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Results tagged with at.algebraic-topology
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user 363264
Homotopy theory, homological algebra, algebraic treatments of manifolds.
3
votes
1
answer
230
views
Example of a non $\pi_1$-injective, degree one, self-map of a three-manifold
All manifolds will be assumed to be closed, oriented, and connected.
Let $f\colon M\to M$ be a map of degree $\pm 1$. It is not hard to show that $\pi_1(f)$ is surjective.
What is an example of a non …
- 28.2k
8
votes
1
answer
279
views
Non-compact three-manifolds with the same proper homotopy type are homeomorphic?
I am looking for some literature with some (counter) examples of the following fact (though I don't know if the fact is true or not):
Let $M, M'$ be two non-compact connected $3$-manifolds with the s …
- 106
7
votes
1
answer
234
views
If the number of ends of Freudenthal space is infinite, then its space of ends is homeomorph...
I don't know whether this is the right place to discuss a part of someone's thesis or not. If it is wrong, let me know; I will delete my post.
I am reading this thesis.
Corollary 4.1.15. on page 63 sa …
- 2,090
5
votes
0
answers
163
views
The fundamental group of the complement of badly embedded open $n$-ball in $\Bbb R^n$
Let $\mathcal D^n$ be an open subset of $\Bbb R^n$ such that $\mathcal
D^n$ is homeomorphic to $\{x\in \Bbb R^n:|x|<1\}$. Suppose $\Bbb R^n\setminus \mathcal D^n$ is path-connected. How bad can
$\pi …
- 1,097
5
votes
1
answer
210
views
How many pairwise non-homeomorphic non-empty closed subsets of the Cantor set are there? [duplicate]
My question is more or less related to basic set theory. But I don't know even that. Apologies if I added the wrong tags.
Motivation: How many non-compact (planar) surfaces are there upto homeomorphi …
- 10.6k
2
votes
0
answers
104
views
Vanishing of Goldman bracket requires simple-closed representative?
Let $\Sigma$ be a connected oriented surface, and $[-,-]\colon \Bbb Z\big[\widehat\pi(\Sigma)\big]\times Z\big[\widehat\pi(\Sigma)\big]\to Z\big[\widehat\pi(\Sigma)\big]$ be the Goldman Bracket. Note …
- 1,097
5
votes
0
answers
130
views
Earliest known proof of "Any degree one self-map of an orientable connected finite-type non-...
I attended a talk where the speaker said the following is due to Nielsen. I searched here and there but couldn't find the corresponding paper, if any. So, what is the earliest known proof of the follo …
- 1,097
2
votes
1
answer
149
views
Isotopic homeomorphisms of surface induces same map on the space of ends
Let $\Sigma$ be a non-compact orientable connected two-manifold without boundary. Let $f,g\colon \Sigma\to \Sigma$ be two homeomorphisms. Suppose there is a homotopy $H\colon \Sigma\times [0,1]\to \Si …
- 28.2k
1
vote
1
answer
143
views
Transversal pre-image of a small enough trivial tubular neighborhood contains a trivial tubu...
A similar post on MSE without answer.
Let $f\colon M'\to M$ be a smooth map between two orientable closed smooth manifolds and $S$ be a smoothly embedded closed orientable submanifold of $M$ of co-dim …
- 7,842
2
votes
1
answer
291
views
In which dimensions is it true that every topological ball embedded by a smoothly embedded s...
I asked a question on MSE with no answer. Here is my question in the generalized version.
Question 1: Suppose we are given a connected three-manifold $M$ (possibly non-compact, or non-orientable) and …
- 19.4k
4
votes
1
answer
184
views
Inequivalent free $\Bbb Z/n\Bbb Z$-actions on orientable compact bordered surface
Let $S_{g,b}$ denote the orientable connected compact surface of genus $g$ with $b$ boundary components. A group homomorphism $\varphi\colon G\to \text{Homeo}^+(S_{g,b})$ is said to be free $G$-acti …
- 1,424
7
votes
1
answer
265
views
Stallings' binding tie
I came to know that the statement below could be proved using Stallings' binding tie argument, though I have no reference article proving the statement by the binding tie argument. Can anyone help me …
- 28.2k