All Questions
Tagged with gt.geometric-topology at.algebraic-topology
1,145 questions
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homotopy equivalence between configuration spaces on non-homeomorphic spaces
(1). Let $D^m$ be the closed $m$-disc in $\mathbb{R}^m$. For each $k$, does the $k$-th configuration space on $D^m$ homotopy equivalent to the $k$-th configuration space on $\mathbb{R}^m$
$$
F(D^m,k)\...
- 2,223
1
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342
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Geometric representatives of homology classes of manifolds
Is it true that for even dimensional differentiable manifold $M^{2n}$ all singular homology classes in dimension less than $n$ can be represented by a submanifold?
- 11
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256
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Is this a "new" terminology in homology/cohomology theory?
I have the following question. For our research purpose, we have introduced the following concept:
Let $f:X\to Y$ be a continuous, disrecte and open mapping between two locally compact metric spaces. ...
- 1,881
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403
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Functors with Mayer-Vietoris Sequences
Let $F$ be a contravariant functor from some category of spaces (e.g. smooth manifolds or (compact?) topological Hausdorff spaces), to Abelian groups. Assume that for any open sets $U, V \subseteq X$ ...
- 9,853
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99
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PL or projective PL map on the links of a PL manifold
Let $M$ be a PL manifold and $f: M\rightarrow M$ be a PL homeomorphism. Suppose that $f(x)=x$ for some vertex $x$. Is the restriction map of $f$ on the links of $x$ also PL? Someone claims that this ...
- 1,373
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266
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What invariants are of great concern in the field of 3-manifolds and why? How much do we know about them? [closed]
I am curious about 3-manifolds though I know little.
Here I am trying to know what invariants people in this field are interested in.
The following are what I have known and what I particularly want ...
- 121
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211
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Toral decomposition
I have a couple of questions on the following theorem:
Theorem. (Jaco, Shalen)
Let $M$ be a compact, irreducible, orientable 3-manifold with incompressible boundary. There exists a collection $\...
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121
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Existence of open dense subset in a Lie group
Let $G=S\cdot R$ be a Levi-Malcev decomposition of a connected complex Lie group
and $\Gamma$ a discrete subgroup of $G$ such that the subgroups
$\Gamma_g := \Gamma\cap (gSg^{-1})$ are finite for all ...
- 645
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0
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366
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Question on Steenrod realizability problem
René Thom proved that for any topological space $X$, any integral homology class of $X$ (of any degree) has an odd multiple that is the pushforward of the fundamental class of a smooth compact ...
- 257
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365
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Killing homotopy groups by removing subsets
Let $X$ be a locally finite CW-complex and let $U$ be an open subset of $X$. Given a non-zero homotopy class $x\in\pi_i(U)$ say, is it possible to find a closed subset $Z\subset U$ whose removal from $...
1
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1
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526
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Discrete subgroups of isometry group of proper metric space
Let $X$ be a proper metric space and consider its isometry group $\mathrm{ISO}(X)$ endowed with the compact-open topology. Let $G$ be a subgroup of $\mathrm{ISO}(X)$.
Consider the following ...
- 13
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1k
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Again about Bing's house with two rooms [duplicate]
Possible Duplicate:
How to show that the “bing’s house with two rooms” is contractible?
I don't know why my question is closed? here, I make my question clearly, when "hollowing ...
- 187
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1
answer
801
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Can every 3-manifold be triangulated? [closed]
One of my classmates was telling me that it is an open question whether every 3-manifold can be triangulated. This was rather surprising. He said that the question as far as he remember is settled ...
- 87
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1
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201
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Ambient isotopy of the diagonal submanifold in product space
Given a closed manifold $M^n$ and its $k$-fold product space $M^n\times\cdots\times M^n$,Can the diagonal submanifold $\Delta:=\{(m,\cdots,m)\in (M^n)^k\mid m\in M\}$ be isotopied to the submanifold
$...
- 73
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3
answers
2k
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Identifying the orientation bundle uniquely
A nonorientable surface $S$ is homeomorphic to the $k$-th connected sum
$\mathbb{R}P^2 \sharp \ldots \sharp \mathbb{R}P^2$.
For each nonorientable surface $S$ there exists an oriented $2$-fold ...
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1
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284
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Künneth formula and induced map in homologies
Let $X,Y,Z$ be smooth connected manifolds and $f \colon X \times Y \rightarrow Z$ a smooth map. Suppose that we have $H_{*}(X \times Y; \mathbb{Z})$ is isomorphic to $\bigoplus_{p+q=*}(H_{p}(X; \...
- 369
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1
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328
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group actions on fibre bundles
Suppose that we have a group $G$ acting on the spaces $E$ and $B$. Suppose moreover that we have fibre bundles $\xi$ and $\eta$ in the following commutative diagram
If $\xi$ is a trivial bundle, i.e....
- 1,990
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1
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778
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unordered configuration space of pointed space
Let $(X,*)$ be a pointed topological space.
Let $F(X,k)=\{(x_1,\cdots,x_k)\in X^k\mid \forall i\neq j: x_i\neq x_j, \}$.
Let $F(X,k)/S_k$ be the $k$-th unordered configuration space.
Is there an ...
- 1,990
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1
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271
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When do cobordism groups depend on differential structure? [closed]
I heard that cobordism group with structures sometimes depend on differential structure of space.
Do you know any examples or references about this facts?
I want to know when difference occur between ...
- 1
0
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1
answer
511
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Principal bundle associated to a fiber bundle
Let $\pi : E\to B$ be a fiber bundle with fiber $F$ over a finite complex $B$ whose structure group is a compact Lie group $G$. How can we determine the principal $G$-bundle associated to $\pi$? For ...
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1
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483
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Is this manifold orientable? [closed]
Let $C$ be the set of points $(a,b,c,d) \in \mathbb{C}^4$ which satisfy
1) $ \left|a\right|^2+\left|c\right|^2=\left|b\right|^2+\left|d\right|^2 =1 $.
2) $ a\bar{b}+c\bar{d}=0 $
There is a (...
0
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1
answer
328
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Relationship between quotient CW-complexes after attaching cells
I have been trying to prove the following simple-looking result which I require for some work in low-dimensional topology. I expect it is likely true and in a textbook somewhere so any reference or ...
- 187
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1
answer
376
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Relation between trivial tangent bundle $\Leftrightarrow$ certain characteristic classes of tangent bundle vanish [closed]
We know that
framing structure means the trivialization of tangent bundle of manifold $M$.
string structure means the trivialization of Stiefel-Whitney class $w_1$, $w_2$ and half of the first ...
- 447
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votes
1
answer
148
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Unseparability of two linked rings in higher dimensions [closed]
I am not familiar with topology. We know that in $R^3$, we cannot separate two "rings": two copies of $S^1$, if they are "linked".
I wonder that is there any similar results for two copies of $S^1\...
- 9
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1
answer
135
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Local embedding and disk in domain perturbation
Consider say $M=(\mathbb{S}^1\times\dotsb\times \mathbb{S}^1)-q$ ($n$-times). Assume that $B$ is an $n$ disk in $M$ (for instance, thinking of $\mathbb{S}^1$ as gluing $-1$ and $1$, the cube $B=[-\...
0
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1
answer
154
views
Why does $X_0\times S^1\simeq X-X_0$? [closed]
Let $X$ be an $n$-dimensional connected smooth manifold, and let $X_0$ be an embedded $(n-2)$-dimensional compact submanifold of $X$ with the trivial normal bundle. How do we get inclusion?
$$X_0\...
- 563
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1
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395
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Topology of manifolds and transition functions
let me start by describing some examples which may well demonstrate the motivation:
A manifold is obtained by glueing together Euclidean spaces, and there is a transition function on the overlap of ...
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2
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219
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If $i(x,z)\neq 0$ and if $y$ is conjugate of $x$, then what can we say about $i(x*y,z)$?
Let $S_g$ denote the closed oriented surface of genus $g\geq 2$. Let $x,y$ be two different (upto fixed base point homotopy) but freely homotopic curves, i.e. $y$ is a non-trivial element from a ...
- 3,751
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1
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213
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An integrality theorem for immersions of complex projective spaces in the euclidean space
There are three questions:
Please let me know your proof of the following theorem:
If $CP^3$ can be immersed in $R^8$ with an Euler class $W_{2}(\nu)$ for the normal bundle of $CP^3$ respect to $R^8$...
- 165
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1
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205
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Space-time trajectory that cannot be straightened and its braid form
Considering we have the space-time trajectory of multiple particles (or any objects) in the X-Y-Time coordination system. Given a projection direction, we can obtain the braid form of the space-time ...
- 73
0
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1
answer
185
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Borsuk–Ulam theorem on the sphere with expluded poles [closed]
Consider a sphere without two poles $U^2$. Will Borsuk–Ulam theorem still work, i.e. $\forall$ continuous functions $f:U^2 \rightarrow \mathbb{R}^2 ~\exists x \in U^2$ such as $f(-x)=f(x)$?
- 415
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1
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189
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cohomology ring of the fundamental group of unordered configuration space
From the lecture notes INTRODUCTION TO CONFIGURATION SPACES AND THEIR
APPLICATIONS, p. 18, I find:
Os it possible to derive the cohomology ring $H^*(Conf(S,k)/\Sigma_k;\mathbb{Z}_2)$ from the above ...
- 2,223
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1
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296
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Compact Lie groups with only 3 dimensional cohomology generators
Let $M$ be a compact connected semi-simple Lie group. Then by Hopf's Theorem $H^*(M;\mathbb Q)=\Lambda[\omega_1,...,\omega_s]$ where $\omega_i\in H^i(M;\mathbb Q)$ , $i\ge 3$ is odd.
For which $M$, $...
0
votes
1
answer
97
views
Connecting two hypersurfaces in R^{n+1} by embedded curves
Let $M^n$ be a smooth closed embedded hupersurface in $\mathbb R^{n+1}$.
Denote by $D$ the bounded connected component of $\mathbb R^{n+1}\backslash M$.
We assume that $\mathbb R^{n+1}\backslash D$ is ...
- 285
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0
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120
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Topological transversality by dimension
We know that to achieve transverality in the topological category, for example to make a continuous map into a manifold transverse to a topological submanifold, we need the existence of micro normal ...
- 803
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0
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194
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Equivariant cohomology with discrete group action
As far as I know, the equivariant cohomology can be regarded as the generalisation of de Rham cohomology with group action on manifolds. From the literature, the group action is Lie group type. I am ...
0
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0
answers
260
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Another definition of singular homology
The singular homology is defined via standard simplex. Now if I propose another definition of singular homology groups, based on arbitrary simplex, as follows:
Let $X$ be a topological space. A $n$-...
- 781
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0
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266
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Define a characteristic class on a simplicial complex (non-manifold)
Given a simplicial complex with only triangulation and only branching structure, is it enough to define Stiefel–Whitney class?
(Please provide Yes or No answers, and reasonings.)
Given a fixed ...
- 10.5k
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0
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74
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Do adjoining basepoints and/or moduli of spaces affect fixed points nicely?
My question is when will $(X_+)^G$ or $(X/A)^G$ be equal to $(X^G)_+$ or $X^G/A^G$ respectively for $X$ a $G$-space, $G$ a finite cyclic group and $X^G$ the ordinary fixed points. These seem like they ...
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0
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119
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Nullity of the linking matrix of a framed link $L$ equals the first betti number of the manifold obtained by surgery on $L$
I have asked this on mathstackexchange as well. I'm not necessarily asking for a proof, just a hint or a point to the right direction (although I'm not saying that a proof isn't welcome). I'm studying ...
- 101
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0
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134
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when is "fibering" preserved under homotopy equivalence
Suppose I have an oriented $F$ bundle over $B$ with total space $E$ (all of the three are closed manifolds) and i have a closed manifold $E'$ which is homotopy equivalent to $E$.Is there any condition ...
- 259
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148
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There is no quasiregular diffeomorphism from punctured ball into ring (on the plane)
The idea is to use l2 cohomology as a quasiregular map invariant.
It is easy to see that there are closed 1-forms on the ring which are not exact, but it occures that every closed l2-form
$f_1(x,y)dx +...
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0
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850
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Meaning of Regular Neighborhood for Homology Basis Curves in $S_{g,2}$
I have been trying to understand the meaning of the expression "regular neighborhood" in the context described below, but I'm stuck:
We have a collection of curves $c_i$ for $i=1,2,..,n$ embedded in ...
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-1
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1
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307
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Are there results about the group of homeomorphisms of $(T^2-\{*,*\})$ up to isotopy?
I am studying a fiber bundle over circle with fiber $T^2-\{*,*\}$.
Since this is a mapping torus, the group $Homeo(T^2-\{*,*\})/isotopy$ plays an important role.
Are there some existing theorems on ...
- 157
-2
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1
answer
1k
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Component and quasi-component
Let $X$ be a topological space and $x\in X$. Then the quasi-component of the point $x$, denoted by $C_x$, is the intersection of all clopen (closed-and-open) subsets of $X$ which contain the point $x$...
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