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Results tagged with gt.geometric-topology
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user 450
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).
11
votes
Is the composition of two bundle projections necessarily a bundle projection?
No, the composition of two bundle projections needn't be a bundle projection.
It is already not true that the composition of two covering maps is a covering map.
You can find a counterexample in Span …
- 47.5k
4
votes
nowhere vanishing vector field on a manifold
Dear Pengfei, about your supplementary questions:
a) "when is an 1-dimensional subbundle L⊂TM orientable ?" A line bundle is orientable if and only if it is trivial (independently of it being or not …
- 47.5k
0
votes
Is the universal covering of an open subset of $\mathbb{R}^n$ diffeomorphic to an open subse...
Consider the standard embedding of the unit interval in $\mathbb R^2$ viz. $I=[0,1]\times \{0\}
\subset \mathbb R^2$. Let $C$ denote the Cantor subset $C \subset I$ and define $U= \mathbb R^2 - C$, an …
- 47.5k
26
votes
Accepted
What should be taught in a 1st course on smooth manifolds?
I nominate Ehresmann's theorem according to which a proper submersion between manifolds is automatically a locally trivial bundle. It is incredibly useful, in deformation theory for example, but is sa …
49
votes
What are the open subsets of $\mathbb{R}^n$ that are diffeomorphic to $\mathbb{R}^n$
Ad question 1): Yes, all open star-shaped subsets of $\mathbb{R}^n$ are diffeomorphic to $\mathbb{R}^n$.
This is surprisingly little-known and there is a proof due to Stefan Born.
You can find t …
- 47.5k
19
votes
0
answers
312
views
Can one properly embed a differential manifold into numerical space of double dimension? [duplicate]
If $X$ is a $ C^\infty$ differential manifold of dimension $n$, then there exists an embedding $f:X\to \mathbb R^{2n+1}$.
This is a not too difficult theorem due to Whitney, proved in many textbooks. …
- 47.5k