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Let me write $V$ for a finite-dimensional vector space over some field (the field will not play a role), and $\mathsf{P}(V)$ for the poset described in the question, which I consider as a category. Le …

The group $U(1) \rtimes \mathbb{Z}/2$ you describe is nothing but the group $O(2)$ (as $U(1) = SO(2)$). As such I think one can see the spectral sequence for the extension does collapse, and one obta …

No. For a $G$-manifold $M$, taking the signature of the fixed points $M^G$ defines a homomorphism $\phi : \Omega_n^G \to \mathbb{Z}$, as if $W : M_0 \leadsto M_1$ is a cobordism then so is $W^G : M_0^ …

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, …

EDIT: This is wrong, as Jens explains below. It is enough to show that the isotropy groups of the 0-cells are trivial: higher dimensional cells must have smaller isotropy. We may suppose, by restr …

I think $f: M /\!\!/ G \to B\Gamma$ cannot be nontrivial on $\mathbb{Q}$-(co)homology in degrees beyond the dimension of $M$, because I think one can find a factorisation $$f_* : H_*(M /\!\!/ G ; \mat …