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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).
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
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
33
votes
Accepted
Can an oriented closed $n(\geq 2)$-dimensional manifold be smoothly embedded in $\mathbb{R}^...
A closed smooth $n$-manifold embeds into $\mathbb R^{2n-1}$ if and only if the normal $(n-1)$th Stiefel-Whitney class vanishes. This is due to Hirsch-Haefliger in dimensions $\neq 4$ and to Fang in di …
- 29.1k
26
votes
Accepted
Are there non-smoothable homotopy/homology spheres?
Since the Poincaré conjecture is known in all dimensions any homotopy $n$-sphere is homeomorphic to $S^n$ and hence admits a smooth structure.
Any manifold of dimension $\le 3$ admits a smooth struct …
- 29.1k
25
votes
Non embedding of $Y\times Y$ into $\mathbb{R}^3$
Theorem 10 of "An Alternative Proof that 3-Manifolds Can be Triangulated" by R.H. Bing [Ann. of Math. Vol. 69, (1959), pp. 37-65] states that any topologically embedded simplicial 2-complex $K$ in a t …
- 29.1k
20
votes
What are the open subsets of $\mathbb{R}^n$ that are diffeomorphic to $\mathbb{R}^n$
There are several characterizations of manifolds diffeomorphic to R^n when n>4, e.g. an open manifold that is simply-connected at infinity (Stallings), or the image of a degree one proper map from R^n …
- 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
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
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
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
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
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
14
votes
Accepted
Topological embeddings of real projective space in euclidean space
The first page of W.~Massey's paper On the imbeddability of the real projective spaces in Euclidean space states that $\mathbb RP^n$ with $n>1$ cannot be imbedded topologically in $\mathbb R^{n+1}$ be …
- 29.1k
14
votes
Accepted
Homeomorphism/ homotopy types of non-negatively curved manifolds
As mentioned in comments, the first dimension where an infinite family of pairwise non-homeomorphic closed nonnegatively curved manifolds occurs is $3$ (the lens spaces). The question becomes more cha …
- 29.1k
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 …