41
votes
Elementary proof that $\mathbb{R}^3 \setminus \{p_1,\dots,p_n\}$ is not homeomorphic to $\mathbb{R}^3$
The fundamental group of the one-point compactification of $\mathbf R^3 \setminus \{p_1,\ldots,p_n\}$ is a free group on $n$ generators, for any $n \geq 0$.
- 40.2k
24
votes
Elementary proof that $\mathbb{R}^3 \setminus \{p_1,\dots,p_n\}$ is not homeomorphic to $\mathbb{R}^3$
Let $P_n$ be the property for a Hausdorff topological space $X$: for every compact subset $K$ of $X$, there exists a compact subset $L$ of $X$ such that $K\subset L$ and $X-L$ has exactly $n$ ...
- 63.9k
24
votes
What happens to a closed manifold to ensure it is homeomorphic to a torus $T^{n}$?
For dimensions $n \geq 5$, the answer is yes. First, note that $M$ is homotopy equivalent to a torus since it must be a $K(\mathbb{Z}^n,1)$. Second, Hsiang-Wall show in "On Homotopy Tori II" ...
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19
votes
What happens to a closed manifold to ensure it is homeomorphic to a torus $T^{n}$?
This answer is intended to give references of the cases for the case $n \leq 4$. In dimensions $n \leq 2$ this is covered in a first topology course so there are two interesting cases.
In dimension 3,...
- 9,580
14
votes
Are there invariants of cell complexes similar to the Euler characteristic?
In a certain restricted setting the answer is that the Euler characteristic is the only such invariant. This is not a complete answer to the question since more general things are allowed. However, it ...
- 7,875
13
votes
Accepted
Proof of the stable homeomorphism conjecture
For your first question, yes Kirby did prove the n-dimensional annulus conjecture AC$_n$ for $n>4$ in the 1969 Annals paper that you cite. The only reason this might not be clear from reading the ...
- 20k
11
votes
Accepted
Homeomorphisms vs Borel automorphisms
Let $\bar X = X\cup \{\infty\}$ be the 1-point compactification of a discrete space $X$. Then the autohomeomorphisms of $\bar X$ are all permutations $p$ of $X$ (extended by $p(\infty)=\infty$), ...
- 14k
10
votes
Accepted
Are any embeddings $[0,1]\to\mathbb{R}^3$ topologically equivalent?
As igorf pointed out in the comment, the answer to the first question is 'no'. A quick counterexample is by looking into the complement of Fox-Artin arc, which is not simply connected. See figure ...
- 2,083
8
votes
Accepted
Extension of homeomorphisms
It sounds like you're looking for something like the Klee trick. If $K,K' \subset \mathbb{R}^n$ are compact and homeomorphic, it gives a construction of a self-homeomorphism $\phi$ of $\mathbb{R}^{2n}$...
- 8,167
8
votes
Accepted
Induced homeomorphism from a quasi-isometry between hyperbolic spaces
Properness is already needed to have a well-defined boundary at infinity, i.e., with a topology not depending on the chosen base point. This is Proposition III.3.7 in Bridson-Haefliger, which builds ...
- 10.4k
8
votes
Is an open subset of a rigid space rigid?
The answers to (1) and (2) are "no" in general for metrizable spaces.
Let $C$ be a connected compact metric space with more than one point, with the property that the only continuous self-maps of $C$ ...
- 35.2k
7
votes
Extending homeomorphisms from closed countable sets to S^2
Yes. It follows from the Cantor case. First, it is no restriction to assume that $A,B$ are compact totally disconnected (e.g., countable) subsets of the plane and we want to prove that any ...
- 63.9k
7
votes
Accepted
Transitive homeomorphisms of Erdős spaces
The answer to both questions is affirmative.
Theorem 1. The complete Erdos space $\mathfrak E_c$ has a self-homeomorphism whose every orbit is dense in $\mathfrak E_c$.
Proof. We use a known result ...
- 41.8k
6
votes
Accepted
generators for the handlebody group of genus two
As stated in Ian Agol's answer, the mapping class group of a handlebody $B$ maps onto $Out(\pi_1B)$. This is easy to show by lifting known generators for the automorphism group of a free group. ...
- 20k
6
votes
Do mixing homeomorphisms on continua have positive entropy?
edit
I realized there is a simpler construction that achieves the same (examples of top. mixing zero-entropy homeos on $S^3$): Instead of the bi-directional flow through cuboids, have a flow from ...
- 6,652
6
votes
Is an open subset of a rigid space rigid?
For the third question: If you assume that $X$ is zero-dimensional (i.e., it has a basis consisting of clopen sets), then the answer to question 1 is yes.
Proof: Suppose $U$ is an open, non-rigid ...
- 18.5k
5
votes
Accepted
homeomorphisms induced by composant rotations in the solenoid
In this paper of J.Kwapisz I have found the following
Theorem 1. Any homeomorphism $h$ of the dyadic solenoid $S$ is isotopic to the "affine" homeomorphism of the form $g:x\mapsto \pm(2^n x+b)$ for ...
- 41.8k
5
votes
Accepted
Exotic homeomorphisms of a cube
The answer is no and a suitable counterexample in any dimension $n\geq 2$ was constructed by Oxtoby and Ulam. This is Theorem 12 p. 919 in:
J. C. Oxtoby, S. M. Ulam, Measure-preserving homeomorphisms ...
- 28k
4
votes
Extending homeomorphisms between compact metric subsets
In the "positive" direction, there exist some deep results (called Z-set unknotting theorems) on extensions of homeomorphisms between Z-sets of Menger manifolds, see Theorem 4.1.18 in the book "...
- 41.8k
4
votes
Accepted
Why can any open subset $U$ of $\mathbb{Q}^\infty$ be written as disjoint union of basic clopen subsets?
We can first express the open set $U$ as a countable union of basic clopen subsets $A_n=\{(q_k)_{k\in\mathbb{N}};i_{n,k}<q_k<j_{n,k} \text{ for $k=1,\dots,n$}\}$, where $i_{n,k}$ and $j_{n,k}$ ...
- 10.6k
4
votes
Why are homeomorphism groups important?
Of course the automorphism group of any mathematical object is interesting if the object itself is interesting. But that doesn't single out the homeomorphism group . Let me give some examples.
Let $...
- 7,583
3
votes
Extending homeomorphisms from closed countable sets to S^2
$A\cup B$ is countable and closed, so there is a point $n\in S^2\setminus (A\cup B)$ with an open disk centered at $n$ that does not meet $A\cup B$. Using stereographic projection from the pole $n$, ...
- 271
3
votes
generators for the handlebody group of genus two
Edit: This answer was wrong - I misremembered the result. See Allen Hatcher's answer. Of course, the references in the second paragraph still stand.
Those two operations give trivial outer ...
- 68.8k
2
votes
Do mixing homeomorphisms on continua have positive entropy?
The simplest counterexample would be the identity map on a one-point space.
- 4,164
2
votes
Extending homeomorphisms between compact metric subsets
As you can see from the comments the answer is: hardly ever.
As mentioned above the case of one-point sets necessitates the space being homogeneous. But that is not enough, say in $\mathbb{R}$ when ...
- 11.4k
2
votes
Accepted
Are there invariants of cell complexes similar to the Euler characteristic?
As pointed out in the comments, every characteristic class in $H^d(BO(d), G)$ provides a $G$-valued locally computable invariant of $d$-manifolds, by pulling back via the classifying map of the ...
- 3,001
2
votes
Are there invariants of cell complexes similar to the Euler characteristic?
Meanwhile it seems to me that the discrete analogues to all Stiefel-Whitney numbers of $d$-dimensional manifolds are invariants of this type:
First, for every $n$ there is a rule to color every $d-n$-...
- 3,001
2
votes
Homeomorphisms vs Borel automorphisms
For every uncountable Polish space $M$ (hence every non-discrete metrizable manifold with countably many components), the groups $\mathrm{Homeo}(M)$ and $\mathrm{Borel}(M)$ are non-isomorphic. ...
- 63.9k
2
votes
Is an open subset of a rigid space rigid?
Here is a counterexample to the first question, a planar, non-separating one-dimensional locally connected continuum.
Let $A_1$ be the line segment in the $xy$-plane $\mathbb{R}^2$ with end points $(...
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