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Let $MV_F$ be the $\mathbf{A}^1$-homotopy category over the field $F$. Let $H$ be the homotopy category of spaces, and let $H_{/S^1}$ be the homotopy category of spaces over the circle. When $F = \ma …

Let $X$ be a hyperbolic 5-manifold. Can there be any class in $H^2(X;\mathbf{Z})$ that is torsion and whose square in $H^4(X;\mathbf{Z})$ is not zero? For example, are there hyperbolic 5-manifolds t …

If $M$ is an oriented $d$-manifold, let $D(M)$ denote the top exterior power of $H^*(M,\mathbf{C})$. Then $D(M_1 \amalg M_2) = D(M_1) \otimes D(M_2)$. Is there a good recipe for a map $D(M) \to D(N) …

If you triangulate your space refining the stratification, a constructible sheaf is given by the data of a vector space $V_{\sigma}$ (a stalk at the barycenter, say) on each simplex $\sigma$ and a res …

To be safe, let me assume the cohomologies are taken with coefficients in a field, like $\mathbf{C}$. Let $I' \subset I$ be the indices for which $U_i$ is nonempty. The incidence algebra of $I'$ is …

A while ago I heard that there was no known triangulation of the Grassmannian of 3-planes in 6-space. To believe a statement like that, you have to be a little bit ungenerous about what you mean by " …

I don't know a very general answer. Your duality on cell complexes resembles Verdier duality and has a local nature, but some of these equivalences aren't like that. E.g. $K = (0 < 1 < 2)$ and $K^{o …

Let R be the 2-periodic complex K-theory spectrum, or any other naturally occuring 2-periodic E-infty ring spectrum. The suspend-once functor gives an autoequivalence of the category of R-module spec …

Here's another proof based on the structure of the flag variety $G/T$ of $G$. A compact Lie group $G$ has a maximal torus $T$, and $G$ is a principal $T$-bundle over the quotient $G/T$. Borel showed …

No. Counterexamples were first constructed by Winkelmann, as quotients of $\mathbb A^5$ by algebraic actions of $\mathbb G_{\text{a}}$. I learned this from Hanspeter Kraft's very nice article availa …

Let C be the category of homogeneous G-manifolds; the hom sets have a natural topology so you can consider C as an infinity-category. The equivariant homotopy category is the category of contravarian …