All Questions
2,364 questions with no upvoted or accepted answers
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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 $P_k^n\rightarrow k$ ...
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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.
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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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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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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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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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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 ...
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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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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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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 ...
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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)$$ of (discrete) groups, we have a first quadrant spectral sequence with $E_2$ page
$$...
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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 ...
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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) ...
- 9,340
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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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51
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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\...
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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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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\...
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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.
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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 ...
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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. ...
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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 ...
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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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(The Homotopy type of the) lifting of homeomorphism of Grassmanian
For $k<n$ put $FM_{k\times n}$ for the space of all $k\times n$ full rank matrices with real or complex entries. Note that the permutation group $S_{n}$ has an obvious action on this space ...
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372
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Role of determinant of the matrix corresponding to $i$-th Homology group.
I was thinking about the proof of the Lefschetz's Fixed point theorem and the ingeniuty of the Hopf's Trace formula, i.e. associating the trace of the matrix for deciding about the fixed points. Now ...
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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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304
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generalisation of the universal coefficient spectral sequence
Suppose I have a bounded chain complex $C_{\ast}$ over the group ring $\mathbb{Z}G$ for a finite group $G$. In topology we are usually in a situation when $C_{\ast}$ is a complex of projective modules....
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restriction to the boundary in Morse theory
Given a compact manifold with boundary $(M,\partial M)$. There is a natural pullback map
$$ H^*(M) \to H^*(\partial M) $$
I'm wondering if there is a reference that:
1) constructs this map in ...
- 1,331
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403
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Functors with Mayer-Vietoris Sequences
Let $F$ be a contravariant functor from some category of spaces (e.g. smooth manifolds or (compact?) topological Hausdorff spaces), to Abelian groups. Assume that for any open sets $U, V \subseteq X$ ...
- 9,853
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45
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Equivalence between the definitions of Reidemeister Coincidence number
Given two applications $f$ and $g$, denote by $R (f, g)$ the set of Reidemeister classes determined by $f$ and $g$ (according to the algebraic definition, on the induced on fundamental groups). And $\...
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PL or projective PL map on the links of a PL manifold
Let $M$ be a PL manifold and $f: M\rightarrow M$ be a PL homeomorphism. Suppose that $f(x)=x$ for some vertex $x$. Is the restriction map of $f$ on the links of $x$ also PL? Someone claims that this ...
- 1,373
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Twisted calibrations and sufficient conditions on homology of sub-manifolds
I think my question is somehow easy to solve, but I'm not very familiar with algebraic topology, so I'm not able to figure out the solution for myself. I'm working on a problem in metric geometry and ...
- 111
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200
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When is equivariant cohomology generated by equivariant Euler classes?
Let $X$ be a smooth complex projective variety acted upon by algebraically by a complex torus $T$. Let $F_1,\ldots,F_n$ be the connected components of $X^T$ and assume that the restriction map $$H_T^*(...
- 4,920
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193
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Homology of a higher-dimensional full torus with a certain disk removed
Let $X=S^2\times B^5$ and let us consider an embedding of an
open $5$-disk in $X$ with its boundary
glued by a map of degree $2$ to some $\{
s_0\} \times S^{4}\subseteq \partial X$ . The embedding ...
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284
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Metalinear frame bundle on sphere or $\mathbb{C}P^n$
Let $M$ be a smooth manifold. A complex metalinear frame bundle $\tilde F(P)\to M$ of a rank $n$ complex vector bundle $P\to M$ is a principal $ML(n,\mathbb{C})$-bundle together with a covering map $f:...
user21574
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46
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spherical map of fixed points?
Let $B = \{\, x \in \Re^m : \|x\| \le 2 \,\}$, and let $f : B \to B$ be a continuous function whose set of fixed points is $S^k = \{\, x \in B : \|x\| = 1, x_{k+2} = \cdots = x_m = 0 \,\}$. Can it ...
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187
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Gysin sequence for $\mathbb S^3$ bundle
Let the group $G=\mathbb S^3$ act semi freely on a paracompact space $X$. Then exercise 12 of G.E. Breadon's book , Introduction to compact transformation groups pg 169 asks to derive the following ...
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