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Results tagged with gt.geometric-topology
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user 99414
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).
8
votes
1
answer
216
views
Non-additivity of intersection forms
Given two oriented $4k$-manifolds $X_1$ and $X_2$, Novikov additivity tells us that
$$
\sigma(X_1 \sharp X_2) = \sigma(X_1) + \sigma(X_2).$$
More generally, if we glue the boundaries of two such man …
7
votes
1
answer
421
views
Computing $\pi_2$ of the complement of a 2-knot or spaces with aspherical splittings?
As far as I know, it is an open question if the complement of a ribbon disk $D^2 \subset B^4$ is aspherical. In reading "Some remarks on a problem of J.H.C Whitehead" by Howie, it is noted that there …
- 11
9
votes
1
answer
765
views
Intuition for torsion of a chain complex and application to lens spaces
I have read a bit about the torsion of an acyclic complex. One of my concrete hopes was that I could understand why $L(7,1)$ and $L(7,2)$ are not homeomorphic - I am under the impression that classif …
- 34.7k
6
votes
1
answer
321
views
Homology spheres bounding homology balls but not embedding into $S^4$
Are there any examples of integer homology spheres $Y^3$ that bound smooth integer homology balls but that do not smoothly embeded into $S^4$?
- 9,580
3
votes
0
answers
59
views
Nonuniqueness of Heegaard surfaces for submanifolds of $S^3$
Let $M^3$ be some compact submanifold of $S^3$ with connected boundary. I am interested in the failure of the analog of Waldhausen's theorem for $M^3$ - namely, I would like examples of such $M$ toge …
- 5,349
9
votes
1
answer
417
views
Non-isotopic homology spheres in $S^4$ with equal complements?
Are there two diffeomorphic smoothly embedded homology 3-spheres $M_1^3, M_2^3 \subset S^4$ that have diffeomorphic complements but such that $M_1$ and $M_2$ are not isotopic? I would be interested in …
- 68.9k
4
votes
1
answer
363
views
h-cobordisms between non-simply-connected 4-manifolds
Let $M_0^4$ and $M_1^4$ be two closed smooth 4-manifolds and let $M$ be an $h$-cobordism between them (i.e., a compact smooth 5-manifold with boundary the disjoint union of $M_0$ and $M_1$ and with th …
- 204
4
votes
0
answers
76
views
Tangles whose unknottedness realize boolean functions?
Let $f : \{0,1\}^n \to \{0,1\}$ be a Boolean function. Denote the two possible simple single crossing tangles by $T_0$ and $T_1$ (your choice for which is which). Is there some "generalized $n$-tang …
- 5,349
5
votes
1
answer
295
views
Set of proper homotopy classes of arcs in a manifold
Let $M^n$ be an $n$-manifold with nonempty boundary and let $\partial_0 M$ be a specific connected component of $\partial M$. I am interested in the set of continuous maps $f : [0,1] \to M$ such that …
- 10.9k
8
votes
2
answers
321
views
Manifolds with trivial mapping class group and large $H^1$?
Are there smooth closed manifolds $M^n$ in every dimension $n \geq 3$ with trivial mapping class groups and with $H^1(M^n;\mathbb{Z}/2\mathbb{Z})$ arbitrarily large?
I am under the impression that "ge …
- 3,399
4
votes
0
answers
194
views
3-manifold proof of Grushko's theorem
Grushko's theorem says that given an epimorphism $\phi: F \to G_1 * G_2$ where $F$ is a finitely-generated free group, there exists. subgroups $F_1$ and $F_2$ of $F$ so that $F = F_1 * F_2$ and $\phi( …
- 5,349
4
votes
1
answer
239
views
Realizing groups as the fundamental group of graphs of groups allowing non-injective maps?
In the definition of a graph of groups it is assumed that the maps from the edge groups to the vertex groups of injections, however in what follows I will also be interested in the case where the maps …
- 38.6k
7
votes
1
answer
363
views
Decidability of knot equivalence in general 3-manifolds? Surface equivalence?
Given a closed orientable 3-manifold $M^3$ and two knots $K_1$ and $K_2$ in $M$, is there an algorithm to decide if $K_1$ and $K_2$ are isotopic? Is there an algorithm to decide if there is a homeomo …
6
votes
1
answer
272
views
Characteristic class that cannot be represented by disjoint tori
Is there a simply-connected smooth closed 4-manifold with a characteristic class $x \in H_2(X; \mathbb{Z})$ such that $x$ can not be represented by a disjoint union of tori in $X$?
I would not know ho …
- 19.4k
4
votes
1
answer
194
views
Minimal genus of characteristic surfaces?
Let $X^4$ be a simply-connected closed smooth 4-manifold. Then every element $x \in H_2(X; \mathbb{Z})$ can be represented by an embedded orientable surface and the minimal genus of such a surface is …
- 10.9k