All Questions
2,364 questions with no upvoted or accepted answers
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246
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Relation between the pushback closed form of sphere bundle and the pullback closed form of ball bundle
Let $B$ be a closed oriented $n$-manifold, and $\pi_N:N\to B$ be an oriented $m$-dim ball bundle, i.e. each fiber is an oriented $m$-dim ball(disk) $D^m$. We have a sphere bundle $\pi_\partial:\...
- 1,915
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305
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The mapping cylinder of a map between spaces that are homotopy equivalent to CW complexes
Suppose $X$ and $Y$ are spaces that are homotopy equvialent to CW complexes, and let $f:X\to Y$ be a continuous map. I am trying to show that the pair $(M_f,X)$ is homotopy equivalent to a CW pair.
I'...
- 213
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96
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Is there terminology for a sum of maps where every map is not null-homotopic?
Consider maps $f_1,\dots,f_m \colon \Sigma X \longrightarrow Y$, where $X$ and $Y$ are CW complexes.
When considering the sum of either the maps themselves
$$\bigvee f_i \colon \Sigma X \...
- 208
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157
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Decomposing the homology of a connected sum of surfaces in a way which highlights the combinatorics of gluing
For each $i = 1,2$ and $j = 1,\ldots,n$, let $C_{i,j}$ be a connected compact oriented surface with $k$ boundary components. For $i = 1,2$, let $C_i = \sqcup_{j=1}^n C_{i,j}$, and let $C$ be the ...
- 7,537
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200
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Question regarding affine fibre bundles
Let $f:X\to Y$ be a morphism of affine varieties such that it is a fibre bundle with fibre $F$. Let $\pi_1(Y)=\Gamma$ be a free group (non abelian) of finite rank and $\pi_1(F)$ is a finite group $G$ ...
- 585
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165
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distance between two orthogonal projection matrices and its covering number
Let $X, Y \in \mathbb{R}^{n\times p}$ such that $\Vert X- Y \Vert_{HS} \leq \delta$ (Hilbert-Schmidt norm). Also, assume that both $X, Y$ have full column rank. Let the orthogonal prpjection operator ...
- 399
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32
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Results on compact slices in a regular foliation
Let $(M,\mathcal{F}$) be a smooth and regular foliation (not necessarily of comdimension 1). I am wondering if there are known (partial) results on the existence of compact, connected submanifolds $F\...
- 133
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93
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What is t-equivalence in function spaces?
In $C_p$-Theory monographs, it is said that two topological spaces $X$ and $Y$ are said to be $t$-equivalent means that $C_p(X)$ is homeomorphic to $C_p(Y)$. Then they also define $u$-equivalences (...
- 31
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123
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Algebraic deformation retract of subvarieties
Let $X$ be a complex projective variety. Let $V_1$ and $V_2$ be two closed subvarieties of $X$. We say $V_2$ admits an algebraic deformation retract to $V_1$ in $X$, if the closed immersion $V_1\...
- 5,901
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133
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What are the obstacles for a complex to be a space of loops?
It is known that any space of loops is an H-space. So my question has two parts:
What are the obstacles for a complex to be an H-space? Is there any hope to somehow reasonably classify/characterize ...
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123
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Cohomology of bisimplicial set is the cohomology of the total simplicial set?
Given a bisimplicial manifold (or set, topological space) there is the bar construction, which assigns to it a simplicial manifold, called the total simplicial manifold.
Now stepping back for a moment,...
- 1,198
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142
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Direct limit of groups rings of finite quotients of a profinite group
Background:
Let $G$ be a profinite group, for $M \leqslant N$ open normal subgroups we have the projection map $p_{M,N} \colon G/M \to G/N$
which induces a transfer map on rational group rings
$$
p_{M,...
- 401
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57
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To partition planar convex regions into n mutually non-congruent convex pieces of equal area and perimeter
This post continues Cutting a spherical surface into mutually non-congruent pieces of equal area.
Question: Given a planar convex region C and an integer n, how does one decide if C can be divided ...
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110
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Zeroth homology of the complement of a closed set
Suppose $F$ is a closed set in $\mathbb{R}^n$ with $n>1$.
Are there some known conditions that must be imposed on $F$ so that its complement in $\mathbb{R}^n$ has a finite number of components? ...
- 411
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35
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Are there local maps of simplicial (co-)cycles on $d$-manifolds beyond cohomology operations?
I'm interested in locally defined maps of cocycles/chains on manifolds of a fixed dimension $d$ which are compatible with cohomology. To be concrete about what "local" means, let me consider ...
- 3,001
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384
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Are elliptic curves infinite loop spaces?
Is the group action on an elliptic curve reflecting the fact that it is a space with with transfers?
Through the recognition principle
$$\mathrm{Mon}_{\mathbb{E}_\infty}(\mathrm{Spc})^{gp}\simeq \...
- 705
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155
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Topology of spinor varieties
I am looking for a reference about the topology of spinor varieties. Is there some place where the Betti numbers, Chern numbers, cohomology ring etc. of spinor varieties is given (including general ...
- 6,995
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284
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A question on existence of gradient vector field on manifold with boundary
Let $M$ be a compact manifold with smooth boundary $\partial M$. Does $M$ admit a gradient vector field $\nabla u$, which has no zeros, i.e. $\nabla u(x)\neq 0$, $\forall x\in M\cup\partial M$?
Thanks ...
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143
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End space of non-compact 2-manifolds described with proper rays
I am wondering if the classification of noncompact surfaces given by Ian Richards can be stated with proper rays instead of nested sequences of connected open subsets with compact boundary. I asked ...
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132
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definition of generic function
what is definition of generic function in following paper ? i need a reference for definition generic function .
"A. Hatcher, W. Thurston, A presentation for the mapping class group of a closed ...
- 119
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243
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When is a topological manifold which is not an almost complex manifold algebraic?
When is a topological manifold which is not an almost complex manifold isomorphic to a real algebraic variety in the sense of locally ringed space?
It is well known that Serre's GAGA theorem solves ...
- 11
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120
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Local description of the universal family $\pi: \overline{\mathbf{U}}_{0,n} \longrightarrow \overline{\mathbf{M}}_{0,n}$
I would like to get an understanding of the notion of geometric fibers of the universal family:
$$\pi: \overline{\mathbf{U}}_{0,n} \longrightarrow \overline{\mathbf{M}}_{0,n}.$$
In fact Knudsen show ...
- 31
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125
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Homotopy of sheaves
On a certain topological space $X$ I want to think about sheaves up to homotopy, i.e., homotopies in the space of sheaves over $X$, and then see what homotopy classes of sheaves I get. Is there a good ...
- 11
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150
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Lifting theorem for finite spaces: replacing perfect normality by normality
In the Lifting theorem for finite spaces (Thm. 3.5, Eric Wofsey, quoted below),
can one relax the condition "$A$ is a closed subset of a perfectly normal $X$" to
"$A\to X$ has the right ...
- 317
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194
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Existence of Morse function on suspension
Let $X$ be a smooth simply connected compact manifold of dimension $n$ with boundary. Let $Y$ be a smooth compact manifold of dimension $n+1$ without boundary such that $H_{i+1}(Y)=H_{i}(X)$(reduced ...
- 61
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138
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Automorphism group of indefinite orthogonal Lie group $G=O(p,q)$ vs that of a double covering group $\tilde{G}$
Previously I mentioned in Automorphism group of a Lie group $G$ vs that of a double covering group $\tilde{G}$: same or not? that the automorphism group of a Lie group 𝐺 may be the same as that of ...
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102
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DA structure of a Dehn twist
I am trying to find the DA bordered homology structure of a Dehn twist.In https://arxiv.org/pdf/0810.0687v6.pdf page 255 bottom the authors tabulate the differentials of the right module(A side) of ...
- 99
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191
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Questions related to Morse theory
I asked this question long back but could not get a satisfying answer. Starting with a finite CW complex $X$ of dimension $n\geq 3$. I have embedded it inside $R^{2n}$. Now take a tubular neighborhood ...
- 1,177
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222
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Cohomology spectral sequence of a CW complex filtered by its skeletons
Let $X$ be a CW complex. Suppose, $$\emptyset\subset X^0\subset X^1\subset \cdots \subset X^p\subset \cdots \subset X= \bigcup_{i=0}^{\infty} X^i,$$
is a filtration of $X$ by its skeletons $X^i$. Now ...
- 191
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86
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Terminology: are there any names for "quotients" of cellular towers in stable categories?
A cellular tower in SH or in a "more general stable homotopy category" is a chain of morphisms $\dots X^{(n)}\stackrel{g^n}{\to} X^{(n+1)}\to \dots$ along with some more data and conditions; ...
- 16.9k
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181
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Homotopy section but no pointed homotopy section
Can anyone give an example of a pointed map $p:(E,e)\to (B,b)$ between connected pointed spaces (reasonably nice, say of the homotopy type of CW complexes) such that $p$ admits a homotopy section but ...
- 35.9k
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218
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How to check a fiber bundle is trivial
Given a smooth fiber bundle $X \to S^1,$ such that the fiber, $F$, is homotopic to $S^2 \vee S^2.$ Is it true that this is always a trivial fiber bundle?
In general, how to check a fiber bundle is ...
- 1,177
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97
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Homotopy type of complement to a union of linear subspaces
Im not sure if this question is appropriate for MO, but I'm looking for a hint about some questions about homotopy type of complement to a union of linear subspaces in vector space $\mathbb{R}^n, \...
- 11
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90
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Regular mapping space vs continuous mapping space for affine schemes
Let $A$ and $B$ be Zariski open affine sub-schemes of $\mathbb{A}_{\mathbb{C}}^{n}$ and $\mathbb{A}_{\mathbb{C}}^m$ respectively. We denote the infinite symmetric product of $B$ by $Sym^{\infty}(B)$. ...
- 5,901
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146
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Do all $\mathbb{E}_{k}$-comonoids in $\mathcal{C}_*$ come from “freely-pointed” $\mathbb{E}_{k}$-comonoids on $\mathcal{C}$?
In Coalgebras in symmetric monoidal categories of spectra, Péroux and Shipley prove the following (Lemma 2.4):
Let $\mathcal{C}=\mathsf{Sets},\mathsf{Top}$, or $\mathsf{sSets}$. The free basepoint ...
- 11.8k
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160
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Contractible four-manifold which admits a decomposition
Let $M^4$ be a noncompact, contractible, smooth manifold. Suppose there exists an exhaustion $M=\bigcup_{i\ge 1} U_i$ by open sets such that (1) $\bar U_i \subset U_{i+1}$ and (2) each $U_i$ is ...
- 891
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History of simplicial complex
It is easy to find the definition of a simplicial complex:
https://en.wikipedia.org/wiki/Simplicial_complex
I am interested in the history and first occurrences of the concept.
When did people start ...
- 479
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129
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Open cone homeomorphic to the Euclidean space
Let $X$ be a topological space and the open cone $C(X)$ over $X$ is defined to be $X \times [0,1)$ with $X \times \{0\}$ identified. Suppose $C(X)$ is homeomorphic to $\mathbb R^4$, can we prove that $...
- 2,535
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139
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Terminology for an kind-of principal fibration
My interest is in topological monoids, but I think the question may make sense (in some fashion) for monoids of sets.
Let $M$ be a topological monoid, and let $X$ be a pointed space that $M$ acts on, ...
- 12.5k
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259
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Spaces homotopy equivalent over the topologist's sine curve
Consider $$T=\left\{ \left( x, \sin \tfrac{1}{x} \right ) : x \in [-1, 0)\cup(0,1] \right\} \cup \{(0,0)\}\subset \mathbb{R}^2$$
with the subspace topology.
Denote $p=(-1, \sin -1), q=(1, \sin 1)\...
- 51
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99
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topological dimension of inverse limit of compact spaces
Let $X$ be a compact metric space with topological dimension ${\rm dim}(X)>0$. Let $f: X\times X\to X$ be a continuous and surjective map. Define a family of maps $f_n: X^{n+1}\to X^{n}$ for $n\ge ...
- 675
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298
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Boundary map in Mayer-Vietoris sequence of cohomology
Suppose $M$ is a 3-manifold with connected boundary. Let $T$ be a tangle in $M$, i.e., $T$ is a embedded connected 1-submanifold whose boundary is on $\partial M$ ($T$ is not closed). Moreover, ...
- 673
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231
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transition in homotopy theory
I guess that the following are true; maybe classical? Is there a reference?
Let $X, Y, Z$ be connected pointed CW-complexes, $f:X\to Y$ and $g:X\to Z$ pointed maps. Assume that
for every $k\ge 1$, ...
- 2,479
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213
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Defining path on the prime spectrum
If $p$ and $q $ are two prime ideals of a commutative ring $R $ such that $p \subseteq q$, then we can easily define a continuous function (a path) $f$ from the unit interval $ [0,1]$ to the prime ...
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151
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Density of $G$-invariant morse functions
Let $G$ be a finite group acting on a compact manifold $M$. Let $f$ be a $G$-invariant smooth function. Can it be approximated by $G$-invariant Morse functions?
- 99
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101
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Rational systole of a manifold
I also posted this question on MSE, but since it may be a delicate question, I decided to post it here.
Given a Riemannian manifold $(M^n,g)$ and an integer $1 \leq k \leq n-1$, the $k$-systole of $M$ ...
- 2,095
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80
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A characterization for a space that is similar to locally connected spaces
Let $X$ be a $T_0$ topological space with the property that there exists a basis $\{O_i\}_{i\in I}$ for $X$ such that for each $J\subseteq I$ the subspace $\bigcap_{i\in J}O_i$ has only finitely many ...
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0
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85
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Vanishing cycles and relative homology of I_n fiber
Suppose $\pi: X \rightarrow D$ is a smooth elliptic fibration with a section over the closed disk without multiple fibers, such that all special fibers are in the interior of $D$. The boundary $\...
- 67
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179
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Zeroth cohomology of tensor product of complexes concentrated in nonpositive degrees
This is probably an easy problem, but I can't find any reference.
Let $V$ and $W$ be cochain complexes over some commutative ring, and assume that they have both cohomologies concentrated in ...
- 2,079
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331
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why is monodromy weight filtration compatible with cup product?
This question is about a statement I took for granted in this question.
If $f: X \to S$ is a moprhism from a complex manifold to a punctured disc then the monodromy operator $T$ is quasi-unipotent, so ...
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