All Questions
Tagged with gt.geometric-topology at.algebraic-topology
1,145 questions
3
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194
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The first Stiefel Whitney class v.s. fermion eta invariant v.s. spin structure v.s. $H^1(M,\mathbb{Z}_2)$
$H^1(M,\mathbb{Z}_2)$ specifies the 1st cohomology class of manifold $M$ (can be regarded as spacetime) with $\mathbb{Z}_2$ coefficient,
it is often to see that we say the 1st Stiefel Whitney class
$$...
- 2,147
3
votes
0
answers
191
views
Skeleton of $\mathcal{G}$-simplicial complex
I'm trying to prove a result with simplicial complex endowed with an action of a groupoid. Let start with a formal definition :
$\textbf{Definition.}$ [$\mathcal{G}$-simplicial complex] Let $\mathcal{...
- 53
3
votes
0
answers
55
views
Infinitely many deformation equivalent Hodge diamonds II
Let $S$ be a connected smooth complex-analytic space. Let $\phi:X\to S$ be a proper holomorphic submersion with connected fibers. Can the fibers of $\phi$ have infinitely many distinct Hodge diamonds?
...
user164740
3
votes
0
answers
86
views
Exotic smooth structures on HK manifolds
An HK manifold is a closed simply-connected Kähler manifold $M$ such that $H^0(M, \Omega_M^2)=\mathbb{C}\omega$, where $\omega$ is a holomorphic 2-form on $M$ which is nowhere degenerate as a skew-...
user164740
3
votes
0
answers
71
views
Holomorphic homeomorphisms
Let $M$ be a connected closed smooth manifold. Consider the group $\mathrm{Homeo}(M)$ of homeomorphisms $M\to M$ endowed with the $C^0$-topology.
If $M$ has a symplectic structure some people study ...
user164740
3
votes
0
answers
98
views
Non-diffeomorphic surface bundles over homeomorphic 4-manifolds
For a smooth manifold $M$ an $M$-surface is the total space of a smooth surface bundle over $M$.
Let $M_1$ and $M_2$ be two homeomorphic closed simply-connected smooth 4-manifolds. Can there be an $...
user164740
3
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0
answers
162
views
Exotic smooth structures on Calabi-Yau manifolds
A Calabi-Yau manifold is a simply-connected closed Kähler manifold with holomorphically trivial canonical bundle and $h^{2, 0}=0$.
If two Calabi-Yau manifolds are homeomorphic are they diffeomorphic?
user164740
3
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0
answers
109
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Kähler manifolds deformation equivalent to projective manifolds
Let $M$ be a closed non-projective Kähler manifold. There are three possibilities
there is a proper holomorphic submersion $f:X\to \Delta$ with $f^{-1}(0)\cong M$ such that the projective fibers ...
user164740
3
votes
0
answers
144
views
All Kähler threefolds embed into a common complex manifold
Is there a closed complex manifold into which all closed complex threefolds admitting a Kähler structure embed?
user164740
3
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0
answers
186
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Cobordism theory of some weird space
Let $G=SU(3)$ and $N=SO(3)$, then $G/N= SU(3)/SO(3)$ = a 5-dimensional Wu manifold $W$.
The $W$ is a homogeneous space (also a quotient space), but not a group.
Previously, I am aware of the ...
- 10.5k
3
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0
answers
75
views
Approximative extension of the autohomeomorphism of the complement of the trivial knot?
Let $S^1\subset \mathbb{R}^3$ be the unit circle and suppose $h\colon \mathbb{R}^3\setminus S^1\to \mathbb{R}^3\setminus S^1$ is a homeomorphism. Clearly it might be that $h$ cannot be extended to $S^...
- 396
3
votes
0
answers
405
views
A user guide to the theory on Corks
I am trying to digest the meanings of the corks from the both:
algebraic topology
and
geometry topology
perspectives.
Studying corks is important for understanding the exotic phenomenon of 4-...
- 10.5k
3
votes
0
answers
170
views
Pairing the Arf with Stiefel-Whitney class
The Arf invariant is a nonsingular quadratic form over a field of characteristic 2.
The form that I looked at was:
$$
S(q)=|H^1(M^2,\mathbb{Z}_2)|^{-1/2} \sum_{x\in H^1(M^2,\mathbb{Z}_2)} \exp[\pi \;...
- 10.5k
3
votes
0
answers
888
views
Quotient space, homogeneous space, and higher homotopy groups
Preparation and my input:
For the quotient space $G/H$, knowing the homotopy
groups of $G$ and $H$ one can determine homotopy groups from the long
exact sequence
$$
...
\to \pi_n(H) \to \pi_n(G) ...
- 10.5k
3
votes
0
answers
75
views
Two questions regarding flat fibre bundles and the corresponding group action on the fibre
Let $F$, $B$ be smooth, closed manifolds and $\phi:\pi_1(B) \rightarrow Aut(F)$ a smooth group action of the fundamental group of $B$ on $F$.
Consider the flat fibre bundle $E_\phi := \widetilde{B} \...
- 404
3
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0
answers
118
views
Weak contractibility of some infinite dimensional metric spaces
Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that:
$X_{n}$ is a regular$^1$ CW-complex of constant local dimension$^3$ $n$, it is of finite type$^4$, ...
3
votes
0
answers
150
views
Integral Homology of GIT Quotients
Is there any example of GIT quotients of linear actions on a Euclidean space ${\mathbb C}^n$ that satisfy the following conditions?
The quotient is compact and smooth.
The homology of the quotient ...
- 1,207
3
votes
0
answers
359
views
Cubical approximation theorem for cubical complexes
A version of the simplicial approximation theorem states that a continuous map between finite simplicial complexes is homotopic to a simplicial map after subdividing the domain.
I have found a claim ...
- 985
3
votes
0
answers
137
views
Intersection patterns of loops on surfaces
Let $a,b$ be to simple closed loops on a surface $S$ with homologically trivial intersection (more generally I'm also interested in the case when $b$ is 1-codimensional). Denote their intersection on $...
3
votes
0
answers
257
views
Braids with an infinite number of strings
Has anyone developed a theory for braids with an infinite number of strings?
- 1,181
3
votes
0
answers
93
views
When closed subsets have finitely many connected componenets
Let $X$ be topological space such that every its closed subset has finitely many connected componenets. Is there any charactrization for such topological space?
- 31
3
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0
answers
173
views
More questions about high-dimensional knot invariants
In a question yesterday I asked about the existence of algebraic invariants for embeddings of n-manifolds into n+2-spheres. The answers in the positive dimension all made certain assumptions about ...
- 1,025
3
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0
answers
310
views
Are there CW structures on homotopy limits of CW maps?
Consider the diagram of finite CW complexes $X \stackrel{f}\leftarrow Y$ where $f$ is a cellular map and note that its homotopy colimit is precisely the mapping cylinder
$$C_H = \frac{X \sqcup (Y \...
- 15.5k
3
votes
0
answers
446
views
When does the normal bundle of a submanifold of Euclidean space admit a flat connection?
Given a smooth submanifold of $R^n$, I was wondering if there is a reasonably simple criterion for deciding whether its normal bundle admits a flat connection. I am not ruling out monodromy in the ...
- 313
2
votes
3
answers
746
views
Two solid N_3 glued by its boundary
Let $N_3$ be the genus three non orientable surface. Do we have an analogous 3d manifold as the solid torus and the solid Klein bottle for $N_3$? I don't see how to extend the ideas related to the 3d ...
- 345
2
votes
2
answers
1k
views
Periodic mapping classes of the genus two orientable surface
Please, any information on the periodic mapping classes of the genus two orientable surface, $O_2$, will be greatly thanked. We had been studying the topological structure of 3d surface bundles and ...
- 345
2
votes
1
answer
3k
views
Does $S^2$ have a trivial normal bundle in any closed orientable manifold?
We know that the middle circle $S^1$ in Mobius band has a nontrivial normal bundle. Now consider the higher dimensional case. Let $M$ be a $n$-dimensional ($n\geq5$) closed orientable manifold and ...
- 1,373
2
votes
1
answer
692
views
Hairy ball theorem for odd-dimensional spheres
Let $\mathbb S^n$ be the $n$-sphere: $$\mathbb S^n=\left\{x \in \mathbb R^{n+1}: \left\|x\right\|=1\right\}.$$The hairy ball theorem can be formulated as follows:
If $n$ is even and $f\,\colon\, \...
- 218
2
votes
1
answer
1k
views
embeddings of product of spheres in Euclidean spaces [closed]
I notice that $T^2=S^1\times S^1$ can be embedded in $\mathbb{R}^3$ as a hypersurface (submnaifolds of codimension 1).
In general,
(1). could the product of spheres $S^{m_1}\times\cdots\times S^{...
- 2,223
2
votes
1
answer
308
views
Does any smooth orbifold can be triangulated by orbi-simplex(triangulation of orbifolds)
every smooth manifold can be triangulated, is it true for orbifold? Is it a known result? If yes, is there any reference?
reply to the comment : G does not need to be any subgroup of Sn , any ...
- 21
2
votes
1
answer
465
views
induced group actions and covering maps on Eilenberg-Maclane space
Let $M$ be a finite $CW$-complex. Let $\Sigma_k$ be the symmetric group acting on $k$-letters. Suppose there is a free action of $\Sigma_k$ on $M$. Then we have a covering map
$$
f:M\to M/\Sigma_k.
...
- 2,223
2
votes
1
answer
165
views
string bordism group and framed bordism group for $d \leq 6$ and $d \geq 7$
Why do the string bordism group and the framed bordism group
coincide the same in dimensions lower than 7 ($d = 0,1,2,3,4,5, 6$)?
Why do the string bordism group and the framed bordism group differ
...
- 10.5k
2
votes
1
answer
267
views
on second cohomology of $S^1$-manifold
Let $M$ be a closed oriented real manifold with a free smooth circle action. Denote $BS^1$ to be the classifying space of principal circle bundles and $ES^1\rightarrow BS^1$ to be the universal ...
- 113
2
votes
1
answer
626
views
Homotopy type of an oriented, closed, simply connected manifold
It is well known that every closed, oriented, simply-connected four-manifold $M$ is homotopy equivalent to a CW-complex consisting on a 0-cell, a wedge of two spheres and a 4-cell.
I was wondering ...
- 2,816
2
votes
1
answer
148
views
positions of regular cubes in Euclidean space with all its vertices without distinction
Let $P$ be the standard regular cube, centered at the origin of $\mathbb{R}^3$ with all its vertices on the unit sphere.
If the vertices are labelled by $1,2,3,4,5,6,7,8$, then the collection of all ...
- 2,223
2
votes
1
answer
229
views
Map from homotopy sphere with lifting property induces surjections on homotopy groups. Is it weak equivalence?
Let $E$ be homotopy equivalent to a $k$-sphere. Let $q\colon E\to X$ be a map such that given any continuous $f\colon C\to X$ from a compact space $C$, there exists (a non-unique) $\tilde{f}\colon C\...
- 393
2
votes
3
answers
614
views
what is the meaning of a curve $C$ representing Identity in fundamental group?
Suppose $M$ is a closed 3-manifold, $C\subset M$ is a simple closed curve which represent identity in $\pi_{1}(M)$. Then $C$ bounds an immersed disk in $M$.
My question is:
When does it bound an ...
- 841
2
votes
2
answers
658
views
Do homeomorphisms of boundary components of 3-manifolds extend to the manifold?
The question that I'd like to answer can be generalized to the following: if $M$ is an orientable 3-manifold and $F$ is a boundary component of $M$ (which may have other boundary components), can an ...
- 2,869
2
votes
1
answer
295
views
In which dimensions is it true that every topological ball embedded by a smoothly embedded sphere is a smoothly embedded ball?
I asked a question on MSE with no answer. Here is my question in the generalized version.
Question 1: Suppose we are given a connected three-manifold $M$ (possibly non-compact, or non-orientable) and ...
- 1,097
2
votes
2
answers
151
views
How to prove that $\phi: \;\mathrm Mod(S_g)\to \mathrm Sp(2g, \mathbb{Z})$ is an epimorphism? [duplicate]
How do I prove that homomorphism $\phi : \; \mathrm{Mod}(S_g)\to \mathrm{Sp}(2g, \mathbb{Z})$ (induced by the action of mapping class group of a surface on integer homologies of a surface) is an ...
2
votes
1
answer
372
views
isotopy classes of embeddings of the torus
Let's consider $S^1$-bundle $E$ over a 2-manifold $M$. How many isotopy classes of embeddings of the torus $\mathbb{T}^2$ in $E$?
For each free homotopy classes $\gamma$ of mappings of the circle ...
- 251
2
votes
1
answer
214
views
winding number for outer-pointing normal
While trying to characterize the complexity of a closed differentiable curve (for a path planning application), I've been using a notion which is similar in spirit to the winding number of a curve. ...
- 121
2
votes
2
answers
2k
views
Group action, Fixed point set and Orbit Space
I want to know to what extent is the group action determined by its fixed point data and orbit data, i.e. if $G$ acts on $M$ in two ways with the same fixed point set and orbit space, on what ...
- 257
2
votes
1
answer
633
views
Classification of disk bundle over surfaces
Are there any reference for the classification of orientable disk bundle over a closed surface? I am particularly interested in the case if the surface is $S^2,RP^2,T^2$ or the Klein bottle.
Many ...
- 891
2
votes
1
answer
263
views
Smooth covers rescaling the symplectic form
Let $(M, \omega)$ be a connected closed symplectic manifold of dimension $2n$.
Assume there exist smooth covering maps $\phi_k:M\to M$ such that $\phi_k^* \omega=\sqrt[n]{k}\omega$ for all $k\geq 1$. ...
user164740
2
votes
2
answers
552
views
Is the following 3-manifold irreducible?
We start with product manifold $ X := S_g \times (0,1)$, where $S_g$ is a closed, orientable surface of genus $g \geq 1$. Since this has $\mathbb R^3$ as its universal cover, $X$ is irreducible. Now $...
- 1,255
2
votes
3
answers
490
views
Is true that $\left[\frac{\hat{A}(\mathbb HP^m)} { \hat{M}(\mathbb HP^m) }\right]_{4m} = 0$?
There are two questions:
How to prove that in general
$[\hat{A}(\mathbb HP^m)]_{4m} = 0$
It is possible to verify it for low values of $m$.
How to prove that in general
$\left[\frac{\hat{A}(\...
- 165
2
votes
1
answer
671
views
How to calculate Dr. Curt McMullen's expanding eigenvalues for totally degenerate groups?
What is required in order to derive the expanding eigenvalues of Dr. Curt McMullen's torus orbifold bundles over the circle and the corresponding totally degenerate groups, as presented in Section 3.7 ...
- 23
2
votes
1
answer
365
views
Correspondence between fundamental group and geometric properties of $X$
At the time of studing some algebraic topology I was wondering about the following.
Let $X$ be a topological space and $\pi_1(X)$ be its fundamental group.
If we assume some algebraic property of $\...
- 613
2
votes
1
answer
243
views
Subdivision of geometric simplicial complex
Let $\{v_0,v_1,\cdots,v_n\}$ be $n+1$ points in $\mathbb{R}^N$ which are geometrically independent. We define their convex hull to be a geometric simplex. Using this we can define geometric simplicial ...
- 141