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Results tagged with gt.geometric-topology
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user 20787
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).
29
votes
Kirby's torus trick
I highly recommend Allen Hatcher's preprint in which he employs the torus trick for the pedestrian task of proving existence and uniqueness-up-to-isotopy of smooth structures on every 2-manifold. The …
- 15.4k
20
votes
Accepted
Periodic automorphisms of free groups
No such automorphism exists. Every finite order automorphism of a finite rank free group has a nontrivial fixed conjugacy class. To see why, you can represent the free group automorphism as a simplici …
- 15.4k
19
votes
Fundamental groups of noncompact surfaces
I just ran across this question, and thought I would give a precise version of the proof Ilya suggested. I believe I learned this proof in Richie Miller's topology course, Michigan State University, 1 …
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17
votes
Are there some other notions of "curvature" which measure how space curves?
There is a long history of curvature conditions known as "small cancellation conditions" which apply to group presentations, or if you prefer to the Cayley 2-complex of the presentation. These are com …
15
votes
Accepted
Universal covering of compact surfaces
You can build a certain covering space of the surface $S$ rather explicitly as a nested union of closed discs $D_1 \subset D_2 \subset D_3 \subset \cdots$, each contained in the interior of the next, …
- 15.4k
13
votes
Why are Fuchsian groups interesting?
Fuchsian groups, particularly those which are cocompact, form the tips of several big mathematical icebergs. To put this less metaphorically, several discoveries about Fuchsian groups, obtained using …
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13
votes
Compelling evidence that two basepoints are better than one
In my proof that mapping class groups are automatic, Ann. of Math. (2) 142 (1995), no. 2, 303–384, I used a theorem from ECHLPT "Word Processing in Groups" which says that if a groupoid is automatic …
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12
votes
Accepted
Regarding the Thurston norm and the ways that a three-manifold can fiber over the circle
The answer is: yes if the rank of $H_2(M;\mathbb{Z})$ is $\ge 2$; and no if the rank is $1$ because in that case there is up to isotopy a unique connected surface bundle structure on $M$. The proof us …
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11
votes
A direct proof of the Harer-Zagier recursion enumerating the ways to paste a 2n-gon to get a...
How about
Goulden, I. P.; Nica, A.
A direct bijection for the Harer-Zagier formula.
J. Combin. Theory Ser. A 111 (2005), no. 2, 224–238.
or one of the references therein?
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11
votes
Can we determine which monodromy of surface gives a fibered knot?
Form the mapping torus $M$, check whether $H_1(M;\mathbb{Z})=\mathbb{Z}$ and is generated by a loop on the torus boundary. If not, it isn't a fibered knot complement. If so, do Dehn filling to produce …
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11
votes
Homeomorphic but Non-Conjugate Mapping Tori
Counterexamples are easily constructed using the Thurston norm. In fact, any example of a fibered, oriented, closed 3-manifold $M$, with a fiber of genus $\ge 2$ and with pseudo-Anosov monodromy, and …
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11
votes
Accepted
Flips of triangulations on non-orientable surfaces
My article "Tiling the measured foliation space of a punctured surface", Trans. Math. 306 no. 1 (1988) contains a proof of this fact in the case of oriented surfaces. It is essentially the same as Hat …
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10
votes
Self-covering spaces
Adding further counter-examples to the discussion, compact hyperbolic manifolds in all dimensions are ruled out by the fact that Gromov's "simplicial volume" is nonzero (being proportional to hyperbol …
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10
votes
Accepted
Invariant free factor of a free group
There is a proof attributed to Peter Scott in Lemma 6.0.6 of "The Tits alternative for Out(F_n) I: Dynamics of exponentially growing automorphisms", MR1765705. The proof uses the Kurosh subgroup theor …
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10
votes
What are some of the big open problems in 3-manifold theory?
Cannon's Conjecture: Every finitely generated word hyperbolic group with Gromov boundary $S^2$ has a finite normal subgroup whose quotient is the fundamental group of a closed hyperbolic 3-orbifold.