Let $f :X \to Y$ be a submersion between smooth projective varieties over $\mathbb{C}$ and let $\alpha \in Z^k(X)$ be an algebraic cycle of $X$. Is is true that for all odd numbers …

Firstly I apologize that I am a physicist, with a relatively unrigorous math training. My approach of the problem can be Feynman style. Below $Z$ is the integer $\mathbb{Z}$, and $ …

In "Théorie des Faisceaux", Godement states the following theorem due to Leray (Theorem 5.2.5, page 209). Let ${\mathcal M}=(M_i )_{i\in I}$ be a locally finite closed covering o …

Hi, Is there any known sequence such that the sum of a combination of one subsequence never equals another subsequence sum. The subsequences should have elements only from the …

Let $I$ and $J$ be two ideals in $A$. Show that $\operatorname{Tor}_{1} (A/I, A/J) =\frac {I \cap J} { IJ} $ and $Tor_{2} (A/I, A/J) =\ker(I \otimes_ {A}J \to IJ )$. The first …

We know that group cohomology $H^2(G,U(1))$ consists of 2-cocycles $\beta(A,B)\in U(1)$ corresponding to elements in the group $H^2(G,U(1))$, where $A\in G,B \in G$. Note that $\be …

We know that: Torsion product Tor$^R_1(,)$, which maps two modules, $M_1$ over $R$ and $M_2$ over $R$, to a third module $M_3$ over $R$: $M_3$ = Tor$^R_1(M_1,M_2)$. See for examp …

In arXiv:math/0607544 (following conjectures in arXiv:math/0505662), Rasmussen constructs a family of spectral sequences (the "d_N differentials"), starting at the HOMFLY homology …

Suppose that $G$ is a group acting on a fibre bundle $(F,E,B)$ by bundle automorphisms. In this case, the action automorphisms $E\to E$ give the integral homology $H_\ast(E;\mathbb …

I am interested in whether the transgression maps for group cohomology and group homology are related via a version of the universal coefficient theorem. Let $G$ be a group, $H$ a …

Let $F: \mathcal{A} \to \mathcal{B}$ be an exact covariant functor of abelian categories and let $$\mathscr{C}: A \to A \to B \to A$$ be an exact couple in $\mathcal{A}$ with corr …

Hi all, on the forum page http://www.groupsrv.com/science/about506648.html one can read the following (i cut out nonimportant parts): Question: Does anyone know any condition (no …

Let $X$ be a smooth variety over a field $k \subset \mathbb{C}$ and $Z$ a smooth subvariety. Let $U=X-Z$. I'm trying to understand what information do the Leray spectral sequences …

While preparing some lecture notes, I had a basic point of confusion come up that I haven't been able to settle. The $BP$-Adams spectral sequence (or $p$-local Adams-Novikov spect …

I would like to compute the first few stable homotopy groups of $RP^2$. I first thought to use the Atiyah-Hirzebruch Spectral Sequence, (see Davis & Kirk, pg. 242). Here is w …