connes' Thom isomorphism is itself given by a $KK$-equivalence (see for example Blackadar's book "K-theory for Operator Algebras" - Theorem 19.3.6). This means there are …

As David Handelman already pointed out, by continuity of $K$-theory we obtain that $K_0(C(X)) \cong \bigoplus_{\mathbb{N}} \mathbb{Z}$ and $K_1(C(X)) = 0$. Since $C(X)$ is commutative, it lies in the …

This is an awkwardly backwards question, but bear with me here: Suppose I have a graded ring $R$ with unit, which has an invertible element $u$ in degree $2$. The multiplicative formal group law $f(x …

There is a paper by Jonathan Rosenberg with the title "The K-homology class of the Euler characteristic operator is trivial" see here, which proves that the only information contained in the class of …

There is a version of $KK$-theory for Banach algebras, which was developed by Lafforgue. There also is a paper titled Banach KK-theory and the Baum-Connes conjecture, which is probably relevant for th …

Well, I am by no means an algebraic geometer and maybe this is not even helpful, but anyway: If you have a bundle $G^\perp$, such that $G \oplus G^\perp \simeq \underline{\mathbb{C}^n}$ is trivial an …

Maybe I am mistaken, but I think the actions they consider should be such that the projection map $\pi \colon P \to X$ is equivariant with respect to $G$. So, if the action of $G$ on $X$ is trivial (a …

By the Atiyah-Jänich theorem the K-group $K^0(X)$ for a compact space $X$ may be represented as $[X, U(Q)]$, where $Q = B(H)/K(H)$ is the Calkin algebra and $H$ is a separable infinite dimensional Hil …

I read in a paper by Christopher Douglas that third cohomology twists of $K$-theory may be interpreted as TMF-classes via a map $K(\mathbb{Z},3) \to TMF$, which is related to String orientations. How …

Consider the following two ways of getting the zeroth space in the $K$-theory spectrum $BU \times \mathbb{Z}$: 1) Take the groupoid of finite dimensional complex inner product spaces with isometries …