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Results tagged with gt.geometric-topology
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user 39747
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).
38
votes
Accepted
Simply-connected rational homology spheres
In dimension 4, we have the following:
Simply-connectedness implies that $H_1(M)=0$. The condition that $M$ be a rational homology sphere implies that $H_2(M), H_3(M)$ are finitely generated torsion g …
- 10.8k
7
votes
Higher homotopy groups of irreducible 3-manifolds
It's wrong for finite fundamental group, as then the universal cover is closed and has nonvanishing $\pi_3$ by Hurewicz. It's true for infinite fundamental group, again by Hurewicz applied to the univ …
- 10.8k
15
votes
Accepted
Hairy ball theorem for odd-dimensional spheres
The Lefschetz fixed point theorem implies that any $f: S^n \to S^n$ without fixed points has degree $(-1)^{n+1}$. But an even map $S^n \to S^n$ has even degree, since it factors as
$$
S^n \xrightarrow …
- 10.8k
15
votes
Accepted
Identifying two definitions of orientation on a vector space
Here's a direct way to relate the two:
One more structure is additivity of orientations. For $V, W$ of dimensions $n,m$, we have a canonical pairing
$$
\Lambda^n V \otimes \Lambda^m W \cong \Lambda^{n …
- 10.8k
0
votes
Probability of random geodesics on the half-sphere intersecting
Away from a zero set, $(a,b,c,d)$ gives you crossing geodesics if and only if $(a,b,c,-d)$ doesn't. It follows that the probability is exactly $0.5$, since the map $(a,b,c,d)\mapsto (a,b,c,-d)$ is mea …
- 10.8k
5
votes
Accepted
how to prove the $n$-times self-product of a map is null-homotopic
Yes, this is true.
Your map $B\Sigma_k\rightarrow BO(k)$ gives rise to a map $B\Sigma_k\rightarrow BO$, i.e. an element in the reduced K-theory group $\tilde{ko}^0(B\Sigma_k)$.
Now that whole group i …
- 10.8k
9
votes
Accepted
Abstract simplicial complexes - Reference for an elementary definition of mapping degree for...
I think the right generality to restrict to is the following:
Let $n$ be a positive integer, and let $X$ and $Y$ be $n$-dimensional oriented simplicial complexes, with the following properties:
Every …
- 10.8k