All Questions
Tagged with gt.geometric-topology at.algebraic-topology
1,145 questions
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About Hopf invariant
I am trying to understand properly the Hopf invariant. So, one way to compute the Hopf invariant from a smooth map $f:S^3\to S^2$ consists on taking two regular values $p,q\in S^2$ and computing the ...
- 81
4
votes
0
answers
241
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Non-spin 5-manifold and $2^2$-Bockstein homomorphism
The $2^2$-Bockstein is $\beta_4$ is associated to
$$0\to\mathbb{Z}/2\to\mathbb{Z}/{8}\to\mathbb{Z}/{4}\to 0,$$
(The $2^n$-Bockstein homomorphism
$$\beta_{2^n}:H^*(-,\mathbb{Z}/{2^n})\to H^{*+1}(-,\...
- 10.5k
4
votes
0
answers
90
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On decidability of a homeomorphism with a prescribed pushforward
This is a refinement of my older question
A homeomorphism with a prescribed action on the fundamental group - decidable or not?
The problem under considreation is the following. Let $M,N$ be two ...
- 6,891
4
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0
answers
105
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Homology of Torelli subgroup groups of automorphism groups of free products
For a group $G$, let $G^{*n}$ denote the $n$-fold free product. There is a natural map $Aut((\mathbb Z/k\mathbb Z)^{*n}) \mapsto GL_n(\mathbb Z/k\mathbb Z)$. Is it known if the group homology of the ...
- 965
4
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173
views
Finiteness for 2-dimensional contractible complexes
While thinking about graph-complex and related operadic stuff, I found a quite interesting (at least for me) question. However, I'm a novice in the algebraic topology, so I'm unable to resolve it by ...
- 1,842
4
votes
0
answers
74
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Determinant twist and $Pin _{\pm}$ structure on $4k$-dimensional bundles [Reference request]
Consider the automorphism $\varphi$ of $O(2n)$ given by $\varphi (A)=det(A)\cdot A$. The induced map in cohomology $H^*(BO(2n))$ sends $w_2$ to $w_2+(n+1)w_1^2$ (a proof is given at the end of the ...
- 3,051
4
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221
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Contractibility of regular CW sphere minus open star
Let $S$ be any regular CW decomposition of (a space homeomorphic to) the $n$-sphere, and consider a cell $\sigma$ of dimension $d \in \{0,\ldots,n\}$. Let $S'$ be the regular CW complex which remains ...
- 15.5k
4
votes
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269
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Action of the mapping class group on separating spheres in a connected sum of aspherical 3-manifolds
Let $M$ be a connected sum of $g$ closed aspherical 3-manifolds $M_1, \ldots, M_g$. [Update: I also assume that all the $M_i$-s are diffeomorphic, i.e. $M$ is a connected sum of copies of the same ...
- 3,203
4
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answers
576
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Topological version of two results in smooth Morse theory
Morse theory is generally presented in the DIFF category. However, there is a version of Morse theory in TOP (see the post Morse theory in TOP and PL categories? for references).
It is well known ...
- 2,136
3
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1
answer
493
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Four-dimensional vector bundles over $S^4$, intuition?
I know that $\pi_3(SO(4)) = \mathbb{Z} \oplus \mathbb{Z}$. We can choose an explicit identification as follows: given $(i, j) \in \mathbb{Z}$, we have a map $\phi: S^3 \to SO(4)$ which sends a unit ...
- 31
3
votes
3
answers
769
views
Reducible 3d torus bundles
Here reducible means that the mapping class for the fiber is a reducible auto-homeomorph in the sense of Nielsen-Thruston. So,
could anyone give me a hint to classify them?
In contrast, do you agree ...
- 345
3
votes
1
answer
431
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Detecting a PL sphere and decompositions
Question 1. A PL $n$-sphere is a pure simplicial complex that is a triangulation of the $n$-sphere. I'm looking for a computer algorithm that gives a Yes/No answer to whether a given simplicial ...
- 903
3
votes
3
answers
537
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Fundamental group of a generalized connected sum
Let $M$ and $N$ be two $n$ dimensional connected closed manifolds with $n \ge 3$, and let $S$ be a $(n-1)$ dimensional closed submanifold common to $M$ and $N$. Consider the connected sum of $M$ and $...
- 311
3
votes
1
answer
1k
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characteristic classes of homotopy equivalent manifolds
Let $M,N$ be two manifolds of different dimensions. Suppose $M\simeq N$, i.e. $M$ is homotopy equivalent to $N$. Do the Stiefel-Whitney classes of the tangent bundles of $M$ and $N$ equal
$$
w(TM)=w(...
- 2,223
3
votes
1
answer
536
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On definition of surgery [closed]
I am a beginner in surgery theory. I have started learning with ALGEBRAIC AND GEOMETRIC SURGERY by Andrew Ranicki.
On page 4 of the book he defines surgery :
Denition 1.2 A surgery on an $m$-...
- 1,303
3
votes
2
answers
1k
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Unreduced Suspension Isomorphism
Tending to a lecture on homotopy theory, the following question occured to me (is that a correct sentence?):
Given a pointed space $(X,x)$, is the UNREDUCED suspension map $S:\pi_k(X,x) \rightarrow \...
- 333
3
votes
1
answer
912
views
Homotopy classes of maps
This is a reference request.
A theorem of Hurewicz (published in Beiträge zur Topologie der Deformationen. IV. Asphärische Räume, Proc. Akad. Wetensch. Amsterdam, volume 39, deel 2 (1936), 215-224, ...
- 31
3
votes
2
answers
191
views
Embedded submanifold in a cylinder
Let $M^n$ be an $n$-dimensional topological closed manifold. Suppose there exists an embedding $i:M \to M \times [0,1]$ such that $i(M)$ is contained in the interior of $M \times [0,1]$ and separates $...
- 891
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1
answer
293
views
Are there compact flat fiber bundles with "truly" infinite structure group?
Let $p: E \rightarrow B$ be a flat fiber bundle with fiber $F$ where $E$, $B$, $F$ are compact, smooth manifolds.
I am looking for a counterexample for the following statement:
$E \cong \widetilde{B}...
- 404
3
votes
2
answers
628
views
Pseudo-manifolds and homology
Is there a good reference for the proof that the cobordism group of pseudo-manifolds is isomorphic to the singular homology group?
I was looking for a more geometrical definition of homology and ...
- 145
3
votes
2
answers
465
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Branched coverings over orbifolds with reflector lines
It is well known that if $F\to B$ is a $n$-finite branched covering over an orbifold with cone-points then the orbifold Euler's characteristics are related via $\chi(F)=n(\chi(B)-\sum_i^r\frac{a_i-1}{...
- 345
3
votes
1
answer
375
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Boundaries of subsets of simply-connected domains
I am trying to prove the following assertion: Let $B$ be a simply-connected set, and let $B' \subsetneq B$ be a proper open connected subset. Then, there exists a point $x \in \partial B'$ of the ...
3
votes
1
answer
553
views
Dold-Lashof construction and classifying space functor
Let $G$ and $H$ be two connected Lie groups. By the Dold-Lashof construction the classifying space $BHom(G,H)$ is well-defined (similar to the Milnor construction).
Is there a relation between $BHom(...
- 33
3
votes
1
answer
615
views
Knots in 3-manifolds
Consider a closed $3$-manifold $M$ and a knot $K$ in $M$.
Is it necessarily true that $\pi_2 (M \setminus K) = 0$?
If not, are there any conditions on $M$ and/or $K$ to ensure the above 2nd homotopy ...
- 77
3
votes
1
answer
205
views
Prove or disprove that there exists no $G$ structure with its bordism group $\Omega_1^{G} =\mathbb{Z}/N$ for $N>2$
It can be found that there are the following bordism group $\Omega_0^{G}$ at $d=0$ and 1 dimensions by requiring $G$ structure for $d$-manifolds:
$$
\Omega_0^{SO} = \mathbb{Z} , \quad \Omega_1^{SO} = ...
- 2,147
3
votes
2
answers
202
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f vectors of simplicial complexes homeomorphic to n dimensional spheres
In dimension 2, the euler poincare formula restricts the incidence properties of edges in a triangulation of a surface.
Are there analogous generalizations for higher dimensions, like elaborations ...
- 2,630
3
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1
answer
552
views
Is the Action of the mapping class group transitive on embedded arcs?
Let S be a surface of genus g with some parked points (n of them). Assume $n \geq 2$ and fix two of the marked points. Consider the set of embedded arcs going between these two special points. The ...
- 27.5k
3
votes
1
answer
236
views
Example of a non $\pi_1$-injective, degree one, self-map of a three-manifold
All manifolds will be assumed to be closed, oriented, and connected.
Let $f\colon M\to M$ be a map of degree $\pm 1$. It is not hard to show that $\pi_1(f)$ is surjective.
What is an example of a non ...
- 1,097
3
votes
1
answer
508
views
Handlebody decomposition of $L(2,1)\times S^1$
I wish to know the handlebody decomposition of $L(2,1)\times S^1$ in terms of Kirby diagrams, where $L(2,1)\cong RP^3$. And if possible, is there a general recipe for getting the handlebody ...
- 125
3
votes
1
answer
124
views
Homotoping diffeomorphism to a $J$-holomorphic one
Let $M$ be a closed simply-connected smooth manifold. Assume $M$ admits at least one almost complex structure.
Is any diffeomorphism $M\to M$ homotopic as a continuous map to a $J$-holomorphic ...
user164740
3
votes
2
answers
409
views
Alexander duality and homology equivalence
While reading the paper of Kauffman and Taylor "Signature of links" I found the following situation.
In the proof of Theorem 2.6 they suppose that two links $L_1, L_2\in \mathbb{S}^3$ are ...
- 521
3
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1
answer
296
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How to compute $\pi_0$ of $Maps(S^1, \Omega^2({S}^2, p))$
Denote by $\Omega^2({S}^2)$ the space of DOTTED maps from the $2-$sphere $S^2$ onto itself. And consider its FREE loop space $X=\mathcal{L}(\Omega^2({S}^2))=Maps(S^1, \Omega^2({S}^2))$. I think that $\...
- 81
3
votes
1
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557
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the "Kahn-Priddy map" and "multiplicative $p$-local equivalence"
The following is a part of a paper that I need to understand
I totally do not know the argument. Could you explain? Thanks.
Let $\Sigma_n$ be the $n$-th symmetric group and $\Sigma_\infty$ be the ...
- 1,990
3
votes
1
answer
284
views
Does every canonical decomposition of the intersection form come from a canonical homology basis?
Take a closed surface $X$ of genus $n$. By a canonical homology basis, I will mean a set of $2n$ homologically independent simple closed curves $\{\alpha_1,\ldots,\alpha_n,\beta_1,\ldots,\beta_n\}$, ...
- 333
3
votes
1
answer
469
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simplicial complex equipped with barycenric metric is complete [closed]
Consider a simplicial complex $C$. On its support $$|C|=\lbrace \alpha = \sum_{v\in C}\alpha_{v}v \mid 0\leq \alpha_{v} \leq 1 , \sum_{v\in C}\alpha_{v} =1\mbox{ and }v|{\alpha_{v}} \neq 0\mbox{ is a ...
- 31
3
votes
1
answer
828
views
Moise's Theorem and the Fundamental Domain of a $3$-Manifold
I'm currently researching the relationship between Moise's Theorem (Every closed $3$-manifold has a triangulation) and other properties of manifolds. In particular, I'm trying to learn about the ...
- 1,441
3
votes
1
answer
361
views
Lectures on triangulations of manifolds by Robion Kirby
I was looking for the book mentioned in the title. Seemingly it was not published, but copies are available in several mathematical libraries. Google books does not provide preview.
I am wondering if ...
- 103
3
votes
1
answer
267
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In what sense is every element of $H_2(G)$ "represented by a free action on some surface"
(This is a cross-post of this unanswered math.stackexchange question)
In Edmond's 1982 paper Surface Symmetry II, at the bottom of page 145, he writes:
"Corollary - If $G$ is a split nonabelian ...
- 7,527
3
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1
answer
351
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Is there a degree one map from a product $B\times S^1 \to \#_n S^2 \times S^1$ for any n
For any $n \geq 1$, let $\Sigma_n$ denote the closed orientable surface of genus n. In http://arxiv.org/abs/1202.6302, the authors showed that for any $n$, there is a degree two, $\pi_1$-surjective, ...
- 113
3
votes
2
answers
186
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equivariant embeddings from the k-th configuration space to the k+1-th configuration space
Let $S$ be a closed, orientable surface in $\mathbb{R}^3$ and $S'$ be the manifold $S\setminus\text{one point}$. Let $F(S',k)$ be the $k$-th (ordered) configuration space on $S'$. It is claimed in ...
- 1,990
3
votes
1
answer
309
views
Triangulation of moduli space.
I am recently reading the paper "Natural triangulations associated to a surface" by B.H. Bowditch and D.B.A. Epstein (http://www.sciencedirect.com/science/article/pii/0040938388900080#), where ...
- 1,713
3
votes
3
answers
316
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Existence of sequence of examples of braking 'Cancellation law in homeomorphic products'
I know there are manifolds (with or without boundary) $A$ and $B$ such that $A\times C$ is homeomorphic to $B\times C$ but $A$ is NOT homeomorphic to $B$.
My question is (in the Diffeomorphism ...
- 1,101
3
votes
1
answer
958
views
When does an antipodal map on a manifold extend to the antipodal map on a spheres
So I have been mulling the following question over in my head for awhile now, and want to see if anyone else might have any ideas.
Begin with $M$ a manifold and suppose that $M$ has an antipodal map $\...
- 5,191
3
votes
1
answer
133
views
Geodesic laminations on the 4-punctured sphere
Let $S_{0,4}$ be the 4-punctured sphere equipped with a hyperbolic metric of finite volume (thus all punctures are cusps). Consider $\gamma$ to be a simple geodesic of $S_{0,4}$ (not necessarily ...
- 133
3
votes
1
answer
200
views
Does the group of compactly supported diffeomorphisms have the homotopy type of a CW complex?
It is known that the group of diffeomorphisms of a compact manifold with the natural $C^{\infty}$ topology has the homotopy type of a countable CW complex. See for instance this thread: Is the space ...
- 491
3
votes
1
answer
260
views
Can such a set be simply connected?
$\newcommand\R{\mathbb R}$Let $U$ be an open subset of $\R^2$ such that the point $(0,0)$ is on the boundary of $U$. Let $f\colon[0,1]\to\R^2$ be the path that starts at $(0,0)$ and moves with a (say) ...
- 128k
3
votes
1
answer
125
views
Cohomology of the coned off space
Let $X$ be a compact manifold with boundary $\partial X$ with $
\dim X\setminus \partial X=n$. Moreover, $X$ and $\partial X$ are both aspherical. Then what's the $H^n(X\cup_{\Sigma\subset \partial X} ...
- 199
3
votes
1
answer
155
views
Surface separating the boundary of a cylinder
Let $M^2$ be a connected closed surface. Suppose there exists an smooth embedding from a connected closed surface $N$ into the interior of $M \times [0,1]$ such that $N$ separates $M \times \{0\}$ and ...
- 891
3
votes
1
answer
115
views
Hadamard-like product on orientable surfaces
Denote by $C$ the category of connected closed orientable surfaces.
Is there a functor $F:C\times C\to C$ such that $b_1(F(S\times S'))=b_1(S)b_1(S')$?
- 41
3
votes
1
answer
139
views
Bordism invariants vanishes in a lifted twisted $Pin^- \times Spin$-structure
It looks to me that the bordism group
$$\Omega_3^{SO} (B(O(2) \times SO(3))) \tag{1}$$
(whose Pontryagin dual for the manifold generator) contains at least a nontrivial invariant:
$$
w_1(O(2))\big(...
- 2,147