All Questions
Tagged with gr.group-theory at.algebraic-topology
69 questions with no upvoted or accepted answers
2
votes
0
answers
119
views
Crossed homomorphism as morphism in the ambient category
Suppose we are given a crossed-homomorphism $\phi:G\to A$ (and an action $\alpha$ of $G$ on $A$)
$\phi(ab)=\phi(a)+\alpha(a)(\phi(b))$. Now, unless the action is trivial, this is not a homomorphism ...
- 199
2
votes
0
answers
164
views
Triviality of map $(\Sigma \theta)^*$
We know that there is a cofibration sequence
$$S^{4n+1}\xrightarrow{\theta}\Sigma^{4m-1} Q_{n-m} \rightarrow \Sigma^{4m-1} Q_{n-m+1} \rightarrow S^{4n+2}\xrightarrow{\Sigma\theta}\Sigma^{4m} Q_{n-m}.$$...
2
votes
0
answers
83
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Euler class of extension of free nilpotent groups
Fix some $n \geq 2$. For $k \geq 1$, let $N_k$ be the free $k$-step nilpotent group on $n$ generators, i.e., the quotient of the free group $F_n$ by the $(k+1)^{\text{st}}$ term $\gamma_{k+1}(F_n)$ ...
- 21
2
votes
0
answers
107
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Homology functors and weak cofibers
I'm looking at a remark in the paper
Kainen, Paul C., "Weak Adjoint Functors", Mathematische Zeitschrift 122 (1971).
It is supposed to prove that generalized homology functors fail to ...
- 81
2
votes
0
answers
106
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Minimal symmetry of a fibre bundle
Let $F \to E \to B$ be a topological fibre bundle with fibre $F$ and base $B$. It can be characterized by a map $B \to BAut(F)$. If it can also be characterized as a map $B \to BG$ (or say $G$ is a ...
- 5,230
2
votes
0
answers
167
views
Any abelian group embeds into a Chow group
Let $G$ be an abelian group. Must there exist a perfect field $k$, a smooth projective geometrically connected $k$-scheme $X$ and an integer $i\geq 0$ such that $G$ embeds into the integral Chow group ...
user164740
2
votes
0
answers
311
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The subtlety with (an algebraic phrasing of) the Whitehead conjecture?
The Whitehead conjecture states that if $X$ is a $2$-dimensional aspherical simplicial complex and $Y \subset X$ is a connected sub-complex then $Y$ is aspherical. This can be re-phrased in terms of ...
2
votes
0
answers
1k
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Lifting of group homomorphisms
I asked this question a few days ago on math stackexchange but didn't get any answer so I thought I post it here too (see here):
In my first course on algebraic topology I heard about the following:
...
- 721
1
vote
0
answers
60
views
Restricted wreath product as fundamental group of a space with coinciding Reidemeister and Nielsen numbers
I am studying a group $\mathbb{Z}_n \wr \mathbb{Z}^k$, where $\wr$ denotes the restricted wreath product:
$$
\mathbb{Z}_n \wr \mathbb{Z}^k = \bigoplus_{x\in\mathbb{Z}^k}(\mathbb{Z_n})_x\rtimes\mathbb{...
- 137
1
vote
0
answers
113
views
Question on models for $EG$ for a $G$-CW complex
I am having trouble finding information on a definition in P. Hanham's PhD thesis paper. recall that given a discrete group $G$ a $G$-CW-complex $X$ is a CW-complex equipped with a topological $G$ ...
- 233
1
vote
0
answers
132
views
Nilpotency of topological groups
A group $G$ is said to be nilpotent if $G$ has a central series of finite length, that is, a series of normal subgroups
$$
\{1\} = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_n = G
$$
...
- 901
1
vote
0
answers
57
views
$\omega$-nilpotent cover of a recurrent surface
Theorem. Any $\omega$-nilpotent cover of a recurrent Riemannian manifold is Liouville.
$\omega$-nilpotent ($\Gamma=\bigcup_{i=1}^{\infty}Z_{i}$, $Z_{i}$ normal in $\Gamma$, where $Z_{n+1}$ maps to ...
- 391
1
vote
0
answers
137
views
Acyclicity of covering space
Suppose we have some 2-dimensional non-aspherical finite CW-complex $K$ with $\pi_1(K)=G$. Is there any sufficient condition on $H\leq G$ (and maybe on the group $G$ itself) which allows to conclude ...
- 1,129
1
vote
0
answers
112
views
When are graphs of cohomologically complete groups cohomologically complete?
A group $G$ is cohomologically $p$- complete if the canonical map from $G$ to it's pro$-p$ completion $\hat G^p$ induces an isomorphism on cohomology $H^\ast_{cont}(\hat G^p, \mathbb{Z}_p) \rightarrow ...
- 11
1
vote
0
answers
238
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Twisted homology of free products
Let $G_1$ and $G_2$ be groups and let $M$ be a vector space equipped with actions of $G_1$ and $G_2$. The free product $G_1 \ast G_2$ thus acts on $M$. How can one compute the twisted group homology ...
- 11
0
votes
0
answers
220
views
The largest value of $k$ for $\mathbb{Z}^k$ to be embedded in $GL(n,\mathbb{Z})$
This is just a question originated from This conversation (commented by Moishe Kohan). I tried to prove those two assertions but I don't know where to start:
If H is a free abelian subgroup of $SL(n, ...
0
votes
0
answers
194
views
Equivariant cohomology with discrete group action
As far as I know, the equivariant cohomology can be regarded as the generalisation of de Rham cohomology with group action on manifolds. From the literature, the group action is Lie group type. I am ...
0
votes
0
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378
views
Isomorphism of invariants and coinvariants over a field
Let $G$ be a finite group with normal subgroup $N$ acting on a vector space $V$ over a field $k$ in which the order of $N$ is invertible. Denote $H:=G/N$. The composite map $V^N \to V \to V_N$ and $\...
- 617
0
votes
0
answers
163
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Presentation complex of a finite perfect group and its features
Let $G$ be a finite perfect group and consider $X_G$, its presentation complex. I have the following questions:
Is there any special property of $X_G$ due to the group's perfectness?
What can we say ...
- 1,177