# Tag Info

Accepted

### Are long exact sequences in homology a special case of spectral sequences?

To answer the question in the title : yes, long exact sequences can be derived from the degeneracy of some spectral sequences. In fact, there are several spectral sequences leading to the long exact ...
• 12.6k
Accepted

### Wang Sequence for the circle $S^1$

Here is the Mayer-Vietoris argument : Let $N,W,S,E$ be the north, west, south and east pole on the circle. Let $U=S^1\setminus\{E\}$ and $V=S^1\setminus \{W\}$. This is an open covering of $S^1$ such ...
• 12.6k
Accepted

### Understanding Adams spectral sequence and Pontryagin-Thom isomorphism intuitively

Your questions seem very broad, and would probably be better served by a textbook reference. I'll try to give some perspectives that might help. Given two spectra $X$ and $Y$, we would compute the ...
• 11.1k
Accepted

### Question about isomorphism given by the Serre Spectral sequence

In the Serre spectral sequence the edge homomorphism $$H_p(X;M)\to E^\infty_{p,0}\subset E^2_{p,0}=H_p(Y;M)$$ is the map induced by $X\to Y$. If $F$ is acyclic then the Serre spectral sequence ...
• 3,869
Accepted

### Grading of Cech-de Rham cohomology

You are quite right that $H_D^n$ is single-graded. However, there is a filtration$$H_D^n = F_0 \supset F_1 \supset F_2 \supset \ldots \supset F_n \supset 0$$on $H_D^n$, and $H_D^{p, n - p}$ is defined ...
• 1,432

### Calculating the extraordinary cohomology of $\mathbb{C}P^n$

Your condition should be that the orientation is an element of the reduced $E$ cohomology of $CP^\infty$, restricting to a generator of $\pi_0(E)$. Look at the reduced AHSS. The elements with ...
Accepted

Accepted

• 1,356

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

In general, there is a way to classify extensions of group $1 \to N \to G \to Q \to 1$ for given $Q$ and $N$. You need a group morphism $\phi$ from $Q$ to $Out(N)$ (the outer automorphism group of ...
• 11.6k

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

Just choose an open, normal subgroup of a compact group, eg. $$1\to p\Bbb Z_p\to \Bbb Z_p\to \Bbb Z/p\to 1$$ So long as the upstairs group is Hausdorff, the open subgroups are exactly the closed ...
• 37k
Accepted

### fibrations and map of spectral sequences.

You do get an induced map between the $E_r$ pages for $2 \leq r \leq \infty$, but a priori the map on the $E_\infty$ page coincides with $f^*$ only up to filtration. Here's a (somewhat silly) ...
• 11.1k
Accepted

### Serre classes and the Serre spectral sequence

What about the bundle $\mathbb{Z}/2 \to S^n \to \mathbb{R}P^n$ where $n>0$ is even? Then for all $k>0$ the groups $H_k(\mathbb{Z}/2)$ and $H_k(\mathbb{R}P^n)$ are all in the Serre class of ...
• 9,380

### Derived proofs of elementary homological algebra theorems?

I have not read it too much, but in the beggining (1.3) of this the author relates derived stuff, Künneth and spectral sequences. Thus it seems that at least this short paragraph might be something ...
• 9,040