All Questions
8 questions with no upvoted or accepted answers
18
votes
0
answers
1k
views
What is the strongest nerve lemma?
The most basic nerve lemma can be found as Corollary 4G.3 in Hatcher's Algebraic Topology:
If $\mathcal U$ is an open cover of a paracompact space $X$ such that every nonempty intersection of ...
- 81
15
votes
0
answers
716
views
Is this "Homology" useful to study?
In the usual singular homology of a topological space $X$, one consider the free abelian group generated by all continuous maps from the standard simplex $\Delta^{n}$ to $X$.
Now we can ...
- 356
11
votes
0
answers
331
views
If an additive group of $\Bbb R^2$ contains a smoothly deformed circle, is it necessarily all of $\Bbb R^2$?
It can be shown that if an additive subgroup of $\Bbb R^2$ contains the unit circle, then it is necessarily all of $\Bbb R^2$. Does this also hold for a suitably smoothly deformed unit circle?
...
- 2,069
5
votes
0
answers
249
views
Aspherical space whose fundamental group is subgroup of the Euclidean isometry group
Let $M$ be a smooth, compact manifold without a boundary, with its universal covering $\tilde{M} = \mathbb{R}^n$. If there exists an injective homomorphism $h: \pi_1(M) \rightarrow O(k) \ltimes \...
- 661
4
votes
0
answers
425
views
Non-triviality of map $S^{24} \longrightarrow S^{21} \longrightarrow Sp(3)$
Let $\theta$ be the generator of $\pi_{21}(Sp(3))\cong \mathbb{Z}_3$, (localized at 3).
How to show the composition
$$S^{24}\longrightarrow S^{21}\overset{\theta}\longrightarrow Sp(3)$$
is non-trivial ...
3
votes
0
answers
282
views
Commutator length of the fundamental group of some grope
A popular way to describe a grope as the direct limit $L$ of a nested sequence of compact 2-dimensional polyhedra
$L_0 \to L_1 \to L_2 \to \cdots$
obtained as follows. Take $L_0$ as some $S_g$, an ...
- 2,083
2
votes
0
answers
164
views
Triviality of map $(\Sigma \theta)^*$
We know that there is a cofibration sequence
$$S^{4n+1}\xrightarrow{\theta}\Sigma^{4m-1} Q_{n-m} \rightarrow \Sigma^{4m-1} Q_{n-m+1} \rightarrow S^{4n+2}\xrightarrow{\Sigma\theta}\Sigma^{4m} Q_{n-m}.$$...
0
votes
0
answers
163
views
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