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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).
10
votes
Accepted
Isometries between spherical space forms
Yes, diffeomorphic spherical space forms are isometric. This famous result of Georges de Rham can be found in [de Rham, G.
Complexes à automorphismes et homéomorphie différentiable.
Ann. Inst. Fourie …
- 29.1k
9
votes
When a compact topological manifold with boundary is a ball?
I wish to address the case $n=4$. This would follow if any $4$-dimensional h-cobordism between $3$-spheres were trivial but at the moment I am not sure how to prove this. There is however a different …
- 29.1k
15
votes
exotic differentiable structures on manifolds in dimensions 5 and 6
Any PL-manifold of dimension $\le 7$ is smoothable, and the smooth structure is unique in dimensions $5,6$. See e.g. remark 6.7 in Rudyak's paper for details.
EDIT: To explain the above, the smooth …
- 29.1k
43
votes
Accepted
Examples of non-diffeomorphic smooth manifolds with diffeomorphic tangent bundle
Here are examples of non-diffeomorphic closed manifolds with diffeomorphic
tangent bundles:
3-dimensional lens spaces have trivial tangent bundles, which are diffeomorphic if and only if the lens sp …
- 29.1k
16
votes
the criteria for 3-dim manifolds diffeomorphic to $\mathbb{R}^3$
I am not sure what Schoen and Yau had in mind. Any open contractible $n$-manifold that is simply-connected at infinity is homeomorphic to $\mathbb R^n$. This is due to Stallings [1] if $n\ge 5$ and
to …
- 29.1k
35
votes
Accepted
If a manifold suspends to a sphere...
Suppose $M$ is a closed $n$-manifold whose suspension is homeomorphic to $S^{n+1}$.
Removing the two "singular" points from the suspension gives $M\times \mathbb R$, while
removing two points from $S^ …
- 29.1k
13
votes
Accepted
4-dimensional h-cobordisms
I think both questions are open. The somewhat sad state of affairs is that there are nontrivial TOP 4d s-cobordisms that are either nonsmoothable or not known to be smoothable, and there are smooth 4d …
- 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
17
votes
Simply-connected rational homology spheres
A complete answer can be found in a paper by D. Ruberman
Null-homotopic embedded spheres of codimension one: a simply-connected rational homology $n$-sphere that is not homeomorphic to $S^n$ exists if …
- 29.1k
8
votes
Accepted
Subgroups of Gromov's hyperbolic groups
The answer is no. The case when $H$ isn't finitely generated is trivial (think of the free groups and its commutator subgroup), examples when $H$ is finitely generated are due to Rips (see the famous …
- 29.1k
14
votes
circle action on sphere
Classification of circle actions on (standard and on exotic) spheres is a classical activity in transformation groups, see e.g. the article of Schultz in the collection "Group actions on manifolds", p …
- 29.1k
1
vote
homotopy type of complement of subspace arrangement
Since you arrangement is so simple, there are must be some purely topological methods to check asphericity (or non-asphericity) of its complement. This is a well-developed subject I do not know much a …
- 29.1k
7
votes
Are there unique geodesics in the NIL and SOL geometry?
Despite their non-uniqueness, a lot is known about geodesics of left-invariant metrics on Heisenberg groups. For example, Jang-Park [Conjugate points on 2-step nilpotent groups, Geom. Dedicata 79 (200 …
- 29.1k
2
votes
Existence of sequence of examples of braking 'Cancellation law in homeomorphic products'
Take any closed $4$-manifold with infinitely many smooth structures, and multiply it by a torus. The product has only finitely many smooth structures, in fact any manifold $M$ of dimension $\ge 5$ ha …
- 29.1k
9
votes
Is every connected metrizable locally path connected space a length space?
This is not an answer but a long comment with references for the locally compact case. It is proved in [Tominaga-Tanaka, Convexification of locally connected generalized continua. J. Sci. Hiroshima Un …
- 29.1k