All Questions
9,056 questions
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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 ...
- 356
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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
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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
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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
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81
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Possible directions of saddle connections
Let's consider a Riemann surface $X$ of genus $g\ge 2$ and $q$ a holomorphic quadratic differential on $X$. A natural parameter on $X$ is a chart for which $q=dz^2$. A $\theta$-trajectory is a maximal ...
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101
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coefficient of homology of configuration spaces over real projective spaces
In the slides Characteristic Classes of Surface Bundles
and Configuration Spaces, Miguel A. Xicot'encatl, page 38, what is the coefficient of the following homology?
Could the coefficient be an ...
- 1,990
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113
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Rational homotopy groups of unordered configuration spaces of the torus
Is there any computations or investigation about the rational homotopy groups of unordered configuration spaces of the torus?
Any help is welcome
- 189
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175
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Question concerning computing $\pi_1(\mathbb R^{3}-B)$ in Alexander Horned Sphere
I was studying an example of the Alexander Horned Sphere on page 171 of Allen Hatcher's book. The example computes the fundamental group $\pi_1(\mathbb R^{3}-B)$ of the complement of the sphere in $\...
- 111
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269
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Can one prove the poincare duality for projective scheme by proving it for projective space?
It's well known the relationship between Poincare duality and Thom isomorphism（I mean cohomology purity $R^q i^! F=0$ if $q\neq c $ ) $\quad $
$Rf_!Ri_!=R(f|_Z)_!$ where f is $P_k^n\rightarrow k$ ...
- 388
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151
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Semicovering and homotopy lifting property
Has a semicovering map ( local homeomorphism + unique path lifting property ), the homotopy lifting property? Clearly it has the homotopy path lifting property.
- 11
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126
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cohomology ring of compact submanifolds of Euclidean spaces
Suppose we have a compact $m$-dimensional submanifold $M$ of $\mathbb{R}^N$ and we want to know the cohomology ring $H^*(M;\mathbb{Z})$.
Let $\epsilon>0$ and a $m$-dimensional finite simplicial ...
- 1,990
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220
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Looking for Uehara, Massey article
Not sure if this is the right place to ask this kind of a question. But I cannot find the following article:
Uehara, Hiroshi; Massey, W.S. The Jacobi identity for Whitehead products.
Algebraic ...
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96
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Is there another equivalence relation on based maps between spheres which form the same graded ring as the homotopy groups?
Let $\sim$ be an equivalence relation on continuous based maps from $S^k$ to $S^n$, where $k$ and $n$ range over the positive integers.
Suppose that
Given maps $f, f^\prime: S^k \to S^n$ and $g, g^\...
user2529
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138
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Acyclicity of covering space
Suppose we have some 2-dimensional non-aspherical finite CW-complex $K$ with $\pi_1(K)=G$. Is there any sufficient condition on $H\leq G$ (and maybe on the group $G$ itself) which allows to conclude ...
- 1,129
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122
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Alexander Duality in the complex plane
Thanks to Alexander duality, we know that for each compact subset $K$ of $\mathbb{C}$ there is an isomorphism $$H_1(\mathbb{C} \backslash K) \simeq \prod_{i \in CC(K)} \mathbb{Z},$$ where $CC(K)$ is ...
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142
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Augmentation of the sphere spectrum
I am wondering if it is sensible to talk about the augmentation ideal of the sphere spectrum in the category of spectra, as well as the `submodule of decomposables', whose construction comes from the ...
- 3,173
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278
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Homology of spherical braid groups
By the spherical braid group, I mean the fundamental group of the configuration space of distinct unordered points in $S^2$. I am wondering what is known about the group homology of the spherical ...
- 116
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117
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Cofibre of the $n$-fold transfer $\mathbb{R}P_+^{\wedge n}\to S^0$
I want to know what is known about the cofibre of the $n$-fold transfer map $\mathbb{R}P^{\wedge n}_+\to S^0$, for $n>1$. I am happy to know of any specific example worked out. The case $n=1$ is ...
- 3,173
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95
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Three-manifolds related by degree 1 map, whose products with the two-sphere are diffeomorphic
Suppose $M$ and $N$ are compact oriented smooth 3-manifolds such that there is an orientation-preserving diffeomorphism between the products $F:M \times S^2 \to N \times S^2$. Further, there are ...
- 778
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114
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When Max(R) is Hausdorff space? [duplicate]
Let $R$ be reduce commutative ring with identity (a commutative ring such that $a^n$=0 ($a\in R$) implise $a=0$) and $Max(R)$ be the set of all maximal ideals of $R$. The hull-kernel (or Zariski ...
- 11
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222
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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
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494
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maps from labelled configuration space to section space / iterated loop space
In the paper Mapping class group and function spaces: a survey, F. Cohen, M.A. Maldonado, 2014, page 3, Section 3:
for a $m$-manifold $M$, consider the disc bundle $D(M)$ in the tangent bundle $T(M)$...
- 2,223
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214
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Orientation form on the blow up of a Kaehler manifold
Let $(X,\omega)$ be a complex Kaehler manifold of (complex) dimension $d$, and let $Y\subset X$ a complex submanifold of dimension $k$. Evidently $[\omega]^d\in H^{2d}(X,{\mathbb{R}})$ is always ...
- 878
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164
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Functors similar to $H^i(\cdot)$
Suppose $T$ is a contravariant functor from the category of pointed topological spaces to the category of abelian groups, then we have homomorphisms $\alpha\colon T(X)\times T(Y)\to T(X\times Y)$ and $...
user39380
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284
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A question about the Leray-Serre spectral sequence
Suppose $F \to E \stackrel{p}{\to} B$ is a fibration with $B$ simply connected. The $E_2^{p,q}$ page of the Leray-Serre spectral sequence is given by $H^p(B;H^q(F))$. Suppose futhermore that $k$ is a ...
- 2,830
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385
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Homotopy type of a CW complex
The only dimension in which not every compact manifold is homeomorphic to a CW complex is $4$. Does every such manifold have the homotopy type of a CW complex?
user74319
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187
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First to note the relation between Stasheff polytopes (associahedra) and compositional inversion?
In my answer to MO-Q: Enumerative geometry and nonlinear waves, I outline the relation between the refined face polynomials of the Stasheff polytopes (associahedra) and the partition polynomials for ...
- 10.5k
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627
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Hochschild-Serre spectral sequence
The Hochschild-Serre spectral sequence says that for a short exact sequence $$1 \to G \to H \to K \to 1 \quad (1)$$ of (discrete) groups, we have a first quadrant spectral sequence with $E_2$ page
$$...
- 11.9k
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54
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Lattice-isotopic essentialization of arrangements
I'm working on a problem related to
$\textbf{Randell's isotopy theorem}$ for complex hyperplane arrangements. I have a question which seems quite obvious. However, I haven't found a rigorous proof ...
- 741
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403
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Morphism of modules of sections and pullback bundles
I'v asked this question on StackExchange but unfortunately nobody answered. I thought that maybe it would be more apropriate to post it here:
so suppose that we have a morphism $\theta: \Gamma(B,E_1) ...
- 9,330
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99
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Name for condition on map of cancellative monoids
Let $M,N$ be cancellative monoids with identity $\epsilon$ and suppose that $k\colon M\rightarrow N$ is a function such that
$k(\epsilon)=\epsilon$
for all $a,b\in M$, there exists $v\in N$ such that ...
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0
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118
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Explicit calculation of G-CW(V) structure of a G-space
I know explicitly the $Z/6$-CW($ξ^2$)-complex structure of $D(ξ^2)$, where $ξ$ is the non-trivial irreducible representation of $Z/6$ without fixed points. I am looking for an explicit calculation of ...
- 745
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Homotopy injection between the unit ball in the Euclidean n space and an n-dimensional metric AR
Let $D^n$ be the closed unit ball in $\mathbb{R}^n$. Given a compact, $n$-dimensional, AR(Absolute Retract) metric space $X$, must it happen that either $X$ embeds in $D^n$ or $D^n$ embeds in $X$?
...
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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?
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200
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Equivariant Homotopy
Let $G=\mathbb{Z}/2\mathbb{Z}$ be $\{\pm1\}$ and let there be two $G$-spaces given: $X=$ The surface of a cylinder including its boundary circles and $S^4$. That means we two G-actions $f_1:G\times X\...
- 396
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300
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Classifying Spaces and Eilenberg-Maclane objects in the category of simplicial rings
[Skip down to the bottom for a correction] Let's work over a field k, assume it is as nice as you need it to be.. Suppose I have an ordinary (edit: commutative) affine group scheme G = Spec(A) over k, ...
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662
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Intuitive Approach to Sheaf and Cech Cohomology [closed]
Sheaf and Cech cohomology $H^*(X,\mathcal{F})$ (which give the same result when applied to good enough topological spaces) are a useful generalisation of the concepts of de Rham and Dolbeault ...
- 2,091
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102
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Global topological equivalence of Morse functions
Two Morse functions $f$ and $g$ are called topologicaly equivalent if there are diffeomorphism $h$ of the source and orientational-preserving diffeomorphism $k$ of the target such that $f=k\circ g\...
- 75
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128
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Properties of "incomplete finite simplicial complexes"
Definition: We say that $K'$ is an incomplete finite simplicial complex if there exists a finite simplicial complex $K$ such that $|K'|=|K|\backslash Y$ where $Y$ is a union of some open faces of K.
...
- 7,609
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184
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A naturality question concerning the universal coefficient spectral sequence
I am reading Hillman's book "algebraic invariants of links" and on page 20 he mentions the following universal coefficient spectral sequence.
Let X be a connected finite CW complex.Let $H$ be a ...
- 1,033
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96
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Loop Motion Planning Algorithms
Happy New Year.
In a similar spirit of question Motion planning algorithm, we consider a path connected topological space $X$, and equip its free loop space $X^{S^1}$ with the open compact topology. ...
- 189
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132
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A question about a manifold in an $n$-dimensional Alexandrov space with curvature bounded below [duplicate]
Suppose $M$ is an $n$-dimensional Alexandrov space with curvature bounded below(maybe with boundary), subspace $A\subset M$ is an $n$-dimensional manifold without boundary. Then whether every point in ...
- 285
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140
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Homotopical nilpotency of self homotopy equivalence
Given a topological space $X$, ${\rm aut}(X)$ denotes the monoid of the homotopy self equivalences of $X$, that are maps $f: X\rightarrow X$ which admits a homotopy inverse. ${\rm aut}_1(X)$ denotes ...
- 189
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279
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Testing the faithfulness of group homomorphisms by testing on the level of induced Lie Algebras
Let $G$ be a group and let $\Gamma_G(k)$ be the $k$th term of the
lower central series of $G$. For each $k\geq 1$, set
$\mathrm{gr}_k(G)=\Gamma_G(k)/\Gamma_G(k+1)$ and
$$\mathrm{gr}_*(G):=\...
- 1,108
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120
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Computing $\text{Tor}_*^{R_G} (\mathbb{Z}, \mathbb{Z}) $ for a compact Lie group $G$
Let $R_G$ be the representation ring of $G$ a connected, simply connected Lie group, $I_G$ the augmentation ideal and $\mathbb{Z}=R_G/I_G$. $R_G$ acts on $\mathbb{Z}$ via $V \cdot n = (\dim V ) n$. I ...
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