Assume that $G$ is a compact Lie group and $M$ is a smooth oriented manifold on which $G$ acts freely. Then the orbit space $M/G$ is a smooth manifold with dimension $dimM-dimG$. I …

We can map from set of coverings over $X$ to symmetric group $\mathfrak{S}_n$ via monodromy (if we fix a loop at the basepoint). Also we can consider braid group $Br_n(Y)$, allow s …

Let us first precise the question : let $T$ be a torus, $\alpha : T \to \mathbb{C}$ be an irreducible character. I am interested in the $T$-equivariant Euler class of the ($T$-equi …

Let $X$ be a topological space, and $Y,Z$ be subspaces. My question is a bit vague and open-ended: when is it the case that, if $Y$ and $Z$ represent the same (nonzero) cycle in ho …

Suppose that I have a complex projective variety $X$ endowed with an algebraic action of a complex torus $T$. Suppose also that the set $X^T$ of fixed points is finite. I would lik …

Hello to everyone, I am studying the properties of combinatorial model categories, following the exposition given by Jacob Lurie in Higher Topos Theory ([HTT] from now on), in sec …

Let $\pi:E\longrightarrow B$ be a $G-$vector bundle and $s:B\longrightarrow E$ be an equivariant smooth section such that $s^{-1}(0)$ is compact, where $G$ is a compact Lie group. …

Let $X$ be a quasi-projective variety over the complex numbers, equipped with an action of a linear algebraic group $G$. Is there a natural mixed Hodge structure on its equivaria …

I would like to know if there is an explicit example of a finitely presented group that can not be realised as the (topological) fundamental group of a normal complex quasi-project …

In the equivariant Atiyah-Singer index theorem, when $G$ is a compact group acting on a manifold $M$ and $R(G)$ is the representation ring of $G$. We have the analytic index $$ \te …

Let $M$ be a smooth $G-$manifold, where $G$ is a compact Lie group. From the result of S.Illman that $M$ admits an equivariant triangulation. Moreover, we can construct the equivar …

Let $M$ and $N$ be two smooth compact, oriented manifolds and $X\subset M$ an oriented submanifold of $M$ of dimension $k$ (not necessarily closed). Suppose in addition that $\bar …

This question is specifically related to the spectra $X(n)$ used in Devinatz, Hopkins and Smith's proof of the nilpotence conjectures, but any general answer in terms of the Thom i …

Let a compact Lie group $G$ act on a manifold $X$. Let $\mathbb Q$ be the field of rational numbers and assume that cohomology $H^{\ast} (X)$ with $\mathbb Q$-coeffiecients is fi …

Let $X$ and $Y$ be two complex manifolds of dimension 2 and let $\varphi:X\rightarrow Y$ be an isomorphism. I have read that the action of $\varphi^*:H^2(Y,\mathbb{Z})\rightarrow …