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As $AG$ is noetherian of dimension 1*, finitely-generated projective $AG$-modules are classified by their rank and determinant, and so $$K_0(AG) = \mathbb{Z} \oplus Pic(AG).$$ Proposition 10.3.8 of to …

No, you can't generally do this even with the added assumptions. The bundle $\tau = TS^{n-1} \to S^{n-1}$ is non-trivial for $n-1 \neq 1,3,7$, but it is stable after (one) trivialisation. Hence $\tau …

This seems to be Theorem 3.8.6 in Lurie's DAG-VI, which he says is also in Paolo Salvatore, "Configuration spaces with summable labels", Cohomological Methods in Homotopy Theory. Progress in Mathemati …

There are two questions you could ask: about the space of immersions $Imm((M, \partial M), (N, \partial N))$ of $M$ in $N$ taking the boundray to the boundary, where the boundary is allowed to move, o …

In my paper ``Generalised Miller--Morita--Mumford classes for block bundles and topological bundles" with Johannes Ebert, we construct a block $\mathbf{HP}^2$-bundle $\pi: E^{20} \to S^{12}$ which can …

I don't believe this is true. Let $(G, H) = (\Sigma_3, C_3)$ and $f : C_3 \to \Sigma_3$. Then your square says that $$H^1(C_3;\mathbb{Z}/3) = \mathbb{Z}/3 \longrightarrow H^1(\Sigma_3;\mathbb{Z}/3) = …

This probably depends on your definition of homotopy colimit, but it you mean ``the geometric realisation of the simplicial replacement" then it seems to me that $X /\ \!/_h M$ is homeomorphic to $X \ …

We have $H^*(BO, \mathbb{F}_2) = \mathbb{F}_2[w_1, w_2, \ldots]$, where the $w_i$ are the Stiefel--Whitney classes. If $f: \mathbb{RP}^\infty \to BO$ classifies the reduced universal real line bundle, …