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Results tagged with gt.geometric-topology
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user 172802
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).
9
votes
Accepted
Boundaries of subsets of simply-connected domains
It seems if you take $B=\mathbb{R}^2$ and $B'$ the complement of the closure of $\Big\{\big(x,\sin\big(\frac{1}{x}\big)\big);x\in(0,\infty)\Big\}$ this is a counterexample. (Added bonus: $B'$ is also …
- 10.6k
4
votes
Accepted
Subset in $[0,1]^k$ with positive density
Apparently not. Let $\gamma=1/3$ and choose some $\varepsilon<\frac{1}{4}$.
For any $k\in\mathbb{N}$ we can consider the set
$$A=\left\{(x_1,\dots,x_k)\in[0,1]^k;\;\sum_{i=1}^k\lfloor2x_i\rfloor\equiv …
- 10.6k
7
votes
Accepted
For a closed Riemannian manifold $M$, must the set of points with non-unique closest points ...
More generally, for any closed subset $S$ of a complete manifold $M$, the set of points $x$ at whose minimal distance to $S$ is attained at more than point has measure $0$.
Indeed, consider the distan …
- 10.6k
2
votes
Accepted
Estimating the volume of a convex shape in higher dimensions based only on normal sections
Those constants don't exist for any $d\geq4$, here is an idea of why.
For each $\varepsilon>0$ let $A_\varepsilon=\{(x_1,\dots,x_d)\in[-1,1]^d;\lvert (d-1)x_d-\sum_{i=1}^{d-1} x_i\rvert\leq\varepsilon …
- 10.6k
40
votes
2
answers
2k
views
Can the nth projective space be covered by n charts?
That is, is there an open cover of $\mathbb{R}P^n$ by $n$ sets homeomorphic to $\mathbb{R}^n$?
I came up with this question a few years ago and I´ve thought about it from time to time, but I haven´t b …
- 10.6k
11
votes
Accepted
Uncountable collections of distinct subsets of an interval (existence)
My comment reposted as an answer:
If the continuum hypothesis holds, then we can give a well order $\prec$ to $\mathbb{R}$ isomorphic to the first uncountable ordinal. And then for each $j\in[-1,1]$ w …
- 10.6k