All Questions
2,364 questions with no upvoted or accepted answers
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123
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Action of the symmetric group on connected sums of manifolds (minus a disk)
Let $M$ be a connected compact topological $n$-dimensional manifold without a boundary and with a CW-structure $M= \bigcup M^i$. We have that
$$ (\#^g M)\smallsetminus D^n \simeq \bigvee_{i=1}^gM^{n-...
- 523
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0
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107
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Classifying map of a simple circle bundle
Let $\mathbb{K}_0 \subset \mathbb{K}$ be two tori (subtori of $(S^1)^n$). We suppose that $\mathbb{K}_0$ is obtained from $\mathbb{K}$ by the following procedure: consider, on the lie algebra $\text{...
- 1,227
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124
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Transgressive elements in Hopf algebra spectral sequence
Let $E^r$ be a commutative Hopf algebra homology spectral sequence over $\mathbb{Z}/p$, i.e. such that every sheet is a commutative Hopf algebra.
I am struggling with proving (probably simple) fact -...
- 1,759
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128
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Lift up characteristic class to chain complex
In derived category, there is a slogan, "cohomology is bad, chain complex is good". In the theory of characteristic classes, we could associate a vector bundle to cohomology classes of the base space. ...
- 677
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73
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Filtrations of spectra related to cellular ones and singular homology
I would like to study filtrations of spectra (i.e., objects of the "topological" stable homotopy category $SH$; a filtration of a spectrum $E$ is a sequence of compatible maps $E_{\le i}\to E$) whose ...
- 16.9k
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96
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What are the "ouverts convenables" used to prove Brieskorns lemma?
In the proof of Brieskorns lemma, see 3.3 here, Brieskorn mentions that we take "ouverts convenables" satisfying some properties, but, as far as I can tell, never specifies what these opens actually ...
- 2,788
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163
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Whitehead Theorem for maps
Let us consider two simply-connected CW complexes. Combining the theorems of Whitehead and Hurewicz we have that a map between them is an equivalence if and only if its induced map on integral chains ...
- 517
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90
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Cobordism of an annulus with a non-vanishing vector field
Let $M$ be a compact three-dimensional manifold with corners, which is a cobordism of the two-dimensional annulus. In particular, the codimension one boundary of $M$ consists of two copies of the ...
- 778
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147
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Example of open manifold with no free integer homology non-homeomorphic to a ball
I would like to state that if an open oriented even-dimensional (complex) manifold $M$ is such that $dim(H_k(M,\mathbb{Z}))=0$ for $k>0$, and 1 for $k=0$, then $M$ is homeomorphic to an open ball.
...
- 333
1
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427
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On Lefschetz theorem and sum of Betti numbers as lower bounds for fixed points
Let $M$ be a closed manifold with holomorphic cell decomposition (if it is complex), or at least with only even cohomology. In particular, its Euler characteristic is equal to the sum of its Betti ...
- 1,227
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468
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Path lifting property of holomorphic unbranched map
Suppose $X$ is a Riemann surface and $ a\in\ X $ suppose $ \phi\in\mathcal O_a $ is a holomorphic function germ at $a.$ According to the theorem 7.8 of Forster's book Lectures on Riemann surfaces on ...
- 632
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49
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Weighted cancellation norm of a word computation
A symmetric set without identity $S$ is a set with a bijective function $inv : S \rightarrow S$ with no fixed points such that $inv(inv(x)) = x$ for any $x \in S$.
We say that two disjoint pairs $\{...
- 111
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80
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Is $Iso(V)$ a deformation retract of $GL(V)$ when $V$ is a finite dimensional linear normed space
Assume that $V$ is a finite dimensional real or complex normed linear space. Let $Iso(V)\subset GL(V)\subset L(V)$ be the space of linear isometric endomorphisms, invertible endomorphism and ...
- 366
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142
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Comparing Different Notions of Unicoherence in the Plane
Unicoherence is a generalization of simple connectedness that has been useful in topology in one and two dimensions. It is also a fundamental concept in shape theory, and thus has relations to Cech ...
- 439
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173
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Modeling scientific theories with category theory (or, how to represent a biological system categorically)
Suppose I wanted to compare Linnaean classification, which arranges species by similarity in ranked taxa, to modern phylogenetic systematics, which appeals to descent with modification and branching ...
- 87
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408
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Nice proof of the Reidemeister-Singer’s theorem?
Is there a nice proof (preferably with pictures) of the Reidemeister-Singer theorem? I'd prefer some classical methods, perhaps in a book or lecture notes?
I want to learn how things are done.
- 1,465
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207
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free $S^1$ action on $\mathbb{R}P^n$ and $\mathbb{C}P^n$
I want to construct free $S^1$ action on $\mathbb{R}P^n$ and $\mathbb{C}P^n$.
For $n=2m-1$, consider $S^n ⊂ C^m$. Then $S^1$ freely act on $S^n$ by $(ξ, (z_1 , z _2 , · · · , z _m )) → (ξz_1 , ξz_2 ,...
- 395
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110
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Triangulation induces morphism of Cochain Complexes
Let $X$ be a topological space, $R$ a ring, $n \in \mathbb{N}$ natural. Let $S_n(X, R) = \bigoplus_{s: \Delta_n \to X} R$ where the $s: \Delta_n \to X$ are the singular n-simplices, therefore ...
- 6,048
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84
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example of smoothing of quotient surface singularities with maximal Milnor number
In Wahl's paper "Smoothing of normal surface singularities", he shows that smoothing in the Artin component of a quotient surface singularity has the maximal Milnor number in the versal family. ...
- 31
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59
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Integer valued signature of $4n$ dimensional orbifolds
Let $M^{4n}$ be a smooth oriented $4n$-dimensional manifold without boundary. Then we have an intersection form in $H^{2n}(M^{4n},\mathbb R)$ and such a form has signature $(n_+, n_-)$.
Question. I ...
- 14.3k
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273
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Mayer-Vietoris for local system cohomology
We define local system cohomology of $(X,M_\rho)$, $\rho:\pi_1(X)\to Aut(M)$, as the cohomology associated to the complex
$$Hom_{\mathbb{Z}[\pi_1]}(C_0\tilde X,M_\rho)\to Hom_{\mathbb{Z}[\pi_1]}(C_1\...
- 2,788
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92
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How are representations of SU(2) identified with non-intersecting curves in a disk? (Rumer-Teller-Weyl)
Apparently (e.g. see bottom left of first column on p3 of this manuscript) the representations of $SU(2)$ can be identified with diagrams of non-intersecting curves (unoriented embedded $1$-manifolds) ...
- 607
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317
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Fixed point set of torus action is discrete and infinite?
Let $T=(\mathbb{C}^*)^k$ act holomorphically on a smooth quasi-projective complex algebraic variety $M$.
Can the fixed point set $M^T$ be discrete and infinite?
I think the answer is no because ...
- 297
1
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105
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The inverse image of a Noetherian topological space
A topological space $X$ is called Noetherian if
closed subsets satisfy the descending chain condition, equivalently,
the open subsets satisfy the ascending chain
condition.
Let $A$ and $B$ be ...
- 11
1
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112
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Automorpshism of a free abelian group of a free product
Let $G=A_1*A_2*...*A_n*F_k$ be a free product, where each $A_i$ is free abelian of different rank (rank at least 2), and $F_k$ is a free group. Consider an arbitrary automorphism $\phi$ in $Aut(G)$, ...
- 11
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205
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Sheaves and isomorphisms with chain complex of singular chains (Sheaf Theory, Bredon)
Let $\Delta_{\ast}(X,A)$ (resp. $\Delta_{\ast}^c(X,A)$) be the chain complex of locally finite (resp. finite) singular chains of $X$ modulo those chains in $A$.
How to show that the homomorphism of ...
- 11
1
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125
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Transition functions and twisting functions
A principle bundle over a topological space can be given by transitions functions. On the other hand, in the simplicial world principle bundles can be given by twisting functions. Is there an explicit ...
- 2,312
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409
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Rational homotopy and l-adic cohomology
In rational homotopy theory there is a basic construction which, given a prime number $l$ and a $CW$-complex $X$, produces a localized space $X_l$ equipped with a map $X\rightarrow X_l$ that induces ...
user95283
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222
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The lattice subgroup quotient
Let $G$ a finite group and $L(G)$ it's lattice of non trivial subgroups. Is it true that the quotient space $|L(G)|/G$ is contractible, where $|L(G)|$ is the geometric simplicial complex associated to ...
- 933
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152
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Stone cech compactification of a zero dimensional topological space
Let $X $ be a zero dimensional topological space, that is, a topological space with a basis of clopen sets. Is there any characterization for the ston cech compactification for such a space?
- 11
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109
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Empty regions on the second list of unstable Adams spectral sequence
Define $\phi(n) = 4n - 2$. Is there a proof that on the second list of unstable Adams spectral sequence (for all spheres) there are no elements in squares $(n, m)$ such that $m < \phi(n)$. The ...
- 1,129
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62
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Sampling in a polyhedral complex
Assume one is given a polyhedral complex $P$ in $\mathbb{R}^n$. Now consider picking uniformly at random a $D \subseteq \{0,1\}^n$. Is there way to upper bound the probability that $D$ (a subset of ...
- 2,246
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139
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Cohomology algebra of a fiber bundle
Given a fiber bundle $(E,B,p,F)$ with path connected base $B$ and fiber $F$, both closed smooth manifolds of finite dimensions. The standard tool to compute the cohomology of the total space $E$ is ...
- 63
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251
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Copylefted introduction to topology
Is there a textbook in topology with a copyleft license?
$$ $$
- 45k
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80
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Conical resolution with torsion in the 1st homology group
By a conical resolution I mean a resolution of singularities $\pi:X\rightarrow Y$ where $X$ and $Y$ are endowed with an action of $\mathbb G_m$ (commuting with $\pi$) such that $Y$ is contracted to a ...
- 790
1
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206
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Analog of Gauss-Bonnet formula for principal bundles over manifolds with boundary
The Gauss-Bonnet formula gives a topological invariant as an integral over a local density on the given manifold. In particular, when there is a boundary, GB formula has to be supplemented by a ...
- 21
1
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497
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Invariance of combinatorial/geometric euler characteristic
I am trying to read and understand the paper:
TARGET ENUMERATION VIA EULER CHARACTERISTIC INTEGRALS
by YULIY BARYSHNIKOV AND
ROBERT GHRIST.
And I am having trouble with a statement. First of all, ...
- 909
1
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0
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89
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On the Multi-Compression theorem of Rourke and Sanderson
Suppose $M$ is a compact manifold so that it admits an embedding $f:M\to N\times\mathbb{R}^l$ with a splitting of the normal bundle as $\nu_f\simeq\nu\oplus\epsilon^l$ where $\nu$ is some $k$-...
- 3,173
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209
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Simplicial approximation theorem and related stuff
I would like to ask for a list of references that explain the simplicial approximation theorem, barycentric subdivisions and Kan $Ex^{\infty}$ functor in details, fully exploiting the language of ...
- 1,403
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305
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Steenrod's geometric view of fiber bundles
In Dan Freed's notes pp.4 beneath (12.24) he comments that, let $P \to M$ be a principal $G$-bundle and $E = (P \times F)/G = P \times_G F$ its associated fiber bundle. The fibers of $E \to M$ are ...
- 457
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133
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Standard proof that cyclic ordering of edges is preserved under planar graph homotopy?
I have several questions about the following theorem statement:
Thm: Let $G = (V, E)$ be a planar graph, and let $\varphi_0 : G \rightarrow \mathbb{R}^2$,
$\varphi_1 : G \rightarrow \mathbb{R}^2$ be ...
- 11
1
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144
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Long exact sequence of orbifold homotopy groups for orbifold fibration
I am looking for a reference for long exact sequence of orbifold homotopy groups of an orbifold fibration. There is a paper by W. Chen in the arXiv arXiv:math/0102020. But it is for a very general ...
- 11
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423
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What is the cohomology of $\operatorname{Sym}^g$ of a genus $g$ surface?
Is there any paper that computes $H^*(\operatorname{Sym}^g(F_g); \mathbb{Z})$ or $H^*(\operatorname{Sym}^g(F_g); \mathbb{Q})$?
- 11
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117
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The holonomy groupoid of certain one dimensional foliations of 2 dimensional Euclidean regions
What Is the first fundamental group of each of the following $3$ dimensional Hausdorff manifolds? What about homology groups of these 3-manifolds? Is the first one a contractible manifold?
The ...
- 366
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278
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Trivial cohomology for fibers implies isomorphism on cohomology
Let $f: Y \rightarrow X$ be a map of topological spaces such that for any $x \in X, f^{-1}(x)$ has trivial cohomology for some cohomology theory (in my case, cohomology with rational coefficients is ...
- 4,991
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81
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Homotopy invariant deletions of open faces of simplicial complexes
Given a finite simplicial complex (as a topological space) $\Delta$ and a face $\tau$, suppose we delete the interior of $\tau$ (a point if $\tau$ is a vertex, otherwise homeomorphic to an open ball ...
- 123
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93
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Adjunction of Crossed Module Functors
I am wondering about the following two related questions and don't know if they have already clear answers or not.
1) Suppose that we already know the functor $F \colon \mathcal{C} \to \mathcal{D}$ ...
- 191
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174
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Generalization of the fiber changing trick for principal bundles?
We know that a principal bundle can induce a fiber bundle as follows: if $F$ is a space which admits a $G$-action then a principal $G$-bundle $p: E \to B$ induces a fiber bundle $p: E \times_G F \to B$...
- 457
1
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0
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244
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Fibre bundle and Borel construction of compact groups
if $G$ is any compact group and $H$ is closed subgroup of $G$,
then $G/H\rightarrow X_{H}\rightarrow X_{G}$ is a fibre bundle? ($X_G=X\times _{G}E=\left( X\times E\right)
/G $ is orbit space where $X,...
- 1,367
1
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0
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180
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Behaviour of the Serre spectral sequence on a product of fibrations
Given fibration sequences $F\rightarrow E\rightarrow B$ and
$F'\rightarrow E'\rightarrow B'$,
consider the homology Serre spectral sequence $S$ for the product of fibrations
$F\times F'\rightarrow E\...
- 103