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Results tagged with homological-algebra
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user 124004
(Co)chain complexes, abelian Categories, (pre)sheaves, (co)homology in various (possibly highly generalized) settings, spectra, derived functors, resolutions, spectral sequences, homotopy categories. Chain complexes in an abelian category form the heart of homological algebra.
6
votes
Accepted
Groups with unusual cohomological dimension of direct product
Let $G=(\mathbb{Q},+)$. Then ${\rm cd}(G)=2$ and ${\rm cd}(G\times G)=3$.
- 3,451
4
votes
Dimension of classifying space of a group
No, not even in the case when $G$ has a 1-dimensional classifying space: if $G$ is free of rank at least two, then for any $N$ with $G/N\cong \mathbb{Z}$, $N$ will be free of infinite rank and so $N$ …
- 3,451
6
votes
Is there a finitely presented group with infinite homology over $\mathbb{Q}$?
This answers the other part of your question, not answered by Thompson's group. For each $i\geq 3$ there is a finitely presented group $G_i$ with the property that $H_i(G_i\mathbb{Q})$ is infinite di …
- 3,451
6
votes
Accepted
can the actions of fundamental groups annihilate homology?
There are finitely presented groups that do not have any non-trivial linear representations, so for these groups as fundamental group you are just asking whether the ordinary real homology of $X$ is t …
- 3,451
8
votes
(co)homology of symmetric groups
Here are some comments, including an answer to (3). Firstly, if you want an actual explicit computation of the mod-2 cohomology of symmetric groups $S_n$ for as large an $n$ as possible, you should l …
- 3,451
4
votes
Cellular homology of the universal cover
As Benjamin Steinberg says, it does not work in general, and you can see the problem already for 1-dimensional complexes. Suppose you take a Cayley graph for $G$. This is a graph (or 1-dimensional C …
- 3,451