Articles of spectral sequences

Bockstein homomorphism and Steenrod square

question: What is the relation between Bockstein homomorphism and Steenrod square? For example, can one explain why the following relation works in the case of cohomology group with $\mathbb{Z}_2$ coefficient? For $x \in H^m(M^d,\mathbb{Z}_2)$, $$ \beta_2 x=\frac{1}{2} d x \text{ mod } 2 \in H^{m+1}(M^d,\mathbb{Z}_2) $$ which is the Bockstein homomorphism. It turns out that […]

Why all differentials are $0$ for Serre Spectral Sequence of trivial fibration?

Consider the fibration $F \hookrightarrow F \times B \to B$. I understand that if I take kunneth’s theorem for granted that the group extensions $F_{n-i,i} \to F_{n-i+1,i-1} \to F_{n-i+1,i-1}/F_{n-i,i}$ associated to the filtration $F_{n,0} \supset …F_{n-i,i}…$ are all trivial extensions, and that $d_r=0$ for $r>2$. How can we deduce these two facts without assuming Kunneth’s […]

Cover and extension of a Lie group

We know that $SU(2)$ is a double cover of $SO(3)$, such that $$SU(2)/Z_2=SO(3),$$ through a finite extension $N=Z_2$. For other examples of simply-connected Lie groups such as $SU(2)$, $SU(N)$ or $E_8$, (1) are there nontrivial extensions of $SU(2)$, $SU(3)$, $SU(N)$ or $E_8$? (2) are there nontrivial finite extensions of $SU(2)$, $SU(3)$, $SU(N)$ or $E_8$? Please […]

Examples of group extension $G/N=Q$ with continuous $G$ and $Q$, but finite $N$

Can we have some (new) examples of group extensions $G/N=Q$ with continuous (e.g. Lie groups) $G$ and $Q$, but a finite discrete $N$? Note that $1 \to N \to G \to Q \to 1$. What I know already contains: $$SU(2)/\mathbb{Z}_2=SO(3).$$ $$\frac{\mathbb{R}/{\mathbb{Z}}}{\mathbb{Z}_n}={\mathbb{R}}/{(n\mathbb{Z})}.$$ What else are the examples that you can provide? A systematic answer to obtain […]

Calculate the cohomology group of $U(n)$ by spectral sequence.

Here $U(n)$ is the unitary group, consisting of all matrix $A \in M_n (\mathbb{C})$ such that $AA^*=I$ Problem How to calculate the integer cohomology group $H^*(U(n))$ of $U(n)$? What if $O(n)$ replace $U(n)$? My primitive idea is that: as for $U(n)$, it is a Lie group and can acts transitively on the unit sphere with […]