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Results tagged with gt.geometric-topology
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user 75
Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
35
votes
What are the open subsets of $\mathbb{R}^n$ that are diffeomorphic to $\mathbb{R}^n$
You can certainly have a set diffeomorphic to $\Bbb R^n$ but not star-shaped. For example, for $n=2$, the Riemann mapping theorem implies that any simply connected open set is diffeomorphic to the pl …
- 31.2k
17
votes
Accepted
Punctured 3-manifold
If $\pi_2(M\setminus\{p\})=0$ then $M$ is simply connected (and hence $S^3$ by the Poincare conjecture). To see this, consider the universal cover $q:U\to M$ and let $V=q^{-1}(M\setminus \{p\})$. Th …
- 31.2k
17
votes
All mapping space between CW complexes is a CW complex?
A mapping space $\mathrm{Map}(X,Y)$ between two finite CW-complexes never admits a cell structure if both $X$ and $Y$ are positive-dimensional. If you use the compact-open topology, this essentially …
- 31.2k
14
votes
Accepted
Construction of maps $f:S^3 \to S^2$ with arbitrary Hopf invariant?
You can get them by precomposing with a degree $n$ map from $S^3$ to itself. In particular, this gives an interpretation in terms of the group structure: if $h:S^3 \to S^2$ is the Hopf map (which is …
- 31.2k
13
votes
Accepted
Is every finite subcomplex of a contractible simplicial complex contained in a finite contra...
Let $X$ be any finite complex such that any map $X\to X$ homotopic to the identity is surjective and for which there is a surjective but nullhomotopic PL map $f:X\to X$ (for instance, $X$ could be $S^ …
- 31.2k
10
votes
Accepted
Must a closed totally path-disconnected subset of the sphere have connected complement?
A circular version of the pseudo-arc (where you construct it out of "circular chains" whose ends connect up to each other) is a counterexample. It is connected and totally path-disconnected, and its …
- 31.2k
7
votes
Accepted
Map from homotopy sphere with lifting property induces surjections on homotopy groups. Is it...
Yes, we can conclude that either $q$ is an equivalence or $X$ is contractible. Since any cycle lives in a compact subset of $X$, $q$ will also induce surjections on homology. It follows that $X$ is …
- 31.2k
5
votes
Accepted
is there an anyon structure analogous to spin structure for rank 2 bundle?
I know nothing about the physics you have in mind, but I can tell you about the topology. The classifying space $BSO(2)$ is a $K(\mathbb{Z},2)$, so oriented 2-plane bundles are in bijection with clas …
- 31.2k
3
votes
Accepted
continuous R^2xR^2xR^2/E^+(2) -> R^3 injection?
By translation, fix the first point to be at the origin. Consider the other two points as complex numbers, and take their quotient. As long as the other two points are not both at the origin, this c …
- 31.2k