We show that for a finite group $G$, the $G$-category $\underline{\mathrm{Sp}}_{G,\mathbb{Q}}$ of rational genuine $G$-spectra is parametrized semiadditive, in the sense of Cnossen--Lenz--Linskens, for morphisms of genuine $G$-spaces whose fixed point maps have $\pi$-finite fibers. In particular, this means that the canonical norm map between a colimit and a limit of rational genuine $G$-spectra indexed by such $G$-spaces is an equivalence. This extends the Wirthm\"uller isomorphism from $G$-orbits to a broader class of $G$-spaces and combines the theory of ambidexterity of Hopkins and Lurie with parametrized category theory.
Let $M$ be a closed, smooth or topological $n$-manifold, with $n \geq 4$. We construct a homotopy highly cartesian square relating the space ${\mathcal E}(M (\ell))$ of homotopy self-equivalences of $M$ (in a suitable range) over the Postnikov $\ell$-sections of its stable normal microbundle, and an $(\infty + n)$-fold...
Let $X$ be a connected finite-type CW-complex with fundamental group $G$. We show that the profinite completion $\widehat{G}$ determines the cohomology jump loci $\mathcal{V}^q_s(X,\mathbb{C})$ under two hypotheses: that the loci are finite unions of torsion-translated subtori, as for smooth quasi-projective varieties,...
Let $\mathscr{O}_K$ be a mixed characteristic complete DVR with perfect residue field $k$, and let $\mathfrak{X}$ be a proper formal scheme over $\mathscr{O}_K$ with semistable reduction. Answering a question of Fontaine and Jannsen, a classical theorem of Hyodo and Kato gives a rational identification between the log...
We prove an equivariant Landweber exact functor theorem for abelian compact Lie groups $G$. For a graded module $N$ over the $G$-equivariant Lazard ring $L_G$, we give necessary and sufficient algebraic conditions on $N$ for the functor \[ X\longmapsto (MU_G)_*(X)\otimes_{L_G} N \] to define a homology theory on the ca...
For a finite group $G$, we define a constant spectral $G$-Mackey functor as a genuine $G$-spectrum in which the restrictions between categorical fixed points are equivalences. We further identify the full subcategory of constant spectral $G$-Mackey functors with the category of modules over a certain $\mathbb{E}_1$-rin...
We disprove Bohmann's equivariant generating hypothesis for every nontrivial finite group $G$, even when all $\RO(H)$-graded homotopy groups at every subgroup $H$ are tested. For each fixed $G$ and prime $p$ dividing $|G|$, we construct ghosts on finite $G$-spectra with arbitrarily long nonzero composition powers. We a...
Si-Hao Ma, X. Shi, Zhou-Li Xu et al.· 0 citations
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