# Tag Info

Accepted

Accepted

### Persistence barcodes and spectral sequences

The answer to your question is no, nobody has used persistence to improve the algorithmic efficiency of computing differentials, although of course the relationship between persistence intervals of a ...
Accepted

### Sphere spectrum, Character dual and Anderson dual

The Anderson dualizing spectrum $I_\mathbf{Z}$ can be defined as follows. Consider the functor $X\mapsto \mathrm{Hom}(\pi_{-\ast} X,\mathbf{Q/Z})$ from the homotopy category of spectra to graded ...
Accepted

### What is the relationship between spectral sequences and obstruction theory?

This is a partial answer, but every obstruction theory (in some precise sense) provides you with a spectral sequence (in fact several). Let me clarify what do I mean with obstruction theory. All this ...

### cup product and Steenrod operations in Serre spectral sequence

The behavior of the Steenrod squaring operations in the Serre spectral sequence was determined by Araki and independently by Vázquez (whose article I cannot locate online). However, it's a little work ...
Accepted

### "Rotated" version of the Atiyah-Hirzebruch spectral sequence

Good question. I think the answer is yes. The unnamed spectral sequence is usually referred to as the isotropy spectral sequence. For a group $G$ acting on $X$ and an abelian group $A$ of ...

### To compare the total, base and fiber spaces of two fiber bundles

No. Consider the map from the fibre bundle $$B\mathbb{Z} \to BD_\infty \to B\mathbb{Z}/2$$ to $* \to * \to *$. Here $D_\infty = \mathbb{Z} \rtimes \mathbb{Z}/2$ is the infinite dihedral group. You ...
Accepted

Accepted

### Hodge Numbers and Leray Spectral Sequence

I don't think I defined the Hodge numbers in this way. Rather, the argument in Section 1 shows that the Hodge numbers agree with the dimensions of the terms in the $E_2$ page of the Leray spectral ...
Accepted

### Cohomology ring of a fiberwise join

What you wish to prove is not true. Namely, if $I=\operatorname{ker} f^*$ then $I^2\subseteq \operatorname{ker}(f\ast f)^*$ but the inclusion may be strict. The fibred join construction comes up in ...

### Pullback and homology

Here is a positive answer to a slightly different question. Call a map $X\to B$ "acyclic" if it induces an isomorphism in homology for every coefficient system on $B$. (If $B$ is simply connected ...

### Why does strong convergence of the EMSS imply that Tot commutes with suspension spectrum?

I'm going to avoid the question and answer the edit. Hopkins has some results on this in his Oxford thesis which were announced without proof in his Asterisque paper of 1984. For a quite thorough ...
### Torsion in the integral cohomology of $BPU_{n}$
You may want to have a look at this paper: X. Gu. On the cohomology of classifying spaces of projective unitary groups. arXiv:1612.00506, (link to arXiv) The spectral sequence involving $BSU_n$ ...