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For a space1 $X$, let $\mathcal{L}X = \mathrm{Maps}(S^1, X)$ be the free loop space. Inclusion of constant loops gives a natural map $X \to \mathcal{L}X$. This is not a homotopy equivalence unless $X …

Certain spheres admit nontrivial fibrations, i.e. the Hopf fibrations and the maps to projective spaces. Also, a product of spheres is a sphere bundle in more than one way. Are there manifolds …

I was recently in the situation of having access to the category of $G$-modules (for some group $G$ which I had forgotten), as just a category, i.e. no monoidal structure, together with the forgetful …

For a complex $C^{\bullet}$ of finite dimensional vector spaces, one has a determinant $$|C^\bullet|:= \bigotimes \left(\Lambda^{top} C^i\right)^{(-1)^i}$$ functorial with respect to quasi-isomorphi …

Let $X$ be a manifold. Does the Hochschild homology/cohomology of the category of constructible sheaves on $X$ have a more familiar name?

Let $X$ be a space good enough to have a fundamental class, and $E$ a complex vector bundle on $X$. Let $P$ be some polynomial expression, and say I want to evaluate $P(c_i(E)) \cap [X]$. Is the …

Consider an oriented manifold $X$. To calculate its Euler characteristic, one might integrate the Euler class. Now if $X$ were a complex manifold, and given as a section of some complex bundle $E$ o …

Let $X$ be the sort of topological space for which it makes sense to talk about the intersection homology. Fix a perversity $p$, or just take $p= 1/2$ if you like. Is there some naturally d …

It is a bit unclear what you are asking. The thing you call the geometric genus is certainly not the geometric genus; the literal object you wrote is $\infty$ and if you first compactified the curve …

Such a variety always has Euler number zero. This is because: Due to being cut out by homogenous equations, it is invariant under the $\mathbf{C}^*$ action $t.(x_1,\ldots,x_n) \to (tx_1, \ldots, t …

For singular plane curves, there is a conjectural formula (due to Alexei Oblomkov and myself) in terms of the HOMFLY polynomial of the links of the singularities. For curves whose singularities are t …