# Questions tagged [at.algebraic-topology]

Homotopy theory, homological algebra, algebraic treatments of manifolds.

7,746
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### Computation on Characteristic classes

I am organizing a reading seminar on Characteristic Class. The audience in the seminar is interested Symplectic and Contact manifold. I work in Categorification and would like to compute some ...

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### Extending diffeomorphisms

Suppose we have a diffeomorphism $f:{\mathbb{S}}^n_{+}\to\mathbb{S}^n$ of class $C^1$ of the closed upper hemisphere onto a submanifold of $\mathbb{S}^n$ with boundary.
Question. Is it possible to ...

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### How to learn homotopy theory

I studied some basic algebraic topology (homotopy/homology/cohomology groups). When reading about the Dold-Thom theorem, the fancier and more recent sources sooner or later all started to use homotopy ...

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### Cohomology in a combinatorial way using ribbon graphs

I am interested in studying the cohomology of surfaces.
Let $S$ be a compact orientable connected surface. One possible way is to learn cohomology using differential forms.
Is it possible to approach ...

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### completed tensor products and filtered limit

Let's start with two inverse systems $\{A_p\}_{p \in \mathbb{Z}}$, and $\{B_p \}_{p \in \mathbb{Z}}$ of $C$ modules. Give each $A_p$, $B_p$, and $C$ discrete topology. Consider inverse limit topology ...

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### Types of differential structures on higher dimensional spheres

This problem comes from the smooth Poincaré conjecture:
Is a homotopy equivalent manifold to sphere is differential
homeomorphic to standard sphere?
Since the general Poincaré conjecture has been ...

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### Do there exist smaller simplicial models of barycentric subdivisions?

Let $S$ be a simplicial complex and let $Bary(S)$ denote its barycentric subdivision.
Of course, the geometric realizations of $S$ and $Bary(S)$ are homeomorphic.
However, one issue that arises in ...

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### Why did Ravenel define a ring spectrum to be flat if its smash-square splits into copies of itself?

In appendix A.2 of the orange book, Ravenel defines a ring spectrum $E$ to be flat if $E\wedge E$ is equivalent to a coproduct of suspensions of $E$. (Call this definition (1).) I've seen this ...

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### Is the face poset of a compact intersection of cylinders and half-spaces shellable?

Let the $n$-disk $D^n$ be stratified hemispherically (so there are two 0-dimensional strata at the poles, two 1-dimensional strata for the prime meridian and the international date line, two 2-...

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### Rational G-spectrum and geometric fixed points

For a finite group $G$, how is a rational $G$-spectrum $X$ detected by the geometric fixed point functor $\phi^H$ where we consider the conjugacy class of $H\leq G$? I tried finding a reference for ...

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### Monodromy group action on de Rham cohomology

Let $f : Y \longrightarrow X := \mathbb{P}^1\setminus\{0,1,\infty\}$ be the smooth proper morphism associated to the Legendre family, which is an elliptic fibration of the punctured line, with fibre ...

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### Topology of the moduli space of a 2-dim closed surface

Consider the moduli space $\cal{M}_{\Sigma_g}$ of a 2-dim closed surface $\Sigma_g$ of genus $g$. What is the topology of such a moduli space $\cal{M}_{\Sigma_g}$?
For example, what is $\pi_n ( \cal{M}...

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### Categories associated to digraphs

Let's take a directed graph, or a digraph, $G=(V,E)$ given by a finite set of vertices $V$ and a finite set of edges $E$. We can assume that pairs of vertices can have parallel edges between them and ...

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### Comparing cohomology of a total complex with the cohomology of semidirect product

$\DeclareMathOperator{\Tot}{Tot}$I have the following problem. Let $H$ and $G$ be groups such that $H$ acts on $G$, i.e., there exists a group homomorphism $H\to \mathrm{Aut}(G)$ and let $M$ be an ...

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### Equivalent statement for Borsuk-Ulam theorem

I was going through this paper by Tanaka. In the introduction he says the following
"The classical Borsuk–Ulam theorem can be
restated as the point space is I-trivial."
I am not sure how to ...

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273
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### Construct a 'nice' trivializing cover of universal principal $G$-bundle $EG \to BG$

Let G be a discrete or say for sake of simplicity a finite group. In Hatcher's book Algebraic Topology on p 89 the construction of universal bundle $EG$ carries structure of a $\Delta$-complex whose $...

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### About infinite loop space and $\Omega$ spectrum

Let $A$ is an topological abelian monoid. Also $\pi_0(A)$ is a group and $A$ has $CW$ structure.
$BA$ is a classifying space of the topological abelian monoid.
My purpose is to construct an infinite ...

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### Can information theory characterise a (topological) space?

Consider an objective function f: $\mathbb{R}^n\rightarrow\mathbb{R}$, with a vector of variables $\theta$, i.e. $f(\theta)$, $\theta \in \mathbb{R}^n$. Depending on $f$, there can be interesting ...

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### Why does the tangent classifier classify the tangent (micro)bundle?

Let $\mathcal{M}\mathrm{fld}_n$ denote the $\infty$-category of topological manifolds (without boundary) and embeddings; more precisely, it is the homotopy coherent nerve of the simplicial category ...

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### Composition of 3-braids to obtain braids with trivial closure

Given a 3-braid $b=\sigma_1\sigma_2^{-1}\sigma_1\sigma_2^{-1}\sigma_1$ (which has non-trivial closure), can we find a 3-braid $c$, which has trivial closure (closure results in any trivial knot or ...

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### Non-straightenable multiple space-time trajectories and 'entangled' braid

Considering we have the space-time trajectory of multiple particles (or any objects) in the X-Y-Time coordination system. Given a projection direction parallel to the X-Y plane, we can obtain the ...

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### $M^\wedge_I \to N^\wedge_I$ an isomorphism if $S_P^{-1}M^\wedge_P \to S_P^{-1}N^\wedge_P$ is an isomorphism for all primes $P$ containing $I$

Let $R$ be a Noetherian ring, $I \subseteq R$ an ideal, and $S \subseteq R$ a multiplicative set.
Lemma 2.3 of Adam, Haeberly, Jackowski, and May's paper A generalisation of the Segal conjecture ...

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### Are there infinite number of 3-braids with trivial closure?

Not counting equivalent braids, are there finite or infinite numbers of 3-braids whose closures are trivial knot or links? If the answer is infinite, are there some patterns in those infinite numbers ...

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### Meaning of the first Chern class of the unit tangent bundle of a surface

(This is a fairly basic question, not really research level, except that I am a research mathematician working on other things who is trying to understand more topology for use in my own work.)
Let $\...

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### Higher homotopy local systems

The concept of a local system in algebraic geometry is often described as a locally constant constructible sheaf on a scheme $X$, which is in essence a sheaf whose stalk at a point $x$ comes equipped ...

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### Proper action of $\mathbb{R}^n$ [migrated]

I am trying to prove the following:
I know that there is a proper action of $\mathbb{R}^n$ on $\mathbb{R}^n$, but is it possible to construct a proper action of $\mathbb{R}^n$ on $\mathbb{R}^m$, where ...

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1
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### fundamental group of $X/\mathbb{R}^n$

Suppose with have a topological manifold $X$ and a group $G$, is there a way to compute the fundamental group of $X/G$ in function of $\pi(X)$ and $\pi(G)$?
are there any settings on X that can ...

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### Proper action on product manifold

Suppose that $\mathbb{R}^n$ is the maximal group that can act properly on a manifold $N$ and $\mathbb{R}^m$ is the maximal group that can act properly on a manifold $M$ ( i.e, $\mathbb{R}^{m+1}$ can't ...

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### Space-time trajectory that cannot be straightened and its braid form

Considering we have the space-time trajectory of multiple particles (or any objects) in the X-Y-Time coordination system. Given a projection direction, we can obtain the braid form of the space-time ...

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### What homology theory is calculated by unreduced cubical chains?

For a topological space $X$ and a subspace $A$, let $Q_n(X,A)$ be the group of singular cubical $n$-chains of $X$ relative to $A$ and let $D_n(X,A)$ be the subgroup of degenerate cubical chains. The ...

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### Faithful locally free circle actions on a torus must be free?

Do we have an example of a smooth action $S^1 \curvearrowright T^n$ which is faithful, locally free but not free?
I know such an action must induce an injection $\rho:\pi_1(S^1)\to\pi_1(T^n)$.
Another ...

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### Can a cyclic group of prime order act on a rationally acyclic finite dimensional complex and have no fixed points?

Let $C_p$ b the cyclic group of order $p$, with $p$ a prime.
Is is possible for $C_p$ to act (cellularly) on a rationally acyclic finite dimensional CW complex $X$ with no fixed points?
Standard Smith ...

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### Calculating degree via homotopy

I'm looking for a reference for the following:
Suppose that $f_1,f_2\colon S^n\rightarrow S^n$ are smooth maps. Let $i\colon S^n\rightarrow \mathbb{R}^{n+1}$ be the inclusion, and suppose that $F\...

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### Loop-suspension of degree d map of sphere

Let $f\colon S^n \rightarrow S^n$ be a basepoint preserving map of degree $d \geq 1$. We then get an induced map $\Omega \Sigma(f)\colon \Omega \Sigma S^n \rightarrow \Omega \Sigma S^n$. There is ...

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### Bibliography about vector fields defined by an oriented double covering

I am trying to study the $\omega-$limit set of a trajectory on a connected no oriented manifold so my idea is to use an oriented double covering to lifting the trajectory and analyze its $\omega-$...

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### Poincaré duality

Is the next statement true?
Let $M$ be a non-compact linearly connected oriented topological manifold of dimension $n$, and let $M^+$ be the one-point compactification of $M$. Then there is a ...

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### Equivariant cohomology of symmetric group acting on a product

Let $X$ be a finite CW-complex. The symmetric group $S_n$ acts on the product $X^{\times n}$ in the obvious way. Let $H^{\bullet}_{S_n}(X^{\times n})$ be the (Borel) equivariant cohomology of this ...

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### Exit path categories of regular CW complexes

Given a finite, regular CW complex $X$ (by regular, I mean that the gluing maps $D^n \to X$ from the closed unit ball to $X$ are homeomorphisms onto their image), denote by $S$ the finite partially ...

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### Threefolds with the same Betti numbers and the same Chern numbers

By a threefold, I mean a compact complex manifold of dimension three.
My question is a simple one:
Are there known INFINITELY many non-homeomorphic threefolds that have the same Betti numbers and the ...

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### the Brouwer fixed point theorem for maps rather than spaces

Is there a version for the Brouwer fixed point theorem for maps rather than spaces ?
In other words, for a family of endomorphisms, can the fixed point be chosen continuously, under some assumptions ?
...

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### Does every complex orientable $E_\infty$-ring admit an $E_\infty$ complex orientation?

A ring spectrum $E$ is complex oriented if it is equipped with a ring map $MU\rightarrow E$. It is complex orientable if such a ring map exists.
An $E_\infty$-ring $E$ is $E_\infty$-complex oriented ...

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### Explicit form of boundary operators of topological cones

Let $\Omega$ be a regular, finite, $n$-dimensional CW complex with chain modules $\mathscr{C}_k$ and boundary operators $\partial_k$.
For many problems in computational geometry, a key operation is to ...

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### 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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### Topology of a union of facets of a convex polytope

The following question arose from a survey paper I am writing on
combinatorial reciprocity. Let $\mathcal{P}$ be a $d$-dimensional
convex polytope. Let $\mathcal{Q}$ be a union of facets (codimension
...

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1
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### Ways to prove that $n$-component Brunnian link is nontrivial

The attached image shows a way to construct an $n$-component Brunnian link for any $n\geq 3$. That is, this link is not trivial, but deleting any of its components makes the new link trivial. The ...

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### What are the cohomological dimensions of ${\rm Aut}(F_n)$, ${\rm Out}(F_n)$, ${\rm SL}_n(\mathbb{Z})$ over the rationals ℚ and integers ℤ?

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Out{Out}\DeclareMathOperator{\cd}{cd}\DeclareMathOperator\SL{SL}$For a group $G$, the cohomological dimension of $G$ over the ring $R$, denoted by $\...

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### Applications of the Dold-Kan correspondence

The Dold-Kan correspondence says essentially that simplicial abelian groups and nonnegative chain complexes of abelian groups are equivalent objects. While this is a very natural statement, I am not ...

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### Homology and cohomology of free loop spaces

String topology, as well as Hochschild (co)homology, give a rich perspective on the homology and cohomology of a free loop space $LM$ of a manifold $M$.
Let $k$ be a field and let $M$ be $n$-...

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### The site and the space

There is a (seemingly simple) statement in the literature on sheaf theory, namely,
If $E$ is the site of opens of a topological space $X$, the notion of sheaf over $X$ coincides with that of sheaf of ...

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### Why, if the geometric realisation of a simplicial map $p$ is a (topological) covering map, must $p$ be a (simplicial) covering map?

I essentially am asking for an explanation of the comment under this post by Tom Goodwillie.
In the "Kerodon", Lurie defines a simplicial covering map as follows:
A map $p:E\to X$ of ...