All Questions
Tagged with gt.geometric-topology at.algebraic-topology
1,145 questions
2
votes
1
answer
131
views
Approximate Jordan-Brouwer theorem
This came up when thinking about this question.
It is well-known that the image of a homeomorphic embedding $f:S^n\to \mathbb{R}^{n+1}$ separates the space into exactly two components, one of which is ...
- 5,529
2
votes
1
answer
128
views
Infinitely many deformation equivalent Hodge diamonds
Let $S$ be a connected open complex manifold. Let $\phi:X\to S$ be a proper holomorphic submersion with connected fibers. Can the fibers of $\phi$ have infinitely many distinct Hodge diamonds?
An ...
user164740
2
votes
1
answer
388
views
coproduct of the homology of iterated loop space on spheres
Let $\Omega^{n+1}S^{n+1}$ be the base-pointed $(n+1)$-iterated loop space on the $(n+1)$-sphere. In the paper The homology of $\mathcal{C}_{n+1}$-spaces, $n\geq 0$, F. Cohen, Lecture notes in ...
- 2,223
2
votes
2
answers
305
views
boundary homomorphism in the homotopy exact seqeunce of principal $SO(9)$ bundle over $S^8$
Consider principal $SO(9)$ bundles over $S^8$.They are in 1-1 correspondence with $$[S^8,BSO(9)]\cong \pi_7(SO(9))\cong \mathbb{Z}$$
Now pick up one such bundle $\xi$,we have the long exact sequence ...
- 111
2
votes
2
answers
767
views
Covering seifert manifolds
Let $M$ be a 3-manifold with boundary. If $M$ has an orientable finite cover that is a Seifert fiber space, then is $M$ also a Seifert fiber space?
- 43
2
votes
1
answer
287
views
How does hyperelliptic involution act on the standard generators of the fundamental group of surfaces of genus g with n punctures?
Let $S_{g,n}$ be the surface of genus $g$ with $n$ punctures. We know that $\pi_1(S_{g,n})$ admits a presentation:
$$\left\langle~ \alpha_1,\beta_1,\dots, \alpha_{g},\beta_{g},\gamma_{1},\dots,\gamma_{...
- 79
2
votes
1
answer
177
views
Isotopic homeomorphisms of surface induces same map on the space of ends
Let $\Sigma$ be a non-compact orientable connected two-manifold without boundary. Let $f,g\colon \Sigma\to \Sigma$ be two homeomorphisms. Suppose there is a homotopy $H\colon \Sigma\times [0,1]\to \...
- 1,097
2
votes
1
answer
652
views
Complex manifold defined over $\mathbb{R}$
Let $M$ be a connected closed complex manifold with an antiholomorphic involution.
Must there be an atlas and a choice of a reference point in each chart such that the transition functions are ratios ...
user164740
2
votes
1
answer
163
views
Structure sets for three dimensional surgery
Is there a treatment in the literature of the structure sets relating simple homotopy equivalences to homeomorphisms in the three dimensional case? I am aware that due to the ...
- 2,630
2
votes
1
answer
140
views
Could an inverse of (weak) Morse inequality exists in some special case?
Could an inverse Morse inequality hold in some sense? More precisely I wish the following result to be true:
Problem
$M$ is a smooth simply connected compact manifold, $dim(M)=n$, $f$ is a morse ...
- 697
2
votes
1
answer
345
views
Goeritz matrix and link coloring
For a link $L$ and a prime $p$, $L$ has a $p$-coloring iff $p$ divides the $\operatorname{gcd}$ of the invariant factors of the Goeritz matrix of $L$.
Do you know the elementary proof of this facts?
2
votes
1
answer
403
views
Compute cohomology of flat fiber bundles - does this always work?
Edit: This has been answered in this thread Are there compact flat fiber bundles with "truly" infinite structure group?.
Setting
Let $p: E \rightarrow B$ be a flat fiber bundle with fiber ...
- 404
2
votes
1
answer
192
views
Does $A_{j,k}$ commute with all its conjugates in homotopy braid groups?
Let $\tilde{B_n}$ be the homotopy braid group; namely, in the deformation of braids, a braid string is allowed to intersect itself. Similarly let $\tilde{P_n}$ be the homotopy pure braid group.
I am ...
- 1,108
2
votes
1
answer
198
views
Simple connectedness of $\mathbb{C}P^2$ intersected with an affine subspace
The complex projective plane $\mathbb{C}P^2$ can be thought of as the set of rank one 3-by-3 Hermitian matrices with norm one, i.e., $\mathbb{C}P^2 = \{xx^* : x \in \mathbb{C}^3, x^*x=1 \}$. As such, ...
- 630
2
votes
1
answer
377
views
Homotopy Extension Property (HEP)
I want to show (although Artin gave an ad hoc proof) that if two braids \beta and \beta' are isotopic as braids, then they are equivalent as tangles. I'd like to use the homotopy extension property (...
- 21
2
votes
1
answer
359
views
Induced Map on Sp(2g,Z) is surjective
Let Mg be the Mapping Class Group for $S_g$, the genus-g orientable surface, and consider the action of Mg on $H_1(S_g,\mathbb Z)$ sending f in Mg to m in $Sp^2(2g,\mathbb Z)$ through the induced map ...
- 105
2
votes
1
answer
1k
views
Thom's result and Poincaré duality
I am interested in singularity theory by topology.
I want to understand following results.
$f$ is a smooth map of a closed surface $M$ which has only
fold points and cusps as its singularities.
...
- 129
2
votes
1
answer
380
views
Is it always possible to connect the endpoints of a smooth injective path, so the resulting path is smooth, closed and simple?
Motivation
The motivation for this question arises from the following problem: Is there a closed, simple, and infinitely differentiable path on the (complex) plane such that every straight line has a ...
2
votes
1
answer
484
views
Mapping torus of orientation reversing isometry of the sphere
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\O{O}\DeclareMathOperator\Iso{Iso}$
Let $ f_n $ be an orientation reversing isometry of the round ...
- 4,081
2
votes
1
answer
282
views
Lifting of a proper map in the cover is a proper map
Let $M$ be an orientable surface without boundary$($I am not assuming $M$ is compact, it can be non-compact$)$. Let $\Phi: M\to M$ be a proper homotopy-equivalnce$($A proper homotopy-equivalence can ...
- 265
2
votes
1
answer
142
views
Extending the maps between the bordism groups: NONE exsistence of a certain kind of extended group
Let $M^d$ be a nontrivial bordism generator for the bordism group
$$
\Omega_d^G= \mathbb{Z}_n,
$$
suppose $G$ (like O, SO, Spin, etc) specify the group structure of the boridsm group. The $\mathbb{Z}...
- 2,147
2
votes
1
answer
196
views
(Intersection)-Cohomology of Orbit Spaces of $SO(n)$ acting on spheres.
This is a problem i am thinking for a while but did not find an answer. Maybe one of you knows it.
Let $G:= SO(n,\mathbb{R})$, $G\times \mathbb{R}^n\to \mathbb{R}^n$ the standard representation of $G$...
- 2,554
2
votes
2
answers
463
views
homotopy type of complement of subspace arrangement
I am studying the homotopy type of a space,and i hope it would be a $K(\pi,1)$ space.
now i have find its covering,once we can say the covering is $K(\pi,1)$,so is the space
itself.and the covering is ...
- 157
2
votes
1
answer
179
views
Model structures on simplicial presheaves of piecewise-linear manifolds
Let $\mathbf{PL}$ denote the category of piecewise-linear manifolds. The goal is to embed $\mathbf{PL}$ into a category of simplicial presheaves, endow it with a model structure, and then localize it ...
user528837
2
votes
1
answer
130
views
Space of the trivial long knot in the thickened surface
Let $F$ be a compact oriented surface and $x_0\in F$ a basepoint. Consider the set $\mathcal E=Emb_0(I,F\times I)$ of embeddings $\sigma\colon I\hookrightarrow F\times I$, $\sigma(\partial I)=\{x_0\}\...
- 357
2
votes
1
answer
256
views
Comparing the exit path category and the nerve of a stratified space
Let $P$ be a finite poset, and $X$ be a topological space stratified by $P$, in the sense that $X$ is equipped with a continuous map $X \to P$ in the Alexandroff topology (or equivalently with a ...
- 3,948
2
votes
1
answer
127
views
Preservation of fiberwise normal bundles under fiberwise homotopy equivalences
I am looking to replicate, in the fiberwise setting, the result of Spivak/Wall that the fiber homotopy type of the normal sphere bundle of a manifold is preserved under homotopy equivalences.
A ...
- 5,839
2
votes
1
answer
311
views
Vanishing cycles exact sequence for degeneration of curves
Let $f : X\rightarrow D$ be a family of smooth projective curves over the complex unit disk $D$, degenerating to a nodal curve $X_0$ above $0\in D$.
Let $\eta\in D - \{0\}$ be a general point, and let ...
- 7,527
2
votes
1
answer
130
views
Gluing isotopic smoothings
Let $M$ be a topological manifold which can be written as $M = U \cup V$ where $U$ and $V$ are open. Suppose both $U$ and $V$ admit smooth structures. Also assume that on the overlap $U \cap V$ the ...
- 803
2
votes
1
answer
654
views
Triangulation induces regular CW complex structure
If a topological set is triangulable, dose the triangulation map gives it the (regular) CW complex structure? From definitions, I see it seems to be, but I am not that sure, for may exist some strange ...
- 103
2
votes
1
answer
167
views
On the simply connectedness of Symmetric products and Hilbert schemes of points
My first question is whether $m$-th symmetric product of $\mathbb{C}^{n}$ is simply connected, where $n\geq 3$.
The second question is whether $Hilb^{m}(\mathbb{C}^{n})$ is simply connected, where $n\...
- 399
2
votes
0
answers
116
views
Generalized Stokes's theorem and the relationship of volume to surface area of objects of arbitrary genus
In this post I saw that it could be explained with the Generalized Stokes's theorem why the derivative of the area of a circle is equal to the boundary of the circle (the circumference):
$$C=\frac{d}{...
- 131
2
votes
0
answers
414
views
$$ \left(\frac{\text{Man}^{\text{fr}}}{\text{Cobordism}},\coprod,\times \right)\simeq \left((\text{Fin}^{\simeq},\coprod)^{\text{gp}},\times\right)?$$ [closed]
If we combine a theorem of Pontryagin and the Barratt-Priddy-Quillen theorem we get that both sides of
$$
\left(\frac{\mathrm{Man}^{\mathrm{fr}}}{\mathrm{Cobordism}},\coprod,\times \right)\simeq \left(...
- 705
2
votes
0
answers
147
views
Extension of isotopies
In what follows $M$ will be a manifold (without boundary, for simplicity) and $C\subseteq M$ will denote a compact subset. In the paper Deformations of spaces of imbeddings Edwards and Kirby prove the ...
- 641
2
votes
0
answers
175
views
Minimal first Pontryagin class $p_1=1$?
From Hirzbuch theorem,
the signature of 4-manifold $\sigma = p_1/3$ with the first Pontryagin class $p_1$.
I know that the $\sigma=p_1/3 =1$ so $p_1=3$ for complex projective space $\mathbb{P}^2$.
Is ...
- 447
2
votes
0
answers
55
views
Tangential normal invariant isomorphism
Recently, I was reading the paper "Finite Group Actions on Kervaire Manifold" by Crowley, Hambolton. But I am having problem understanding a definition. Here it is,
In page 15-16 they are ...
2
votes
0
answers
146
views
Explicit S-duality map
$\DeclareMathOperator{\Th}{Th}$
The Thom space of a closed manifold $M$ ($\Th(M)$) is the $S$-dual to $M_+ (=M \cup \{pt\})$. Let $M$ be embedded in $\mathbf R^{n+k}$. I found the duality map from $S^...
2
votes
0
answers
106
views
Lifting homology classes to the unit tangent bundle, a la Johnson
Let $M$ be a oriented smooth closed 2-manifold, and let $\gamma$ be an oriented smooth simple closed curve in $M$.
In Spin structures and quadratic forms on surfaces, Johnson definines a standard way ...
- 371
2
votes
0
answers
105
views
Unstably dualizable maps
Call a map between compact, connected framed $n$-manifolds $f:M \rightarrow N$ unstably dualizable if there exists an $f':N \rightarrow M$ such that the following diagram commutes up to homotopy:
$$\...
- 5,839
2
votes
0
answers
197
views
$4$-manifolds with boundary homotopic to $K(G,1)$
I am not very conversant with the $3$-manifold or $4$-manifold theory. The literature I found so far related to $3$-manifold tells me that $S^1\times S^2$ is "very special". It is prime but ...
- 1,177
2
votes
0
answers
105
views
Vanishing of Goldman bracket requires simple-closed representative?
Let $\Sigma$ be a connected oriented surface, and $[-,-]\colon \Bbb Z\big[\widehat\pi(\Sigma)\big]\times Z\big[\widehat\pi(\Sigma)\big]\to Z\big[\widehat\pi(\Sigma)\big]$ be the Goldman Bracket. Note ...
- 1,097
2
votes
0
answers
109
views
Homotopical generalizations of the isotopy extension theorem
It's a well known theorem that given isotopic embeddings $i_1,i_2: M \rightarrow S^d$, there is a homeomorphism $\phi:S^d \rightarrow S^d$ which restricts to homeomorphisms $\operatorname{im}(i_1) \...
- 5,839
2
votes
0
answers
118
views
Configurations of points in a spectrum
I am wondering if the following construction has appeared in the literature:
Let $X$ be a pointed space. Define the singular configuration space of $X$, $F^*(X,k)$ ,to be $\{(x_1,\dots,x_k)| x_i=x_j \...
- 5,839
2
votes
0
answers
615
views
Fiber bundle orientability vs manifold orientability
This question seems like a pretty straightforward generalization of a result from vector bundles but its been on MSE for over a week with no answers so I'm reposting https://math.stackexchange.com/...
- 4,081
2
votes
0
answers
323
views
Continuous injective functions with dense image
Let $X$ be the set of continuous, injective functions from $\mathbb{R}^n$ to $\mathbb{R}^n$ with dense image; and equip $X$ with the (relative) compact-open topology. What is known about this space?
...
- 5,405
2
votes
0
answers
346
views
regular CW complex and incidence matrices
Suppose that we have a regular CW-complex $X$. I want to define the incidence matrix of $k$-skeleton of $X$ with respect to the $k-1$ skeleton and I wonder what might go wrong in this case.
If it's ...
- 504
2
votes
0
answers
132
views
Normal bundle information in the surgery structure set
Recall that the surgery structure set of a smooth, compact n-manifold $M$ with normal bundle $\eta$ is the set of maps $f:N \rightarrow M$ that are homotopy equivalences up to the relation $f:N \...
- 5,839
2
votes
0
answers
208
views
Retracting to a bigger compact
Consider the topological spaces $X$ with the following property:
For every compact $K\subseteq X$ there is a compact set $L$ such that $K\subseteq L\subseteq X$ and $L$ is a retract of $X$.
Let ...
- 128k
2
votes
0
answers
67
views
Cohomological dimension of closed $G$-invariant subspaces on homology manifolds with a group action
Suppose $G$ is a compact topological group acting on an $m$-homology manifold $M$ over some ring $R$ by homeomorphisms.
Assume that the action of $G$ is effectively finite on a closed $...
- 2,630
2
votes
0
answers
108
views
Cohomology of colored braid groupoids
Consider braids on $n$ strands and pick $n$ distinct labels $1, \dots, n$. There is a groupoid $\mathcal P_n$ whose objects are tuples $(l_1, \dots, l_n)$ of labels and whose morphisms are braids, ...
- 2,533