# Questions tagged [at.algebraic-topology]

Homotopy theory, homological algebra, algebraic treatments of manifolds.

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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 ...
185 views

### 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 ...
1 vote
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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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### 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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### 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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1 vote
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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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1 vote
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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 vote
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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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