All Questions
Tagged with gt.geometric-topology reference-request
361 questions
6
votes
0
answers
163
views
Reference to the theorem about linear bundles
The following fact looks like it is well-known. I can prove it myself, but I would like to know of a reference which has a proof?
Let $M$ be an $n$-manifold, $C\subset M$ a codimension-1 closed ...
- 1,095
6
votes
0
answers
172
views
Does Novikov additivity hold for topological manifolds?
Recall that Novikov additivity of signature of compact oriented smooth $4k$-manifolds is the following statement :
If two manifolds are glued by an orientation-preserving diffeomorphism of their ...
- 14.3k
6
votes
0
answers
813
views
Limit of metric spaces
Let $\{X_n\}_{n\in \mathbb{N}}$ be a collection of T2 topological spaces, with maps $f_n\colon X_n \to X_{n+1}$. These maps are continuous and open. Let $X$ be the direct limit of this system.
Assume ...
- 2,384
6
votes
0
answers
217
views
What is the state of the art in 4-manfold 2-types?
In an old answer to an old question of mine, Peter Teichner commented that it is an open problem to determine which homotopy 2-types arise from 4-manifolds. In some instances we know that a 4-manifold ...
- 35.5k
5
votes
6
answers
913
views
Topological results from geometry
Hi people,
I'm interested in results, such as the Gauß-Bonnet theorem, Fàry-Milnor theorem or classification theorems for manifolds, which give topological properties from geometric considerations. ...
- 516
5
votes
3
answers
505
views
Embedded ribbons and regular isotopy
I'm reading Kauffman's 1990 paper "An Invariant of Regular Isotopy" about knots that are isotopic through only Reidemeister Type II and III moves, which is known as a regular isotopy. His ...
- 217
5
votes
2
answers
1k
views
First appearance of Novikov's additivity theorem
Hi!
Novikov's additivity theorem states that if you glue together two compact oriented 4n-manifolds along a connected component of their boundaries, the signature of the resulting manifold is ...
- 558
5
votes
3
answers
593
views
Who first used the cross-ratio to describe shapes in hyperbolic geometry?
I was reading this Wikipedia article today:https://en.wikipedia.org/wiki/Shape#Similarity_classes
and I realized that it strongly resembles the use of coss-ratios as "shape parameters" in hyperbolic ...
- 3,359
5
votes
1
answer
188
views
Properly embedded surfaces in handlebodies are compressible or boundary compressible?
I've read in a couple of different places (a paper and a blog) the following fact:
if $F$ is a surface, properly embedded in a three-dimensional handlebody of genus at least two, then $F$ is either ...
- 421
5
votes
1
answer
792
views
$K_0$ of integral group ring of cyclic group $\mathbb{Z}/p\mathbb{Z}$
Is there a table for the computation of $K_0(\mathbb{Z}[\mathbb{Z}/p\mathbb{Z}])$?
These groups are also known as ideal class group in number theory.In topology,they are the home of some important ...
- 593
5
votes
1
answer
884
views
solvable word problem without algorithm
Let $G$ be a finitely generated group. I wonder if there are examples where:
1) The word problem is known to be solvable in $G$ but there is no algorithm known.
2) The word problem is known to be ...
- 829
5
votes
1
answer
906
views
Boundaries of relatively hyperbolic groups
When the interior of an n-manifold $M$ has a pinched negative curvature metric of finite volume, then its fundamental group $\Gamma=\pi_1M$ is relatively hyperbolic relative to the parabolic groups $\...
- 10.4k
5
votes
1
answer
439
views
Rotation part of short geodesics in hyperbolic mapping tori
Otal [Sur le nouage des géodésiques dans les variétés hyperboliques. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 7, 847--852.] showed that "short" simple closed geodesics in 3-dimensional ...
- 1,601
5
votes
1
answer
263
views
Pontryagin square, Postnikov square and their consistency formulas
$\mathcal{P}_2$ is Pontryagin square
$$H^{2i}(M,\mathbb Z_{2^k})\to H^{4i}(M,\mathbb{Z}_{2^{k+1}}).$$
$\mathfrak{P}$ is the Postnikov square $$H^2(M,\mathbb Z_3)\to H^5(M,\mathbb Z_9).$$
question (i)...
- 10.5k
5
votes
1
answer
394
views
closed geodesics are dense in both hyperbolic surface and unit tangent bundle of hyperbolic surface
Could you please recommend me some references for proofs of this fact: "closed geodesics are dense in both hyperbolic surface and unit tangent bundle of hyperbolic surface". Thanks in advance!
- 333
5
votes
2
answers
611
views
Reference request: 2-dimensional Schonflies theorem
Does anyone know a reference for the 2-dimensional version of the Schoenflies theorem? To be precise, I'd like a reference for the fact that every continuous, 1-1 map $S^1\rightarrow \mathbb{R}^2$ ...
- 8,803
5
votes
1
answer
283
views
Homology of spherical $3$-manifold group
I have been studying $3$-manifolds recently and I got stuck in the following situation. For lens spaces the below fact is true.
Let $G$ be a finite group acting freely and orthogonally on $S^3$ so ...
- 179
5
votes
1
answer
151
views
Nonexistence of sphere with one conical point [reference request]
It seems to be considered a classical fact that one cannot have a spherical polyhedral/cone-metric on the 2-sphere with precisely one conical point. However, I've never actually seen it proven ...
- 458
5
votes
2
answers
228
views
Reference for Cochran-Orr-Teichner's filtrations on knot concordance
I would be interested in recommendations for easily accessible/somehow simply readable texts to Cochran-Orr-Teichner's filtrations on knot concordance:
Tim D. Cochran, Kent E. Orr, and Peter Teichner....
user160180
5
votes
1
answer
413
views
Casson invariant and Euler characteristic
A slogan I frequently hear is: "the Casson invariant is the Euler characteristic of the Floer homology of flat SU(2)-connections on the integral homology sphere". Is there a single paper/reference ...
- 31
5
votes
1
answer
192
views
Relating bordism groups of different dimensions
Let
$M_d$
be a $d$-manifold generator of a subgroup of bordism group
$$
\Omega_d^{G},
$$
or further generalization
$$
\Omega_d^{G}(K(\mathcal{G},n+1)),
$$
which $G$ is the given structure ...
- 10.5k
5
votes
2
answers
615
views
Conjugacy problem for small braid groups
The conjugacy problem for braid groups $B_3$ and $B_4$ can be solved in polynomial time, it is noted in the paper by Birman, Ko and Lee(2001).
That was a result in 2001. Are there any new results on ...
- 613
5
votes
1
answer
142
views
Short basis in $\pi_1$ on a hyperbolic surface of bounded diameter
First, some terminology. Let $(S,x)$ be a compact surface of genus $g>0$. A standard collection of loops $\gamma_1,\ldots, \gamma_{2g}$ based at $x$ is a collection of loops that cuts $S$ into a ...
- 14.3k
5
votes
1
answer
216
views
Continuity of taking collapse maps
Let $U$ and $V$ be open subsets of $\mathbb R^n$ and let $\mathrm{OEmb}(U,V)$ denote the space of open embeddings of $U$ into $V$ with the compact-opent topology. Let $\bar{U},\bar{V}$ denote their ...
- 666
5
votes
1
answer
291
views
Question about and good reference for Kahn and Markovic result
As far as I understand, the celebrated result of Kahn and Markovic about quasi-Fuchsian immersions of surfaces in hyperbolic 3-manifolds has the following corollary:
Let $M$ be a compact hyperbolic $3$...
- 829
5
votes
1
answer
248
views
Multisignature and homeomorphism type
In classical surgery theory, there is a map
$$L_{n+1}(\pi_1M)\to S(M^n)$$
Element in $L_{n+1}(\pi_1M)$ is realized as surgery obstruction of a surgery problem to $M\times I$ with one boundary piece ...
- 101
5
votes
1
answer
192
views
Reference for a theorem on crossing changes of links
I've recently stumbled upon a paper of Scharlemann on crossing changes:
link text
In particular I am interested in understanding Theorem 2.2 (page 6):
"Theorem: If links A and B
are related by a ...
- 51
5
votes
2
answers
656
views
Homotopy equivalences preserving structure
Suppose I have $X=X_1\cup X_2\cup…\cup X_n$ and $f:X \to Y$ where $Y$ has a similar decomposition.
Suppose I know that
$f | X_{i_1}\cap…\cap X_{i_r} \to Y_{i_1}\cap…\cap Y_{i_r}$ is a homotopy ...
- 51
5
votes
2
answers
406
views
Unknotting tunnels in surface bundles
Given a once-punctured surface $F$ and an orientation preserving self homeomorphism $\phi$, let $M_\phi$ be the bundle over $S^1$ with fiber $F$ and monodromy $\phi$.
In Sakuma's survey article The ...
- 1,601
5
votes
1
answer
179
views
Symmetry conjecture for minimal dilatation pseudo Anosov mapping classes
The conjecture is something like the following:
The minimal dilatation among pseudo-Anosov mapping classes on a surface $S_{g,n}$ is realized by $\rho\circ\omega$ where $\omega$ is supported on a ...
- 105
5
votes
0
answers
289
views
A certain kind of proof of the Hairy Ball Theorem
I'd just like to know if proofs of the Hairy Ball Theorem along the following lines are well-known or even somewhere in the literature.
From a given vector field $V_1$ on $S^2$, form another, $V_2$, ...
- 17.6k
5
votes
0
answers
131
views
Earliest known proof of "Any degree one self-map of an orientable connected finite-type non-compact surface is homotopic to a homeomorphism"
I attended a talk where the speaker said the following is due to Nielsen. I searched here and there but couldn't find the corresponding paper, if any. So, what is the earliest known proof of the ...
- 1,097
5
votes
0
answers
272
views
When do surfaces in $\mathbb{R}^4$ intersect all their translations in one direction?
I am looking for research or references on the following problem.
Let $S$ be a smoothly embedded connected surface in $\mathbb{R}^4$, with or without boundary. Fix some axis in $\mathbb{R}^4$, let $d ...
- 1,763
5
votes
0
answers
228
views
Automorphism groups of cocompact Fuchsian groups as mapping class groups
Let $\Gamma$ be a cocompact Fuchsian group. So it has presentation $$\langle x_1,y_1, \dots, x_g,y_g,z_1, \ldots, z_r \mid [x_1,y_1] \cdots [x_g,y_g]z_1 \cdots z_r=1, \ z_i^{m_i}=1 \rangle$$
for some $...
- 8,401
5
votes
0
answers
233
views
Is there a well-established terminology for polyhedra/polytopes?
I got confused lately. It seems like in the metric context a polyhedron tends to mean an intersection of a finite number of half-spaces, while a polytope is a convex hull of a finite set of points. At ...
- 18.4k
5
votes
0
answers
1k
views
Prerequisites for reading Gregory Perelman's work
What are the prerequisites for understanding the work of Perelman concerning the Poincaré conjecture?
I am referring to the last three papers here.
- 1,594
5
votes
0
answers
461
views
When does a manifold 'deformation retract' into a small neighborhood of some k-dimensional subpolyhedron?
Under which conditions does a m-manifold $M^m$ admit a deformation retraction to a
small neighborhood of some k-dimensional subpolyhedron?
Or, under which conditions is the identity map $id_M$ of a ...
- 722
5
votes
0
answers
265
views
Quotienting disk inside sphere result in sphere
Let $S^k$ be a topological $k$-dimensional sphere. Let $D^k$ be a $k$ dimensional disk that includes in $S^k$. Let
$q: D^k \to D^r$ be a map and $r \leq k$. Let
$$W = S^k \sqcup D^r/\sim$$
where $S^...
- 2,023
4
votes
2
answers
846
views
Fundamental polygons with infinite pairwise identifications
The topology of a closed surface can be constructed
by identifying edges of a fundamental polygon of an
even number $2n$ of edges.
Labeling the edges and using $\pm 1$ exponents to indicate
direction,
...
- 151k
4
votes
2
answers
267
views
Finite models for torsion-free lattices
Let $G$ be a real, connected, semisimple Lie group and $\Gamma < G$ a torsion-free lattice. Then does there exist a finite $CW$-model for $B\Gamma$?
I know this to be true in many instances (e.g. ...
- 1,255
4
votes
1
answer
124
views
Gromov hyperbolicity for (non-geodesic) metrics on the upper-half plane invariant with respect to SL(2, R) action
$\DeclareMathOperator\SL{SL}$Let $d$ be a metric on the upper-half plane $\mathbb H = \{(x,y) : y > 0\}$ which is invariant with respect to the action of $\SL(2, \mathbb R)$ to $\mathbb H$ which is ...
- 195
4
votes
3
answers
382
views
Extending a continuous map over projective space
Let $X = P^{n-1}(\Bbb C)$ ($(n-1)$-dimensional projective space) with $n \geq 3$, and let $K \subset X$ denote a compact subset. I have a bijective, continuous map $\phi:K \to K$ which satisfies the ...
- 356
4
votes
1
answer
362
views
Who first considered constructibility of simplicial complexes?
A simplicial complex of dimension $d$ is called constructible if it is a simplex, or if it is the union of two constructible dimension-$d$ simplicial complexes along a dimension-$(d-1)$ intersection. ...
- 1,500
4
votes
2
answers
355
views
Books for learning branched coverings
I am self-studying branched coverings. I read it from B. Maskit's Kleinian groups book. I want some more references for reading branched covers. In particular, I want to understand how to create ...
- 613
4
votes
1
answer
485
views
Thurston's preprint: "On the geometry and dynamics of diffeomorphisms of surfaces"
W. Veech on Teichmüller curves in moduli space, Eisenstein series and applications to triangular billiards says on the second paragraph of page 579:
"Thurston's original construction [8] corresponds ...
- 637
4
votes
1
answer
1k
views
Cap product à la Poincaré
Recently, it became apparent to me that I was not the only one who always first thought in terms of cap product before actually computing a cup product. There is no denying this is evil, but I found ...
- 4,432
4
votes
1
answer
260
views
Bounds for the crossing number in terms of the braid index?
Is there a lower bound on the crossing number of a knot (resp., link) with braid index $b$?
For knots, I believe the smallest crossing number for braid index 2 is 3, the smallest crossing number for ...
- 9,114
4
votes
1
answer
119
views
Enumeration of three dimensional spherical good orbifolds covered by Nil, sol and E3
Is there in the literature a list of three dimensional spherical, good orbifolds covered by nil, Sol and E3, and their algebraic topological invariants? (Homology, orbifold fundamental group).
- 2,630
4
votes
1
answer
169
views
"Almost embedding" the complete 2-dimensional complex $\mathcal K_7^2$ into $\Bbb R^4$
Let $\mathcal K_7^2$ be the complete 2-dimensional simplicial complex on seven vertices, i.e. it has all $7\choose 2$ edges and all $7\choose 3$ 2-simplices (and no higher-dimensional simplices).
I ...
- 13.6k
4
votes
1
answer
577
views
Reference request for Poincaré–Lefschetz duality as an intersection pairing
I believe the following is well known after talking to some experts, but I am unable to find a reference for the case with boundary.
Fix a field $F$ and an oriented $n$-manifold $(M,\partial M)$. We ...
- 5,849