All Questions
9,056 questions
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Ignore this question [closed]
This question is a hacky way to create some tags for you to use. Move along.
- 24.1k
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138
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Topology of Asymmetric Symmetric Products
Let $X_1,...,X_m$ be connected, simply-connected CW sub-complexes of a CW complex $X$. Let the symmetric group on $m$ letters, $S_m$, act on $P:=X_1\times\cdots\times X_m$ in $X^m$ by permuting ...
- 8,529
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148
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Only finitely many fundamental groups in $M(n,k,v,D)$?
Let $M(n,k,v,D)$ denote the class of compact manifolds with $Ric \ge \left( {n - 1} \right)k,vol \ge v,diam \le D$.In 1990,M.Anderson proved that "There are only finitely many fundamental groups among ...
- 689
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1
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118
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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
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133
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equivariant singular homology
Let $M$ be a smooth $G-$manifold, where $G$ is a compact Lie group. From the result of S.Illman that $M$ admits an equivariant triangulation. Moreover, we can construct the equivariant singular ...
2
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1
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219
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pairs of matrices up to similiarity and vector bundles over punctured torus
I would like to construct 2D vector bundles over the punctured torus, but I don't know a lot of K-theory. Over the square, there can only be the trivial bundle, but now since $\pi_1(\mathbb{T}^2\...
- 22.8k
3
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187
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Topological self-maps of smooth complex hypersufaces in complex projective spaces
This questions if related to a cute article of Beauville where he proves in particular the following theorem:
http://math1.unice.fr/~beauvill/pubs/endo.pdf
Theorem.− A smooth complex projective ...
- 14.3k
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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
1
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1
answer
115
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Is function from topological group to metric space Borel?
Let $G$ be a pseudometrizable compact abelian topological group, $X$ a compact
metric space and $f:X\rightarrow G$ a continuous bijective function.
Suppose there exists $g\in G$ such that if $d_{G}(...
- 307
2
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1
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244
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Increased connectiviry of cross-effect functors on simplicial modules
I'm trying to write an expository manuscript on Bousfield-Kan's Fiber lemma and the relevant constructions. The proof of one needed theorem in the way seems to claim the following:
if $F$ is a functor ...
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218
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Does the group completion theorem apply to the James construction?
In other words, is the natural map $M \to \Omega B M$, for $M=JX$ the James construction on a space, a group completion? (By "group completion" I mean at the level of homology, I am aware of the space ...
- 213
1
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1
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414
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Are evaluation maps for sections of a fiber bundle weak homotopy equivalences?
Given a fiber bundle $p:E\to B$ and a point $x\in B$, is the evaluation map $\varepsilon:\Gamma^0(E)\to p^{-1}(x)$ defined by $\varepsilon(\sigma):=\sigma(x)$ a weak homotopy equivalence when $\Gamma^...
- 143
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252
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Invariant Ideals in Split Hopf Algebroids
Given a split Hopf algebroid $(S,\Sigma)=(S,S\otimes B)$ over $K$, Ravenel leaves as an exercise the proof of the following:
An ideal $J\subset S$ is invariant under the action of the group $\mathrm{...
- 10.4k
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556
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When a quasifibration is a Hurewicz fibration?
In studying quasifibration I have a question.
When a quasifibration $F\to E\to B$ is a Hurewicz fibration?
If $F,E$ and $B$ are CW-complex, it is right?
- 699
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263
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k-theory of $\mathbb{Z}$
I have a doubt.
Borel computed the rank of the higher algebraic k-theory of $\mathbb{Z}$:
$rank(K_n)(\mathbb{Z})= 1$ if $n\equiv1 mod4$, otherwise this rank is equal to 0.
On the other hand Bjorn ...
- 235
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757
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Leray-Hirsch for HOMOLOGY?
Let $E\to B$ be a fibre bundle. The Leray-Hirsch theorem states under suitable assumptions, the cohomology of $E$ is an $H^*(B)$-module generated by suitable cohomology classes in $E$.
Is there any ...
- 1,207
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101
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ball in universal cover belongs to the union of actions on a section?
M is an n-dim manifold. $\pi :\tilde M \to M$ the universal cover of M. $\tilde p \in \tilde M$ a lift of p. We choose a measurable section $j:{B_1}\left( p \right) \to {B_1}\left( {\tilde p} \right)$,...
- 689
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1
answer
258
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Homology of abelian groups and their finite-index subgroups
Fix some $1 \leq k \leq n$. I'm looking for finite-dimensional vector spaces $M_{n,k}$ over $\mathbb{Q}$ on which $\mathbb{Z}^n$ acts such that the natural map $H_k(\ell \mathbb{Z}^n,M_{n,k}) \...
- 13
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293
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semigroups acting as continuous functions on regular rooted trees
Let $T$ be a regular rooted tree. Make $T$ into a metric space by making each edge isometric to the unit interval. What is known about what semigroups can act as continuous functions on $T$ such ...
- 51
3
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1
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361
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Posets of finite sequences are highly connected
I need the following result for an example in a paper I'm writing. It's easy enough to prove, but I'd prefer to just give a reference. Does anyone know one?
Fix $1 \leq k \leq n$. Define $X_{n,k}$ ...
- 44.8k
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241
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Roots of unity in algebraic K-theory
For any commutative ring $R$, the tensor product of (finitely generated, projective) $R$-modules equips the algebraic K-theory $K(R) = K_0(R)$ with the structure of a commutative ring with unit.
For $...
- 4,133
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963
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How to prove that a map is a Serre fibration?
I want to prove that the homotopy groups of some topological space $B$ of interest to me (not a CW complex) are trivial. I have a strategy of proof that consists in introducing another space $E$ that ...
- 14.4k
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1
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200
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what is the image of $\partial( 1_{S^n})$ for the exact sequence for the fibration $X \to E \to S^n$
what is the image of $\partial 1_{S^n}$ where $\cdots \pi_n(S^n)\rightarrow \pi_{n-1}(X) \rightarrow \pi_{n-1}(B)\rightarrow\cdots$
- 699
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1
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499
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How does a chain map induce another chain map on an isomorphic chain complex?
I have 2 ways of defining a chain complex on a manifold, one of which is the cellular chain complex, $C^{CW}_*$. I know that a cellular map $f: X^n \rightarrow Y^n $ such that $ f(X^n) \subset Y^n $ ...
- 1
3
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1
answer
928
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Simple applications of Atiyah-Bott localization
I am looking for some simple and concrete -- but still non-trivial and illustrative -- applications of Atiyah-Bott localization in the context of equivariant cohomology.
Do you know any good ones?
- 21k
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331
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What is the pro-algebraic completion of the free semigroup on one generator?
This question is motivated by an attempt to understand what is going on in Tom's post from a certain point of view.
Let $\mathbb N^+$ be the free semigroup on one generator (so the positive natural ...
- 38.6k
7
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177
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An explicit description of injective fibrations
If $M$ is a combinatorial model category and $C$ a small category, then the category $M^C$ admits an injective model structure in which the cofibrations and weak equivalences are levelwise. I would ...
- 66.8k
12
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661
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Mapping cylinders of fibrations
If $p: E\to B$ is a fibration, is the map $q:M_p \to B$ from the mapping cylinder
of $p$ also a fibration?
I know that it is if $p$ is trivial, or locally trivial; and I know (from Strøm's "The ...
- 12.5k
1
vote
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answers
222
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Loops in CW-complexes and the 2-skeleton
Let $X$ be a path-connected CW-complex. If $\omega: [0,1] \to X$ is a loop in $X$ and $\partial$ the boundary operator in simplicial homology, then $\partial(\omega)=\omega(1)-\omega(0)=0$, i.e. $\...
- 16.2k
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166
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A question of terminology - Unitizations of semigroups
There are at least two standard ways of unitizing a (small) semigroup $\mathbb A$:
(i) We add an identity regardless that $\mathbb A$ is already unital.
(ii) We add an identity only if none is ...
- 10.5k
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1
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225
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Is the Euler characteristic of a certain nonlinear variety related to that of a certain linear variety?
(This is a generalization of a question I posted a week ago.)
I'm looking at a variety sitting inside the algebraic torus $(\mathbb{C}\setminus 0)^n$ generated by the ideal $I = (*x_1^{\alpha_1} + \...
2
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1
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329
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Model categories and cellular maps
A question came up on MSE and it generated, for me, the following question:
When looking at the maps of CW/cell/simplicial complexes do the cellular/simplicial maps have a model theoretic ...
- 3,726
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1
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406
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Are these systems of linear equations always solvable
Let $X$ be a (finite) simplicial complex and let $f$ be a map from the set of its $n$-Simplices to a abelian group $A$, with the property, that every cycle maps to $0$ (extending $f$ linearly).
Let $...
- 11.1k
2
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answers
84
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Zeroth G-equivariant Stable Stem [duplicate]
Let G be a finite group. Can anyone give me a motivation and rigorous proof of the Burnside ring A(G) is isomorphic to the zeroth G-equivariant stable stem ?
- 745
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1
answer
358
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What does the weights of Lie group mean?
Let $\Delta=\{\alpha_1,\alpha_2\}$ be the simple root system
of the exceptional Lie group $G_2$
with $\alpha_1$ is short and $\alpha_2$ is long,
so $\lambda_1=2\alpha_1+\alpha_2,\lambda_2=3\...
- 139
9
votes
1
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266
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Branch cuts of $GL_n^+(\mathbb{R})$
Branch cuts
Let $GL_n^+(\mathbb{R})$ denote the group of $n\times n$ real matrices with positive determinant. Topologically, $GL_n^+(\mathbb{R})$ is connected, and
$$ \pi_1(GL_2^+(\mathbb{R})) = \...
- 13k
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285
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Generators of local homology groups of an isolated critical point
This is a basic Morse theory question:
Let $M$ be a smooth manifold, and $f:M\to\mathbb{R}$ a smooth function with an isolated critical point $x$. Set $c:=f(x)$. The local homology of $f$ is the ...
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571
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Are there cohomology classes on a hyperkähler manifolds which pull back to the Stiefel-Whitney classes on every Lagrangian submanifold?
This is a bit of a stab in the dark but I was wondering if anyone has defined cohomology classes on a hyperkähler manifold which pull back to the Stiefel-Whitney classes on any submanifold which is ...
- 44.7k
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138
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G-graphs and Cayley graphs
with which kinds of group we can make a G-graph(Bretto 2011) which are hamiltonian Cayley graph?
2
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131
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Topological dimension of quotient group determined by the inverse limit of discrete free monoids
Must the natural quotient group of the inverse limit of a sequence of nested discrete free monoids have topological dimension zero?
The question might well be open, but I would be grateful for news ...
- 1,968
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373
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Completion of n-fold Segal spaces
During a reading course about Jacob Lurie's paper about Local TQFTs, I came across the notion of Segal spaces as models for $(\infty,1)$-categories. Looking at things like the "double nerve" for 2-...
- 7,637
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307
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A modified version of K-theory for manifolds ?
If $X$ is a compact smooth manifold, $K^{0}(X)$ can be defined as the algebraic $K_{0}$-group of $C^{\infty}(X)$. In order to do that we use the following equivalence relation: we say that two ...
- 505
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1
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127
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Bibliographical reference needed (characterizing the weak equivalences of a model category)
I need a bibliographical reference for this fact: let $\mathcal{M}$ be a model category such that all objects are cofibrant; then the class of weak equivalences is the class of maps f such that $\...
- 3,901
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241
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Two-point desuspension for augmented chain complexes?
Let $X$ be a chain complex augmented over $\mathbf{Z}$ with augmentation $\varepsilon_X:X_0 \to \mathbf{Z}$. We define $[1](X)$ to be the $\mathbf{Z}$-augmented chain complex such that $[1](X)_0 = \...
- 19.6k
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381
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Is the cap product bilinear?
This is probably a stupid question, so I apologize in advance.
On p. 239 of Hatcher's book, he defines the cap product $C_k(X;R)\times C^l(X;R)\to C_{k-l}(X;R)$ for $k\geq l$, which he claims is $R$-...
- 1
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606
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Historical and terminological questions about Dan Kan's Ex functor and its relation to the classical case of simplicial complexes
Recall that we may define a functor $\xi:\Delta\to \operatorname{Poset}$ sending a simplex $[n]$ to the set of monotone injections $[k]\hookrightarrow [n]$ for $k\geq 0$ (effectively, $k\leq n$ as ...
5
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583
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Cohomology of Real algebraic Varieities
I understand Serre's GAGA theorem as saying that equations over algebraically closed fields can be studied equally from the algebraic and analytic points of view, at least with respect to cohomology.
...
user2529
5
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1
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466
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nerves of crossed complexes, group T-complexes and classifying spaces
A (reduced) crossed complex is, intuitively, a non-abelian complex of groups $\ldots \to G_2 \to G_1 \to G_0$ with a $G_0$ action in such that $G_1 \to G_0$ is a crossed module.
There are a couple of ...
- 35.5k
2
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1
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412
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How can I prove that the derived couple of the homotopy exact couple is an invariant?
I'm working on (yet) an(other) exercise from Mosher & Tangora's "Cohomology Operations and Applications to Homotopy Theory". This one is about the homotopy exact couple, which is defined for a ...
- 6,069
3
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0
answers
107
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Hindman's theorem variant for noncommutative semigroups
The well set proof of Hindman's finite sums theorem applied to noncommutative semigroups yields a sequence of elements such that finite products ordered coherently with this sequence are in one set. ...
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