All Questions
9,056 questions
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p-local space vs p-completion
I am having some trouble understanding the difference between the $p$-completion and a $p$-local space.
If $X$ a simply connected space has all higher homotopy groups finitely generated, then the $p$-...
- 443
0
votes
1
answer
381
views
Maximal group image
How does one prove: if $S$ is a finitely generated Clifford semigroup its maximal group image is actually $S_{e_{n}}$?
- 1
0
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1
answer
129
views
free action on product of two spaces [closed]
Let $G$ be a compact Lie group acting freely on $X\times Y$ , product of two Hausdorff spaces. Is is true that $G$ must act freely on one of the factor spaces ($X$ or $Y$). For example the group $\...
- 119
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1
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187
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Does there exist a fibre bundle $K(\mathbb{Z}_4,1)\rightarrow K(\mathbb{Z}_2,1)$ with fiber $K(\mathbb{Z}_2,1)$? [closed]
Does there exist a fibration $K(\mathbb{Z}_4,1)\rightarrow K(\mathbb{Z}_2,1)$, evidently with fiber $K(\mathbb{Z}_2,1)$?
- 1,129
0
votes
1
answer
94
views
Fixed point property for the projectivization of manifold of fixed rank matrices
Let $M$ be the manifold of all matrices in $M_{n}(\mathbb{R})$ with fixed rank $0<k<n$. The projectivization of $M$ is denoted by $PM$.
Does $PM$ satisfy fixed point property?
- 356
0
votes
1
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588
views
Homology of a finite disjoint union of open cells
Let $X$ be a topological space. Assume that $X$ admits a finite decomposition of the form $X=\bigsqcup\limits_{i=1}^n V_i$ where each $V_i$ is homeomorphic (in the subspace topology of $X$) to an open ...
- 7,609
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1
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194
views
maps $\mathbb{S}^{n} \to \mathbb{S}^{n}$ [closed]
I'm trying to prove that if two (continuous) maps $f, g : \mathbb{S}^{n} \to \mathbb{S}^{n}$ are such that $f(x) \neq -g(x)$ for any $x \in \mathbb{S}^{n}$, then $f$ and $g$ are homotopic. But I can'...
- 139
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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
0
votes
1
answer
97
views
Extending binary operation used by homotopy classes
There is this operation you learn in algebraic topology when working with homotopy groups and loops i.e. paths on a topological space $X$, $p:[0,1]\rightarrow X$ with $p(0) = p(1)$. Basically it is ...
- 2,275
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1
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351
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Birkhoff decomposition vanishing of the Chern numbers
Birkhoff decomposition vanishing of the Chern numbers of the holomorphic line bundles of the Birkhoff-Grothendieck decomposition, is some statement I read off in One of Connes papers. Without going ...
user39066
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1
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425
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A generalization of cochain complex: quasi-cochain complex
It appears that we can generalize cochain complex to quasi-cochain complex, that still allow us to define cohomology.
Definition:
A quasi-cochain complex is a sequence of commutative monoids $M_n$ ...
- 4,826
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1
answer
122
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Effect of crepant resolution on the torsion in homology of a complete intersection 3-fold
Suppose I have a (2,4) complete intersection 3-fold $X\subset\mathbb P_{\mathbb C}^5$ with 118 nodal ($A_1$) singularities (if it simplifies things then assume it's the intersection of the Grassmanian ...
- 2,108
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1
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489
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Does the closure of a smooth algebraic always define a homology class?
Let $X\subset \mathbb{C} \mathbb{P}^{N}$ be a smooth,
algebraic (locally closed) complex
submanifold of $\mathbb{C} \mathbb{P}^N$
of complex dimension $k$. More concretely, $X$ is of the
...
- 3,245
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votes
2
answers
528
views
The First Homology Group of Configuration Space and Knot Theory
Let $\pi_{1}: \text{Top}^* \rightarrow \text{Grp}$ denote the fundamental group functor and let $H_{1}: \text{Top}^* \rightarrow \text{Grp}$ denote the first homology group functor. We can then define ...
- 1,441
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1
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615
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sign of the First chern class fundamental group of Kahler Manifolds
We know by some facts from Kobayashi, if the Kahler manifold $M$ has positive first Chern class, i.e., $c_1 (M)>0$ then $M$ is simply connected. So if $c_1 (M)<0$ under which assumption on $M$ ,...
user21574
0
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1
answer
162
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3-manifolds, cubes with handles
Somebody knows where I can find some proof of the following fact:
If F is compact, connected 2-manifold with nonempty boundery why there exist n=1-X(F) pairwise disjoint properly embedded 1-cells {A1,....
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1
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351
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Understanding a proof in Adams' Stable Homotopy and Gen. Coh
[Question cross posted on stack-exchange]
I'm slowly working through Part III of the book, and I'm scratching my head a bit while reading the proof of Lemma 3.2 (here reproduced):
Let $X, A$ be a ...
- 398
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1
answer
305
views
Are braid links proper links?
Are braid links proper links? Or are the concepts involved unrelated?
- 63
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1
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285
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A Nomenclature Issue : Imprimitive Semigroup?
The following question was asked by me on the forum sci.math.research,
“An imprimitive group is a transitive permutation group with a non-trivial
equivalence relation compatible with the action of ...
- 113
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1
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1k
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About universal coefficient theorem
Let $(X,A)$ be a finite CW-pair $m=p^r$ for some prime $p$. Unspecified coefficient is in $\mathbb{Z}$.
From the universal coefficient theorem, We know that
$H^1(A;\mathbb{Z}_m)=\textrm{Hom} (H_1(A),...
- 698
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1
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485
views
What is a right-handed Dehn twist of a cut curve of a Riemann surface?
Let $\Sigma_g$ be a Riemann surface of genus g, and $C$ is a cut curve of $\Sigma_g$, i.e. an oriented simple close curve.
What is a right-handed Dehn twist of $C$ of $\Sigma_g$?
- 471
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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
144
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Left syndeticity and right syndeticity in nilpotent group
$\DeclareMathOperator\Pf{\mathcal{P}_\mathrm{f}}$Question: Does there exist any reference regarding the study of left and right syndeticity in nilpotent group? More specifically, did anyone introduce/...
- 73
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1
answer
477
views
Proving the induced map on the cohomology is an isomorphism
I was going through a paper by Tanaka where I am stuck at the following map "f" which is given by the composition of these maps. Next, he mentions that the induced map is clearly an ...
0
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1
answer
148
views
packing numbers and configuration spaces of the torus
Let $S^1$ be the unit circle of radius $1$.
For any $k\geq 1$, let the $k$-dimensional torus $T^k= \underbrace{S^1\times S^1\times\cdots\times S^1}_k$ be the $k$-fold self-Cartesian ...
- 1,990
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1
answer
185
views
Regularity of the Cartesian product of varieties
Let $U$, $V$ and $W$ be algebraic varieties of finite dimensions (in the case I am really interested, $U = \mathbb R$ and $V$ and $W$ are defined by a system of homogeneous polynomials in $\mathbb R ^{...
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1
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139
views
Computationally intractable orbit of a monoid action on a finite set
Suppose for each integer $n\geq 1$ we have a submonoid $M_n\subset \mathrm{Self}(\{1, \dots, n\})$ of self-maps of $\{1, \dots, n\}$.
A characterization of $M_n$ is an algorithm that takes an integer $...
- 1
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1
answer
518
views
Zariski closure of image of a polynomial map
Suppose, $f: \mathbb{C}^1 \to \mathbb{C}^n$ is a polynomial map given by $f(t) = (f_1(t), f_2(t),.., f_n(t))$ where $f_i(z) \in \mathbb{C}[z]$. Then is it true that the Zariski closure of image of $f$ ...
- 59
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1
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200
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The rank of a semigroup
Let $S$ be a finite noncommutative semigroup(without identity) with a subset $M$ such that $\langle M \rangle =S$. If every element of $M$ is indecomposable in $M$, i.e. for any $a \in M$, there are ...
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201
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factorization morphism between projectives spaces
Please help me with this doubt:
Let $f:\mathbb{P}^1 \rightarrow \mathbb{P}^2$ be a non-constant morphism. Is there any factorization of $f$ as $$\mathbb{P}^{1} \overset{h}{\rightarrow}\mathbb{P}^{2}\...
- 25
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1
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122
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finding Morse index for the following functional
not sure if this meets the standards here in this forum. I was dealing with the following functional $I(u)=\frac{1}{p}\int_{\Omega}|\nabla u|^{p}dx-\frac{1}{q}\lambda\int_{\Omega}|u|^qdx$ for $p \geq ...
- 157
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1
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439
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A generalization of Jordan-Schoenflies theorem on simple plane curves
The well known Jordan-Schoenflies theorem says: let $C\subset \mathbb{R}^2$ be a closed simple curve. Then there exists a homeomorphism $f\colon\mathbb{R}^2\to \mathbb{R}^2$ such that $f(C)$ is the ...
- 21.8k
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1
answer
296
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Multiplicative monoid of ring modulo units
Let $M = \mathbb{Z}[\phi] \setminus \{0\}$ be the multiplicative monoid of the ring $\mathbb{Z}[\phi]$ with $\phi = \frac{1+\sqrt{5}}{2}$ the golden ratio.
We define the equivalence relationship $x\...
- 943
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1
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424
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Inclusion of closed submanifolds of a manifold
Consider a smooth compact manifold $M$ of dimension $n$, with or without boundary. Choose a submanifold $N$ of $M$ of dimension $k$, where $1 \leq k \leq n - 1$, such that $N$ is either without ...
- 1,407
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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
answer
655
views
How to understand the simple closed curves in torus?
Let $\alpha$ and $\beta$ be two simple closed curves in $T^2$ that intersect each other in one point.
We identify $\alpha$ with $(1,0)\in \mathbb{Z}^2$ and $\beta$ with $(0,1)\in\mathbb{Z}^2$. Let $(...
- 37
0
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1
answer
63
views
Monoid morphisms satisfying a decomposition condition
Let $A$ and $B$ be monoids, let $f\colon A\to B$ be a morphism of monoids. The following pair of conditions emerged naturally in my research:
For all $a\in A$ and $b_1,b_2\in B$ such that $f(a)=b_1....
- 327
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1
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198
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Discreteness of topological category?
In his definition 18.38 Dan Freed requires the maps of a topological category to satisfy the algebraic relation of a discrete category. Is that restriction to discrete category just for convenience ...
- 457
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1
answer
83
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Congruences of abelian monoids which can be extended to (ideal) congruences of polynomials
Some weeks ago I asked the same question at [math.stackexchange][1] but I have not gotten any feedback. The flavour of the question (but see the details later) is about whether to understand ...
- 173
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1
answer
78
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Conjugation Cells [equivariant cohomology]
I'm studying conjugation spaces, I have found in many sources that a conjugation cell is a conjugation space (without a proof). The widest approach that I have found so far is this paper (example 3.5)
...
- 101
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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
answer
178
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$\mathbb{Z}_{2}$ -equivariant vector bundles over manifold of rank-$k$ matrices
Edit: I remove the trivial part of the first version, according to comment of Alex Degtyarev
Let $M$ be the manifold of all matrices in $M_{n}(\mathbb{R})$ with fixed rank $0<k<n$. There is ...
- 356
0
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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
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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
0
votes
2
answers
319
views
Fixed point theorem that does not require the hemi-continuity of the set valued map?
All of the fixed point theorem I have seen (like Kakutani and Brower, Browder) required the set valued map to be hemi-continuous (lower). Is any fixed point theorem that can assure the existence of ...
- 383
0
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1
answer
407
views
Monodromy action on the local system $R^2\phi_*\mathbb{Z}$
Let $\phi: \mathcal{X}\rightarrow B$ be a family of complex manifolds i.e. $\phi$ is a proper submersive holomorphic morphism, i write $X_b$ for the fiber of $b\in B$.
Suppose $dim_{\mathbb{C}}X_b=2$...
0
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1
answer
175
views
Subvarieties with different topology representing the same cycle
Let $X$ be a topological space, and $Y,Z$ be subspaces. My question is a bit vague and open-ended: when is it the case that, if $Y$ and $Z$ represent the same (nonzero) cycle in homology, then $Y$ and ...
0
votes
1
answer
118
views
Homeomorphism between base of conifolds and spheres
Hello
Call $Y^4$ a conifold which satisfies the following condition:
$\mathfrak{Y}(z):=\sum_{\alpha=1}^{3}(z_{\alpha})^{2}=0,$
where $z_\alpha \in \mathbb{C}$. Now intersect $Y^4$ with $S^5$ to ...
- 77
0
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1
answer
305
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Embedding a semigroup into a divisible semigroup
The following is motivated by the fact that I'd like to have a way, much better if canonical, to isometrically embed a normed group into a normed divisible group. But semigroups are a much more ...
- 10.5k
0
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1
answer
94
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What is the way to after finding Cohen-Macaulay semigroup as ring of monomial?
http://people.missouristate.edu/lesreid/reu/2007/PPT/robin.ppt
it said missing part inside and not missing part outside is non Cohen-Macaulay semigroup
which determine whether is missing in the most ...
- 1