All Questions
Tagged with gt.geometric-topology at.algebraic-topology
1,145 questions
12
votes
1
answer
482
views
Characterize spin cobordism invariants in dimer models
The paper by Cimasoni and Reshetikhin http://arxiv.org/abs/math-ph/0608070 shows that one can map problems about spin structures on a Riemann surface into problems about dimmer configurations on a ...
- 875
12
votes
0
answers
408
views
The $\infty$-category of $n$-manifolds and open embeddings determined homotopically from that of topological manifolds?
Let $\mathrm{Diff}_n$, $\mathrm{PL}_n$, $\mathrm{Top}_n$ denote the $\infty$-categories of $n$-manifolds which are respectively smooth/PL/topological, and open embeddings (for instance by taking the ...
- 7,789
12
votes
0
answers
266
views
Simply connected homology cobordisms
I'm looking for interesting examples of a homology 3-sphere $Y$ for which there exists a smooth, simply connected homology cobordism from $Y$ to itself (or simply to another homology 3-sphere $Y'$, ...
- 779
12
votes
0
answers
306
views
Understanding a formula in Ozsvath-Szabo
I'm a beginning graduate student reading Ozsvath-Szabo's foundational paper, Holomorphic disks and topological invariants for closed 3-manifolds. What I have trouble understanding is a formula in ...
- 355
11
votes
4
answers
1k
views
Knot diagrams, sets of moves and equivalence relations
Short version: Does anyone study equivalence classes generated by a given set of "moves" (in the sense of, but not limited to, Reidemeister moves) on the set of knot diagrams?
Yes, I ...
- 17.6k
11
votes
3
answers
1k
views
Computation on characteristic classes
I am organizing a reading seminar on characteristic classes. The audience in the seminar is interested in symplectic and contact manifolds. I work in categorification and would like to compute some ...
- 341
11
votes
2
answers
1k
views
Infinite loop space structure of $BU^+$
It is well-known that $BU^+$ is homotopy equivalent to an infinite loop space where $U$ is the limit of the unitary groups $U(n)$ for $n \rightarrow \infty$ and $+$ denotes Quillen's Plus construction....
- 165
11
votes
1
answer
620
views
Is $SL(n,\mathbb{Z})$ a CAT(0) group?
Is it possible to find a CAT(0) space on which the matrix group $SL(n,\mathbb{Z})$ acts properly discontinuously and cocompactly? Note: when the cocompactness is dropped , it is possible.
- 1,373
11
votes
2
answers
777
views
Presentation of the monoid of surfaces
In the following every surface is assumed to be connected.
I've read that the commutative monoid of homeomorphism classes of closed surfaces is generated by $P$ (projective plane) and $T$ (torus) ...
- 63.1k
11
votes
2
answers
844
views
Existing proofs of Rokhlin's theorem for PL manifolds
I'm looking for a comprehensive reference to existing proofs of Rokhlin's theorem that a 4-dimensional closed spin PL manifold has signature divisible by 16.
I'm specifically interested in direct ...
- 7,842
11
votes
4
answers
2k
views
What is the "right" definition of the homology(cohomology) of an orbifold?
What is the "right" analog in the orbifold case of a singular homology of a topological space?
We can not just take the homology of the underlying space, because it does not contain much information.
...
- 283
11
votes
2
answers
475
views
What is a finite Haken cover of the Seifert–Weber space?
It's known that the Seifert–Weber space (obtained from a dodecahedron by gluing opposite faces with a 3/10 turn) is an example of a non-Haken 3-manifold. Since every closed 3-manifold is virtually ...
- 113
11
votes
1
answer
2k
views
K-theory as a generalized cohomology theory
Which of the statements is wrong:
a generalized cohomology theory (on well behaved topological spaces) is determined by its values on a point
reduced complex $K$-theory $\tilde K$ and reduced real $K$...
- 2,782
11
votes
2
answers
811
views
Higher dimensional Heegaard splittings?
Smooth (closed, connected, orientable) 3-dimensional manifolds are very special, in that for any 3-manifold $M$ there are two handlebodies, $V$ and $W$, of genus $g$ and an orientation reversing ...
- 732
11
votes
2
answers
533
views
Simplicial replacements in smoothing theory
As far as I can tell, ever since Milnor's Microbundles and differentiable structures (1961) paper, whenever people talk about $Diff(\mathbb R^n)$ or $PL(\mathbb R^n)$ or $Homeo(\mathbb R^n)$, they ...
- 44.4k
11
votes
1
answer
388
views
Twisting bordism classes
Let $X$ be a reasonable topological space (I'd be happy to assume that $X$ is a smooth closed manifold) and let $f\colon M^n \rightarrow X$ be a continuous map from a smooth oriented $n$-manifold $M^n$...
- 113
11
votes
1
answer
454
views
Is there a closed 5-manifold $M$ with $w_1(M)w_2(M)\ne 0$?
I'm trying to find generating manifolds for the cobordism group $\mathit{MO}_5(K(\mathbb Z/2, 2))\cong (\mathbb Z/2)^4$, which can be represented as the cobordism group of closed 5-manifolds $M$ ...
- 6,881
11
votes
1
answer
337
views
Growth of Poincaré duality groups
Can one prove that Poincaré duality groups cannot have intermediate growth?
- 4,188
11
votes
1
answer
379
views
Smooth structure on direct product
Let $M$ be the $E_8$ manifold. Is there a closed manifold $N$ such that $M\times N$ is smoothable? What is the smallest possible dimension of $N$?
user164740
11
votes
1
answer
811
views
What is an interpretation of the relation in the cohomology of the pure braid groups?
In 1968, Arnol'd proved that the integral cohomology of the pure braid group $P_n$ is isomorphic to the exterior algebra generated by the collection of degree-one classes $\omega_{i,j}\ (1 \le i < ...
- 2,830
11
votes
1
answer
542
views
Anomalies in the definition of Turaev's TQFT
In his book Quantum invariants of knots and 3-manifolds page 124, Turaev defined a TQFT $\tau$ axiomatically.
For a cobordism $(M, \partial_{-}M, \partial_{+}M)$, a TQFT assignes a $k$-homomorhism $\...
user22741
11
votes
1
answer
309
views
When can a surface in a 3-manifold be isotoped off a knot?
Given a closed (perhaps irreducible) 3-manifold $M$ with an embedded surface $S$ and a knot $K$, what conditions allow the two to be isotoped to be disjoint? Obviously a necessary condition is that $[...
- 187
11
votes
1
answer
331
views
Embedded 2-tori in $S^1\times S^4$
I am interested in understanding the smooth isotopy class of embedded 2-tori in $S^1\times S^4$. Is it true that every two homotopic embedded 2-tori in $S^1\times S^4$ are smoothly isotopic? It would ...
- 129
11
votes
1
answer
167
views
A group of type F that is an extension of type F-by-type F
Let us first recall that a group of type $F$ is a group admitting a compact classifying space.
Let $K$ and $Q$ be groups of type $F$. Consider the family $\mathcal{G}(K, Q)$ consisting of groups $G$ ...
- 489
11
votes
1
answer
478
views
Intersection map giving rise to Poincaré duality
Let $M$ be a smoothly triangulated compact $d$-dimensional manifold. Consider the subcomplex $C_*^{\pitchfork T}(M)$ of smooth singular chains which are transverse to the triangulation. An inductive ...
- 4,990
11
votes
2
answers
936
views
Embeddings without nonvanishing normal vector fields
For which values of $n$ does there exist an embedding of a smooth compact manifold $M\hookrightarrow R^n$ into $n$-dimensional Euclidean space such that the normal bundle to $M$ has no nonvanishing ...
- 2,927
11
votes
1
answer
335
views
Reference request: sheaves on the site of d-manifolds
I believe I know how to prove the following results. I also know to whom to cite fancy-shmancy results that have these as a very special case. My question is: what are the correct citations for ...
- 54.6k
11
votes
0
answers
220
views
On an Artin (?) subgroup of braid groups
While working on something apparently unrelated I encountered a "braid-like" group, which is a relatively geometric subgroup of a braid group and seems to be itself an Artin group. It seems ...
- 42.4k
11
votes
0
answers
335
views
Isotopy on embedded 3-manifolds in 4-manifolds
Suppose that $X$ be an oriented, closed four-manifold. Suppose that $Y$ is an oriented, closed three-manifold smoothly embedded in $X$. Suppose also that $f:X \to X$ is a diffeomorphism that fixes $Y$...
- 3,751
11
votes
0
answers
654
views
What is known about mapping class groups of 4-manifolds?
I am mostly interested in the case when you have a smooth degree $d$ algebraic surface $X$ over $\mathbb C$ and we can define three distinct groups: $\pi_0(\mathrm{Diff}^+(X))$, $\pi_0(\mathrm{Homeo}^+...
- 765
11
votes
0
answers
650
views
Triangulation of manifolds with corners
Let's begin with some definitions:
A (smooth) manifold with corners is a Hausdroff (and second countable if you want) space that can be covered by open sets homeomorphic to $\mathbb R^{n-m} \times \...
- 909
10
votes
3
answers
807
views
Is every (finite) group action on R^n by diffeomorphisms conjugate to a linear action?
I want to know if every smooth (finite)group action on $\mathbb{R}^n$ is conjugate to some linear action.Thank you!
- 257
10
votes
3
answers
843
views
Invariants of high-dimensional knots
In the study of knots in three dimensions, it can be shown that the fundamental group together with a specification of a meridian and longitude form a complete invariant for knots. What is known about ...
- 1,025
10
votes
3
answers
990
views
One question on cup product and torsion elements
As we know, by universal coefficient theorem, $H^{1}(X,\mathbb{Z})$ is torsion-free. My question is:
for cup product $H^{1}(X,\mathbb{Z})\otimes H^{1}(X,\mathbb{Z})\rightarrow H^{2}(X,\mathbb{Z})$
...
- 399
10
votes
3
answers
2k
views
Homotopy type of the plane minus a sequence with no limit points
It is well known that the plane minus $n$ points is homotopy equivalent to a wedge of circles and hence its fundamental group is free on $n$ letters.
Question: Is the plane minus an infinite sequence ...
- 1,329
10
votes
2
answers
499
views
Symplectic structure on the square of a 3-manifold
Let $M$ be a connected closed orientable 3-manifold. Assume $M$ is not the direct product of a surface and the circle.
Can there be a symplectic or Kähler manifold homeomorphic to $M\times M$? I think ...
user164740
10
votes
2
answers
709
views
4-dimensional cohomology $\mathbb{CP}^2$'s
Let $M$ be a closed, smooth $4$-manifold with integral cohomology ring isomorphic to that of $\mathbb{CP}^2$, is it diffeomorphic to it?
- 6,995
10
votes
1
answer
1k
views
Acyclic Finite Groups
A group is called acyclic if its classifying space has the same homology of a point. Examples of acyclic groups include Higman's group with four generators and relations, also ...
- 2,630
10
votes
2
answers
451
views
Group of surface homeomorphisms is locally path-connected
I think the following is true and I need a reference for the proof. (Given a closed surface $S$, i.e. a compact 2-dimensional topological manifold (without boundary), we endow $S$ with a distance ...
- 1,517
10
votes
2
answers
659
views
Homotopy equivalent smooth 4-manifolds which are not stably diffeomorphic?
Recall that two 4-manifolds $M$ and $N$ are stably diffeomorphic if there exist $m,n$ such that
$$M \#_n (S^2 \times S^2) \cong N \#_n (S^2 \times S^2).$$
That is, they become diffeomorphic after ...
- 27.5k
10
votes
2
answers
730
views
Representability of the sum of homology classes
This is probably a very simple question, but I have not found it addressed in the references that I know. Let $M$ be a closed and connected orientable $d$-dimensional manifold ($d\leq 8$) and let $[\...
- 2,816
10
votes
4
answers
578
views
Is the action of $G$ on $H_1(T^n, \mathbb{Z}) = \mathbb{Z}^n$ faithful?
Let $G$ be a finite group of diffeomorphisms of the torus $T^n$ fixing some point $p$, i.e. $p$ is fixed by every element of $G$. I have two questions.
Is the action of $G$ on $H_1(T^n, \mathbb{Z}) = ...
- 101
10
votes
1
answer
988
views
Status of Zeeman's collapsability Conjecture
Zeeman's conjecture in topological combinatorics states that if $K$ is a contractible polyhedron of dimension 2, then $K\times I$ has a collapsible subdivision.
What is the status of this ...
- 2,630
10
votes
2
answers
703
views
When does an even-dimensional manifold fiber over an odd-dimensional manifold?
Are there simple necessary and sufficient conditions for an (oriented) even-dimensional compact smooth manifold to fiber over an (oriented) odd-dimensional manifold (with oriented fibers)?
For ...
- 54.6k
10
votes
1
answer
838
views
Homology of infinite loop spaces $QX$
Let $X$ be a simply connected space. By $Q$ I denote $\Omega^{\infty}\Sigma^{\infty}$. Then $QX$ is an infinite loop space and the homology $H(QX)$ in $\mathbb{F}_p$ is a Hopf algebra over the Dyer-...
- 471
10
votes
2
answers
890
views
Are virtual cubulated groups cubulated?
Suppose $G$ has a finite index subgroup $N$ such that $N$ acts properly and cocompactly on a CAT(0)-cube complex. Does $G$ also act properly and cocompactly on a CAT(0)-cube complex?
Edit: After ...
- 539
10
votes
1
answer
635
views
Free action of SL_2(F_p) on a sphere
Let $p>2$ be prime. Then for abstract reasons the special linear group $\text{SL}_2({\mathbb F}_p)$ possesses a free action on some sphere (one has to check that any abelian subgroup of $\text{SL}...
- 2,756
10
votes
1
answer
659
views
Are there any tests for knowing whether a topological space admits a CW structure?
We know that for n $\ge$ 5, a manifold admits a piecewise linear structure if and only if its Kirby-Siebenmann class vanish and Galewski and Stern showed the existence of a similar invariant to test ...
- 168
10
votes
2
answers
876
views
Are there nontrivial involutions of $S^7\times S^7$ with fixed point set homeo to $S^7$?
The group $\mathbb{Z}_2$ acts on $S^7\times S^7$ by switching the coordinates with fixed point set $\Delta(S^7\times S^7)\cong S^7$. I want to know whether there are some other $\mathbb{Z}_2$ actions ...
- 888
10
votes
2
answers
497
views
Equivariant cohomology of the complement to the arrangement $\bigcup_{i\neq j}\vec x_i = \vec x_j$?
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\Conf{Conf}$Let $V=\mathbb{R}^d$ be a $d$-dimensional (Euclidean) vector space over real numbers.
Let $G=\SO(V)$ be the ...
- 133