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Results tagged with gt.geometric-topology
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user 40804
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).
71
votes
Accepted
Not all manifolds can be triangulated: In which dimensions?
In dimensions up to three, every manifold is triangulable (this is classical). In dimension 4, there are simply connected non-triangulable manifolds (such as the E8 manifold); in fact, a closed 4-mani …
- 9,580
41
votes
Accepted
Exotic $R^4$ as the universal covering space
This is problem 4.79(A) of Kirby's 1995 problem list, contributed by Gompf. It is my impression that it is still open.
There is some small progress: Remark 7.2 in this article observes that their co …
- 9,580
26
votes
Accepted
Critical dimensions D for "smooth manifolds iff triangulable manifolds"
All smooth manifolds are triangulable, as you say. This follows from Morse theory, which dictates that you only need to know how to triangulate (PL) handle-attachments, which one can do by hand. The r …
- 9,580
23
votes
2
answers
840
views
Classification of fake (quaternionic, octonionic) projective spaces
If $X$ is a closed $n$-manifold, a fake $X$ is another closed manifold homotopy equivalent to $X$. There is some interest in classifying manifolds (up to, say, homeomorphism) homotopy equivalent to a …
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21
votes
0
answers
770
views
Is the mapping class group of $\Bbb{CP}^n$ known?
In his paper "Concordance spaces, higher simple homotopy theory, and applications", Hatcher calcuates the smooth, PL, and topological mapping class groups of the $n$-torus $T^n$. This requires an unde …
- 9,580
16
votes
Accepted
Obstruction of spin-c structure and the generalized Wu manifods
Define the Wu manifold $W(n) = SU(n)/SO(n)$, the inclusion $SO \to SU$ given by thinking of $\Bbb C^n = \Bbb R^n \otimes \Bbb C$ (that is, including real matrices into complex matrices). Note that $W( …
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16
votes
0
answers
427
views
Survey of known results on equivariant transversality
Most basic differential topology theorems carry over to the equivariant case with mild modifications; see for instance Wasserman's paper. One thing that fails (more or less obviously) is equivariant t …
- 9,580
15
votes
Accepted
Non-triangulable 4-manifold as a boundary of some 5 manifold
Your questions are answered by Hsu in his paper 4-Dimensional Topological Bordism.
In particular, associated to any closed oriented topological 4-manifold $X$ is a signature $\sigma(X) \in \Bbb Z$ and …
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14
votes
0
answers
336
views
Are there Alexander-Whitney maps in geometric homology?
When $X$ is a smooth manifold, Lipyanskiy defines a chain complex whose homology is isomorphic to singular homology -
let's say $GC_*(X)$ - generated by maps $\sigma: M \to X$ from compact $k$-manif …
- 9,580
12
votes
Accepted
Existence of normal microbundles
Not all locally flat submanifolds have a normal microbundle, but they do stably.
Rourke-Sanderson prove that there is a PL embedding $S^{19} \times I \to S^{29}$ with no topological normal microbundle …
10
votes
Accepted
Fundamental group of the space of immersions of the 2-sphere in 3-space modulo diffeomorphis...
The action of $\text{Diff}^+(S^2)$ on $\text{Imm}(S^2,\Bbb R^3)$ is free. For pick any immersion $i$ and diffeomorphism $f$; given $x \in \Bbb R^3$ $i^{-1}(x)$ is a closed discrete (because $i$ is an …
- 9,580
10
votes
0
answers
182
views
A minimal $\mathbb Z/2$-invariant Morse function on $U(n)$
Consider the group $U(2n)$ of unitary matrices. This has two standard and important homomorphic involutions. The most famous one $A \mapsto \overline A$ is complex conjugation, with fixed-set $O(2n)$, …
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10
votes
Accepted
Topology change induced by small perturbation
The formal statement you are thinking of when you assert "The topology changes only when..." is Ehresmann's theorem: a proper smooth submersion is a fiber bundle, and hence all fibers are diffeomorphi …
- 9,580
8
votes
Accepted
Topological Spin manifolds in dimension 4
The map $\Omega_4^{\text{Spin}} \to \Omega_4^{\text{SpinTop}}$ is taken isomorphically by the signature to the inclusion $16\Bbb Z \hookrightarrow 8\Bbb Z$, so that the groups are abstractly isomorphi …
- 9,580
8
votes
Accepted
Fixed point set of smooth circle action
Given two $n$-manifolds $M_i$ with a circle action and a choice of free orbit $\gamma_i$ we can define the fiber connected sum $(M_1, \gamma_1) \# (M_2, \gamma_2) = (M_1 \# M_2, \gamma)$ by taking a t …
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