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Results tagged with gt.geometric-topology
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user 1573
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
Locally conformally flat
Any open parallelizable $n$-manifold immerses into $\mathbb R^n$ (this is special case of by Hirsch-Smale h-principle), and hence is conformally flat.
There are many examples of conformally flat man …
- 29.1k
13
votes
Accepted
Manifold whose universal covering is a sphere but which is not a space form?
There are lots of fake lens spaces, and fake spherical space forms (search on these keywords). In particular, a construction of fake lens spaces is in chapter 12 of Milnor's "Whitehead torsion". Here …
- 2,821
9
votes
Computation on characteristic classes
The papers "Characteristic Classes and Homogeneous Spaces, I, II" by A. Borel and F. Hirzebruch is a classical resource. In general, homogeneous spaces form a rich class of examples to compute and pla …
- 29.1k
4
votes
Reference for an easy lemma on homeomorphisms of connected manifolds
In [Ancel and Bellamy, On homogeneous locally conical spaces, Fund. Math. 241 (2018), no. 1, 1–15] it is shown that every homogeneous locally conical connected separable metric space that is not a $1$ …
- 29.1k
15
votes
Accepted
Does a compact contractible metric space have a point that is fixed by all isometries?
There are finite groups that act smoothly on a disk without a global fixed point. You can arrange the metric to be isometric, e.g. via the Mostow-Palais embedding theorem, which equivariantly and smoo …
- 29.1k
10
votes
Accepted
Counterexamples to an analog of Cannon's Conjecture which do not arise from manifolds?
The requirement that the manifolds "do not arise at all from negatively curved compact manifolds" is somewhat vague, but here is one known construction.
If one applies Charney-Davis strict hyperboliza …
- 29.1k
9
votes
Accepted
More four-dimensional counterexamples
Borel's conjecture predicts that any homotopy equivalence of closed aspherical manifolds is homotopic to a homeomorphism. The conjecture has been proved for many fundamental groups, see e.g.
The Borel …
- 12.8k
7
votes
Accepted
Torus bundles and compact solvmanifolds
A group $G$ is isomorphic to the fundamental groups of a compact solvmanifold if and only if it fits into the short exact sequence $1\to N\to G\to\mathbb Z^n\to 1$ where $N$ is a finitely generated to …
- 29.1k
10
votes
Is every (finite) group action on R^n by diffeomorphisms conjugate to a linear action?
Take a contractible manifold $C$, multiply it by $\mathbb R^n$, and let the finite group act trivially on $C$, and linearly on $\mathbb R^n$ such that $0$ is the unique fixed point. Then $C\times 0$ …
- 2,821
7
votes
Accepted
Transitive action on non-orientable $ M $ lifts to orientable double cover
There is a general theory for lifting Lie group actions to covering spaces, see Bredon's monograph "Introduction to compact transformation groups", chapter 1, section 9. In the case of orientation cov …
- 29.1k
17
votes
Accepted
Do all spaces doubly covered by $S^{2n}$ have the homeomorphism type of $\mathbb{P}^{2n}_{\m...
One approach to a homeomorphism classification of closed manifolds simply homotopy equivalent to a closed manifold $X$ of dimension $>4$ is to compute the topological structure set $\mathcal S^s_\text …
- 29.1k
10
votes
Contractibility of space of embeddings of a disc
For details I recommend looking at the papers of Yagasaki on arXiv
especially the paper
Homotopy types of homeomorphism groups of noncompact 2-manifolds, Topology and its Applications
108 Issue 2 (20 …
- 35.5k
13
votes
What are some of the big open problems in 3-manifold theory?
Here are two problems on 3-manifold groups (i.e. fundamental groups of compact 3-manifolds) that I find important.
a. Are 3-manifold groups linear?
Comments: Here a group is called linear if it is is …
- 4,713
3
votes
Accepted
Hyperbolization with word-hyperbolic fundamental group
Charney-Davis in Strict hyperbolization showed how to make $N$ locally CAT($-1$), provided $M$ is PL.
Ontaneda in Riemannian hyperbolization showed how to make $N$ a Riemannian manifold of negative se …
- 29.1k
3
votes
PL-embeddings of balls into PL-manifolds
The precise result you want is Theorem 4.20 (page 56) in Rourke-Sanderson's book.
The notations in the statement are on page 50, but they are self-explanatory, e.g., $I^{n,q}$ is the standard disk pa …
- 29.1k