All Questions
2,364 questions with no upvoted or accepted answers
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142
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Fibration exact sequence in homotopy vs spectral sequence in (co)homology
Perhaps this should be obvious but why is it that one may associate to a fibration exact sequences of topological spaces a long exact sequence of fundamental groups, but in (co)homology, one only has ...
- 2,907
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60
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The size of super level sets and the symmetry on a sphere
Let $u$ be a smooth function defined on the sphere $\mathbb{S}^2$, and let $R \in \mathrm{SO}(3)$ be a three-dimensional rotation. Define
$$
S_R = \{x \in \mathbb{S}^2 : u(x) \neq u(Rx)\}.
$$
Suppose ...
- 434
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86
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Projectivity of equivariant K-theory of toric variety
I'm looking at Vezzosi and Vistoli's paper: Higher algebraic K-theory for actions of diagonalizable groups.
In Theorem 6.9, they prove that the $T$-equivariant K-theory of a smooth projective toric ...
- 959
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150
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Connectedness of deleted symmetric product
Let $X$ be a connected Hausdorff space. It is well-known that the $n$-fold symmetric product $\mathcal{F}_n(X) := \{A\subseteq X : 0<|A|\leq n\}$ is a connected space equipped with the Vietoris ...
- 674
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101
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Finding an example if it exists, for a non-contractible and contractible space with special requirement on quotients of their union?
Let $A$ and $B$ be subsets of $n$-dimensional Euclidean space $\mathbb{R}^{n}$, such that $A$ is non-contractible, $B$ is contractible and $B$ is not an one-point set.
I would like to find example(s) ...
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0
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147
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Noether's formula for real algebraic surfaces
Is there a version of Noether's formula for the Euler characteristic of a surface for Real algebraic surfaces?
Specifically, given $X$ a real algebraic compact smooth surface, what is the relationship ...
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85
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Existence of covering space with trivial pullback map on $H^1$
I have seen somewhere the following claim (but can't remember where): let $M$ be a connected orientable closed smooth manifold with $b_1(M)=1$, then there exists a connected covering space $p:\tilde{M}...
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143
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Cohomology ring of $\mathbb{P}(\mathcal{O}(-1)\oplus \mathcal{O})$
Let $\mathcal{O}(-1)$ be the Hopf bundle over $\mathbb{C}\mathbb{P}^\infty$. Let $\mathcal{O}$ be the trivial rank one bundle. Consider the projectivization of the rank two bundle $\mathcal{O}(-1)\...
- 21.8k
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69
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Number of connective orbit types of torus actions
Suppose that topological group $G$ acting on topological space $X$. If the
set $\left\{ \left[ G_{x}\right] :x\in X\right\} $ is finite, where $\left[
G_{x}\right] $ denotes the conjugacy class of the ...
- 1,367
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120
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Topological transversality by dimension
We know that to achieve transverality in the topological category, for example to make a continuous map into a manifold transverse to a topological submanifold, we need the existence of micro normal ...
- 803
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89
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What happens if I take a doubly-free simplicial abelian group?
Suppose that I have a simplicial set $X_\bullet$. I can take the free abelian group generated by $X_\bullet$, $\mathbb{Z}X_\bullet$. But then I can forget that this has an abelian group structure, ...
- 29
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162
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Gluing faces of n-cube
Assuming $C_n$ be the $n$-cube, the intersection of $C_n$ with a supporting hyperplane $H(P, v)$ is called a face or more precisely a $d$-face if the dimension is $d$.
Let $f_0$ and $f_1$ be faces ...
- 53
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58
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Role of basins of attraction in the Morse decomposition
Let $M$ be a differentiable manifold and $F \in \mathcal{X}(M)$. We define a DS by
$$\dot{x}=F(x(t))$$
An ordered collection $\mathcal{M}=\left\{M_{1}, \ldots, M_{l}\right\}$ of compact subsets of ...
- 247
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43
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Intersection of subspace of cyclical rotations with orthant
In an $N$-dimensional real Euclidian space, let an orthant be specified by a vector
$\underline{x}_0 = \{x_1, x_2, \dots, x_N\}$ where the components $x_k$ are binary in the sense that $x_k = \pm 1$...
- 101
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220
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The largest value of $k$ for $\mathbb{Z}^k$ to be embedded in $GL(n,\mathbb{Z})$
This is just a question originated from This conversation (commented by Moishe Kohan). I tried to prove those two assertions but I don't know where to start:
If H is a free abelian subgroup of $SL(n, ...
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136
views
Can you explain to me how to decompose this persistence module and why?
I am learning topological data analysis on my own. I am currently basically watching This course. But there this thing in the course note that I didn't understand.
So for this persistence module:
$$
\...
- 1
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answers
215
views
Null-homotopicness of an inclusion map
Let $K$ and $L$ be simplicial complexes such that 1) $L\subseteq K$; 2) $K$ is homotopic to $S^4$; 3) $L$ is homotopic to $S^6$.
Is the inclusion map from $L$ to $K$ null-homotopic?
Thanks!
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185
views
Interpreting the edges in the Serre spectral sequence
Let $F \hookrightarrow E \stackrel{\pi}{\rightarrow} B$ be a fiber bundle. For simplicity, assume that $F$ is connected, that $B$ is $1$-connected, and that $B$ is a CW complex. Consider the Serre ...
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176
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Is surjectivity in singular homology stable under pullbacks?
Consider the pullback of a (Hurewicz) fibration $p\colon E \longrightarrow B$ along any map $f$ and let $p'$ denote the base change of $p$ in the pullback. Suppose that $H_*(p)$, the induced ...
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256
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Is the equivalence $\mathrm{CRing}^{\mathrm{op}}\simeq \mathrm{AffSch}$ related to the homotopy hypothesis?
At the heart of algebraic geometry lies the op-equivalence between commutative rings and affine schemes, i.e.,
$$\mathrm{CRing}^{\mathrm{op}}\simeq \mathrm{Aff\,Sch}.$$
At the heart of homotopy theory ...
- 705
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181
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Homotopy of the complement of the Alexander Horned Ball
The Alexander horned ball construction gives a closed embedding from the ball into the sphere, $D^3 \hookrightarrow S^3$. Its complement has zero homology but has a non-trivial $\pi_1$. Since the ...
- 563
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69
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Likelihood ratio of non-trivial cycles in an inhomogeneous random square lattice graph embedded on a toroidal surface
Consider a square lattice (random) graph $G$ embedded on a toroidal surface. Each edge $(i, j)$ of the graph has an associated likelihood probability $p_{ij}$. The probabilities $p_{ij}$ come from a ...
- 269
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303
views
Is Baire's theorem stronger than needed for functional analysis?
Many classic theorems in functional analysis involve using Baire's theorem to prove facts about topology that relate to maps between Banach spaces (or, more generally, F-spaces). The application ...
- 109
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180
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Proof of Co-H map the map $f:\Sigma SU(4)\rightarrow \Sigma^2 \mathbb{CP^3}$
How to show the map $f:\Sigma SU(4)\rightarrow \Sigma^2 \mathbb{CP^3}$ is Co-H-map?
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177
views
Conceptual proof of the simple connectedness of a Jordan domain
I have studied in Hatcher (Algebraic topology, p.169) an homological proof of the Jordan theorem. I would like to understand an upgrade of this theorem, the Schoenflies version.
On my way to prove it, ...
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answers
194
views
Equivariant cohomology with discrete group action
As far as I know, the equivariant cohomology can be regarded as the generalisation of de Rham cohomology with group action on manifolds. From the literature, the group action is Lie group type. I am ...
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211
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reference for homology complex projective space
I am looking for references on homology complex projective spaces; or more precisely the classification (if any) of smooth oriented manifolds which have the same homology groups as $\mathbb{CP}^n$.
anon
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381
views
Isomorphism of invariants and coinvariants over a field
Let $G$ be a finite group with normal subgroup $N$ acting on a vector space $V$ over a field $k$ in which the order of $N$ is invertible. Denote $H:=G/N$. The composite map $V^N \to V \to V_N$ and $\...
- 617
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164
views
Presentation complex of a finite perfect group and its features
Let $G$ be a finite perfect group and consider $X_G$, its presentation complex. I have the following questions:
Is there any special property of $X_G$ due to the group's perfectness?
What can we say ...
- 1,177
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answers
54
views
On the fundamental dimension of a polyhedron
Fundamental dimension of a finite polyhedron $P$ is defined as : $$Fd(P)=\min \{ \dim (X):X\; \text{and} \; P \; \text{have the same homotopy type}\}.$$
My question is that: if $A$ is homotopically ...
- 1,182
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answers
107
views
What is the average degree of a d-simplex?
I am a beginner in network topology topics and while I was reading an article about simplicial complexes where the authors had used random simplicial complexes, I came across a formula using "...
- 1
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0
answers
195
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Definition of union of simplicial complex and a subset
(Cross-posted from MSE: https://math.stackexchange.com/questions/4425225/definition-of-union-of-simplicial-complex-and-a-subset)
Consider a simplicial complex $\Delta$ with vertex set equal to some ...
- 521
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337
views
Can someone explain this proof on aspherical manifolds?
I am trying to understand this proof that the fundamental group of an aspherical manifold is torsion free. The proof is lemma 4.1 from Aspherical manifolds at the Manifold Atlas Project. The proof is:
...
- 29
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answers
57
views
Given m vectors in n dimension where m>>>n, how do you find the vectors that define the largest convex hull constructed with the vectors?
Say there are m vectors in n dimensional space (m>>>n).
There exists a largest convex hull defined by a subset of those vectors.
My goal is to describe the space that is strictly inside the ...
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votes
0
answers
42
views
Conditions on a set implying properties on neighborhoods
Suppose $F$ is a closed set in a Euclidean space, and for $\epsilon>0$, let $V_\varepsilon$ be the $\varepsilon-$neighborhood of $F$ i.e. the set of points $x$ having a distance less than $\...
- 411
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0
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148
views
About the theorem of Weierstrass?
Is $E=Vect\{1,x,x^2,...,x^{2^n},...\}$ dense in $C([0,1])$ for the uniform norm?
While looking for a short proof for Weierstrass' theorem, I came across this justification(*) (which shows this result)...
- 4,074
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answers
260
views
Another definition of singular homology
The singular homology is defined via standard simplex. Now if I propose another definition of singular homology groups, based on arbitrary simplex, as follows:
Let $X$ be a topological space. A $n$-...
- 781
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0
answers
266
views
Define a characteristic class on a simplicial complex (non-manifold)
Given a simplicial complex with only triangulation and only branching structure, is it enough to define Stiefel–Whitney class?
(Please provide Yes or No answers, and reasonings.)
Given a fixed ...
- 10.5k
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143
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Betti number of the boundary of a 4-manifold
Let $M$ be a compact $4-$manifold with boundary $dM$. If $M$ has the homotopy type of a wedge of $2-$spheres then is it always true that $b_1(dM)=0$? By $b_1$, I mean first Betti number.
It is known ...
- 1,177
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74
views
Do adjoining basepoints and/or moduli of spaces affect fixed points nicely?
My question is when will $(X_+)^G$ or $(X/A)^G$ be equal to $(X^G)_+$ or $X^G/A^G$ respectively for $X$ a $G$-space, $G$ a finite cyclic group and $X^G$ the ordinary fixed points. These seem like they ...
- 9
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99
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Spaces of $n$-dimensional topological spaces whose fundamental group is given
If we fix a group $G$ and a dimension $n$, we can ask which $n$-dimensional locally path-connected$^*$ (or otherwise sufficiently nice) topological spaces $X$ have $\pi_1(X) \cong G$. Would these ...
- 79
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133
views
Covering map preserved under homotopy equivalence
Given a $m-$sheeted covering map from $p:M^n\to N^n$, where $M,N$ are manifolds of dimension $n$. Suppose $M$ and $N$ are homotopy equivalent to finite CW complexes $X$ and $Y$ of same dimension $k$. ...
- 1,177
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171
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A fibration is a map which has the right lifting property with respect to injections that are weak equivalences
As mentioned in Motivic Homotopy Theory, an alternative criterion for $X\rightarrow Y$ to be a fibration is that the diagram
$$\require{AMScd}
\begin{CD}
A @>>> X;\\
@VV{i}V @VVV \\
B @>&...
- 1,168
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179
views
Adjunction formula for non compact surfaces
Let $M$ be a non compact complex surface and S an embedded compact Riemann surface in $M$.
I already know how to show the following equality of fiber bundle:
$$\Omega^2_{M}|S =\Omega^1_S \otimes N^*_S$...
- 25
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0
answers
170
views
Cohomology ring of 5-manifold generated in degrees 1, 2, 3
Is there a connected closed 5-manifold $M$ such that $\oplus_{i\geq 0} H^i(M, \mathbb{Z})$ is generated by $H^1(M, \mathbb{Z})\oplus H^2(M, \mathbb{Z})\oplus H^3(M, \mathbb{Z})$ but is not generated ...
- 21
0
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0
answers
278
views
Homology of a closed $3$-manifold with balls removed
This question has been posted on MSE with no answers.
Let $M^3$ be a closed, connected and orientable smooth $3$-manifold and let $\mathring{M}$ denote the manifold $M$ with $n$ disjoint open balls $...
- 2,095
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59
views
Every disk in $(S^2 \times S^1) \setminus B$ whose boundary lies in $\partial B$ separates
Let $M = (S^2 \times S^1) \setminus B$, where $B$ is a small open ball in $S^2 \times S^1$. Is it true to assert that every embedded $2$-disk $D \subset M$ such that $D \cap \partial M = \partial D$ ...
- 2,095
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votes
0
answers
62
views
Nontrivial interpretation of dependent type theory in the category of chain complexes
A category $\mathrm{Ch}(\mathbf{Ab})$ of chain complex has a model category structure, which makes the category to interpret dependent type theory.
E.g. a term of a type $A$ is interpreted as an arrow ...
- 101
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0
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215
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Fundamental ring of a circle
Starting with fundamental group, say of a circle, let's reflect back to path groupoid a little. The path concatenation operation is partial, but this can be remedied by focusing on the sets of paths, ...
- 109
0
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0
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119
views
Nullity of the linking matrix of a framed link $L$ equals the first betti number of the manifold obtained by surgery on $L$
I have asked this on mathstackexchange as well. I'm not necessarily asking for a proof, just a hint or a point to the right direction (although I'm not saying that a proof isn't welcome). I'm studying ...
- 101