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Results tagged with gt.geometric-topology
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user 1822
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).
8
votes
Accepted
Branched coverings of non-orientable 3-manifolds
A branched covering must induce a surjection on rational homology, which one can see, for example, by the existence of a transfer homomorphism. So there is no branched covering $S^3 \to \mathbb{R}P^2 …
- 3,376
12
votes
Accepted
Maximal degree of a map between orientable surfaces
The best, elementary, self-contained, and clarifying proof of Kneser's result, including the desired inequality, is due to Richard Skora. See
Skora, Richard, The degree of a map between surfaces, Math …
- 3,376
11
votes
Restriction of a branched cover to its branch locus
It is useful to reformulate the question in its natural differential topology setting, leaving unneeded geometric considerations aside. It is also natural to consider the analog of the problem in all …
- 3,376
5
votes
Accepted
Isotopy of periodic homeomorphisms of a surface along periodic homeomorphisms
The answer is "yes". It follows as part of F. Bonahon's determination of the bordism group of surface diffeomorphisms. See
Bonahon, Francis, Cobordism of automorphisms of surfaces, Ann. Sci. Éc. Norm. …
- 3,376
5
votes
Accepted
Cyclic groups acting on balls, and interior fixed points
There need not be a fixed point in the interior. As in Bredon's book, p. 61, there exist smooth actions of a cyclic group $C_{r}$, of order $r$, without fixed points on $\mathbb{R}^{n}$ for large enou …
- 3,376
8
votes
Accepted
Making spheres shellable
According to the reviewer of
Bruggesser, H.; Mani, P., Shellable decompositions of cells and spheres.
Math. Scand. 29 (1971), 197–205 (1972), MR0328944, "The authors provide a rather ingenious proof …
- 3,376
4
votes
Equivariant handle decompositions
This foundational paper: Arthur G. Wasserman, Equivariant Differential Topology, Topology, 8(1967), 127-150, has section 4 dealing with equivariant Morse theory for manifolds with a smooth action of a …
- 18.7k
20
votes
Do finite groups acting on a ball have a fixed point?
Bob Oliver classified the finite groups that act without a global fixed point on some sufficiently high-dimensional disk. The conditions are somewhat complicated to state. But for finite abelian group …
- 3,376
11
votes
Can an action of a compact Lie group be nontrivial if it is trivial on the boundary?
Yes, it follows that the action of $G$ on all of $M$ is trivial. In brief this follows from what is known as "local Smith theory." Replace M by the union of $M$ and an open boundary collar on which $G …
- 3,376
2
votes
An obstruction theory for promoting homotopy equivalences that are equivariant maps to equiv...
The simplest interesting case would be when G is Z/p, p prime. In this case the main issue is that by Smith Theory (applied to the mapping cylinder rel domain, say) you will only know that the induced …
- 3,376
25
votes
Is every (finite) group action on R^n by diffeomorphisms conjugate to a linear action?
Yes in dimensions ≤ 2 (classical). Yes in dimension 3 via the Geometrization Conjecture (with much earlier work in special cases). No in higher dimensions, with the simplest examples perhaps being cou …
- 3,376
11
votes
Accepted
A conjecture of Montesinos
It is false. For example, there are closed, orientable, aspherical 3-manifolds that admit no nontrivial action of a finite group whatsoever. The first examples were due to F. Raymond and J. Tollefson …
- 3,376
11
votes
Accepted
Ramified cover of 4-sphere
The answer is yes, at least if we interpret your phrase "ramification of order 2" to mean "simple branched covering". See Piergallini, R., Four-manifolds as $4$-fold branched covers of $S^4$. Topology …
- 3,376
13
votes
Accepted
Ramified covers of 3-torus
Note that a branched covering induces an injection of rational cohomology rings, by transfer considerations. Therefore the cohomology of a manifold that is a branched covering of $T^3$ must contain th …
- 3,376