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In the following, $X$ is a finite complex. The Adams operator $\psi^{-1}$ (complex conjugation) acts on $K^0(X)$. After inverting $2$, the group $KO^0(X) \otimes \Bbb Z[1/2]$ maps isomorphically to th …

Hi Dan, welcome to Math Overflow. The group you denote $K_m(X)$ is the image of the relative K-group $K(X,X^{(m-1)})$, which for nice spaces (e.g. finite CW-complexes) consists of equivalence classes …

Reduced $K$-groups are ideals of the standard $K$-groups. $\tilde K(X) \subset K(X)$ is the ideal of virtual-dimension-zero elements. In particular, the reduced K-theory $\tilde K(S^2)$ is not $\mat …

This is not true. One of the main reasons is that, because you've only specified properties about $K^*(X)$, that leaves a lot of room for $H^*(X;\Bbb Z)$ to have information which doesn't ultimately c …

This is really correct: the stabilization isomorphism isn't immediately compatible with the multiplication. This is a problem even in ordinary homological algebra and is largely a consequence of prete …

Inspired by the title of your question, you should look at Twists of K-theory and TMF by Ando-Blumberg-Gepner. For twists of $TMF$, there is a map $K(\mathbb{Z},4) \to BGL_1(TMF)$, and the latter cla …

A personal list. Hope the hypothetical author would not restrict the base to be a field, even perhaps allowing another DGA as base. (Most of the following comments probably assume this.) The flatne …

I want to add a thought about this question. The proof you're suggesting does seem very much along the lines of trying to turn Snaith's method, which shows K-theory is recoverable in some way, and tu …

To my knowledge, there is no such result for THH of a commutative ring. (Could be?) However, we need less for this result; we only need degree 1 elements to square to zero. Every element in $\pi_1 THH …

Here are the first, more straightforward, differentials in the AHSS $$H^p(X;KO^q(\ast)) \Rightarrow KO^{p+q}(X)$$ for real K-theory. Note $KO^q(\ast)$ is $$ \begin{cases} \Bbb Z &\text{if }q = 8k,\\ \ …

There are a lot of computational methodologies from algebraic topology that you can apply here, moving from less to more complicated. Suppose E* and E* is a pair of a generalized homology theory and …

There is a long exact sequence of (reduced) K-groups $$ K^{n-1}(X) \to KO^{n+1}(X) \to^\eta KO^n(X) \to^c K^n(X) \to^f KO^{n+2}(X) \to \cdots $$ The map $c$ is induced by complexification, sending a r …

By applying the thick subcategory theorem you can come up with lots of new definitions of "type", but it is somewhat unsatisfying because the fact that any of them are equivalent is very non-obvious. …

Degenerating at $E_1$ is, as you describe, equivalent to the image of the map $f_1: D_1 \to E_1$ being contained in the image of $g_1^i$ for all $i$. Roughly, this is because the definition of the $d_ …

For a general complex oriented cohomology theory represented by a ring spectrum $R$, there is a "Hurewicz map" from $R$ to its smash product $H\mathbb{Z}\wedge R$ with the Eilenberg-Mac Lane object fo …