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I don't know if the following is exactly what you're looking for. There is a theorem of Madsen-Snaith-Tornehave from 1977 that says that $gl_1 KO$ is equivalent to $K(\mathbb{Z}/2,0)\times K(\mathbb{Z …

I believe you will need to add some twisting to obtain Poincare duality. Specifically, if $w\in H^3(X,\mathbb{Z})$ represents the obstruction the your manifold possessing a $spin^c$ structure, then th …

To help answer Question 1, Milnor proved a local-global theorem for Witt rings of global fields. Recall that The Grothendieck-Witt ring $\widehat{W}(k)$ of a field $k$ is the ring obtained by starting …

In general, these presheaves are not sheaves, even on the etale sites of fields. As an easy example, $K_2(\mathbb{C})$ is non-torsion divisible, but $K_2(\mathbb{R})$ has a $2$-torsion element given i …

As a particular application of algebraic K-theory, let me mention the intersection product on regular schemes. Let X be a regular scheme over spec Z. Then, one can use the Quillen spectral sequence an …

I attempted the following baby version. Namely, I asked myself whether there exists a unitary map $S:H\oplus H\rightarrow H$ such that the induced map $Fred(H\oplus H)\rightarrow Fred(H)$ is $U(H)$-eq …

You can combine Adeel Khan's answer with Proposition 6.9 of my paper with David Gepner to prove that there are these kinds of localization sequences in a great deal of generality. (Note that our propo …

I would suggest the lectures of Friedlander and Weibel: "An overview of algebraic K-theory" in Algebraic K-theory and its applications (Trieste 1997), 1999; MR. The later lectures include the modern p …