Questions tagged [at.algebraic-topology]

Homotopy theory, homological algebra, algebraic treatments of manifolds.

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Unraveling a path in $S^1$ [closed]

Let $\sigma:[0,1]\to \mathbb{R}/\mathbb{Z}$ be a path (i.e. continuous map). Then there exists $\tau:[0,1]\to\mathbb{R}$ such that $\sigma = p\circ\tau$ where $p$ is the canonical projection. I've ...
Luca T. Castrillón's user avatar
2 votes
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A cell complex constructed from singular knots

Let $\mathcal K_n$ be the set of all $n$-singular knots up to isotopy,i.e. an immersion of $S^1$ into $\mathbb R^3$ with $n$ transverse double points that is an embedding when restricted to the ...
Eric Ley's user avatar
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183 views

Construction of the Mayer-Vietoris spectral sequence

Given subspaces $\{U_i\}_{i \in I}$ of a topological space $X$ with $X = \bigcup_i U_i$ satisfying some conditions, there is a Mayer-Vietoris spectral sequence converging to the homology of $X$. Here ...
Francis's user avatar
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Roadmap for Algebraic Geometry/Homotopy Theory/Algebraic K Theory intersection

I’m afraid that this is quite a general question but I am hoping some experts can weigh in. I am a student generally interested in learning more about the intersection of algebraic geometry, algebraic ...
Song Ye's user avatar
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4 votes
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Lifting of map from $S^3$ to itself

My question concerns the lifting of degree $0$ map from $S^3$ to itself. Let us suppose that all maps are smooth here. Looking at $S^3$ as the space of unit quaternions, one way to define degree is ...
Paul's user avatar
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Explicit representatives for Borel cohomology classes of a compact Lie group?

I'm looking for explicit representatives of $H^3_{Borel}(G, R/Z)$, i.e. a measurable function $G^3\to R/Z$ representing a generator of the cohomology group. (Here $G$ is a compact (perhaps simple) ...
Kevin Walker's user avatar
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Is the category of simplicial $R$-modules closed monoidal?

I am trying to understand if the simplicial mapping space for simplicial $R$-Modules (or at least simplicial vector spaces) is adjoint to the (level-wise) tensor product. For simplicity, let me state ...
SetR's user avatar
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About Čech cohomology in transformation groups

I'm starting a study about theory of transformation groups and equivariant cohomology, in what I read several times that Čech cohomology is the most compatible with this theory, but until now I haven'...
Ludwik's user avatar
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Problem 1.8 from Kirby's list

Context I looked through a book called "Problems in Low-Dimensional Topology", where Rob Kirby lists a set of problems. He provides a list of problems, states their conjectures, and ...
Amanuel Jissa's user avatar
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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 ...
Serge the Toaster's user avatar
3 votes
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What should be unipotent de Rham homotopy group?

What exactly should unipotent $\pi_1^\text{dR}$ be conceptually? What formal properties should it satisfy? This seems to be answered by Chen's theorem, which is stated in Corollary 3.269 of Multiple ...
W. Zhan's user avatar
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Hypercube of chain complexes as functor from (Δ^1 )^n to ∞-category of chain complexes

A hypercube of chain complexes consists of $\mathbb{Z}$-graded vector spaces $C_\epsilon$ for $\epsilon\in\{0,1\}^n$ and maps $D_{\epsilon,\epsilon^\prime}:C_{\epsilon}\to C_{\epsilon^\prime}$ for $\...
Faniel's user avatar
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Steenrod operations on classifying spaces

Steenrod operations can be defined for all finite characteristics $p$. The simplest one, when $p=2$, is the Steenrod square. I wonder if the computation for classifying spaces for classical Lie groups ...
UVIR's user avatar
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Reference request: cohomology of BTOP with mod $2^m$ coefficients

I am searching for a reference with information pertaining to the $\mathbb{Z}/{2^m}$ cohomology of ${\rm{BTOP}}(n)$, for $n \geq 8$ and $m=1,2$, where $${\rm{TOP}}(n) = \{f \colon \mathbb{R}^n \to \...
Baylee Schutte's user avatar
7 votes
1 answer
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Computing $\pi_1$ of the complement of a non-singular plane curve

The following is a well-known fact: Theorem. The fundamental group of the complement of a non-singular curve of degree $d$ in the complex projective plane is cyclic of order $d$. This was further ...
Marco Golla's user avatar
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Criteria for extending vector field on sphere to ball

Below is a theorem that is equivalent to Brouwer fixed-point theorem, which I found quite interesting. The proof is in this PDF file. Let $v: \mathbb S^{n-1} \to \mathbb R^n$ be a continuous map, ...
Zhang Yuhan's user avatar
3 votes
1 answer
117 views

Are the two families of Johnson invariants of the Torelli groups related beyond the first one?

$\newcommand{\sp}{\operatorname{Sp}(H)}$ $\newcommand{\gr}{\operatorname{gr}}$ $\newcommand{\id}{\operatorname{id}}$ $\newcommand{\der}{\operatorname{Der}}$ Johnson has defined two families $\tau_k,\...
Adrien's user avatar
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3 votes
1 answer
86 views

Geodesic laminations on the 4-punctured sphere

Let $S_{0,4}$ be the 4-punctured sphere equipped with a hyperbolic metric of finite volume (thus all punctures are cusps). Consider $\gamma$ to be a simple geodesic of $S_{0,4}$ (not necessarily ...
Nelson Schuback's user avatar
5 votes
1 answer
225 views

Double hom with $\mathbb{CP}^\infty$

Pontrjagin duality gives a double dual theorem for "hom with $S^1$", and $S^1$ is $\textbf{B}\mathbb{Z}$ up to homotopy. $\textbf{B} \textbf{B}\mathbb{Z}$, modeled by $\mathbb{C}\mathbb{P}^{\...
Ronald J. Zallman's user avatar
7 votes
1 answer
406 views

Model categories as a tool to resolve size issues for localizing categories

I have a rather basic question about one motivation for introducing model categories in David White's notes, as a possible way to overcome troubles with size issues appearing when localizing a ...
user267839's user avatar
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6 votes
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Applications of $RO(G)$-graded computations outside of equivariant homotopy theory

While writing a grant proposal I faced a problem of justification my area of interest to a broader audience. So I thought it would be nice to ask it here: What are applications/impact of computations ...
Igor Sikora's user avatar
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Does the group of compactly supported diffeomorphisms have the homotopy type of a CW complex?

It is known that the group of diffeomorphisms of a compact manifold with the natural $C^{\infty}$ topology has the homotopy type of a countable CW complex. See for instance this thread: Is the space ...
Yasha's user avatar
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11 votes
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Can you deduce the correspondence between 2D oriented TQFTs and commutative Frobenius algebras from the (framed) Cobordism Hypothesis?

Background I am currently writing an MSc dissertation on TQFTs (and Khovanov homology, but that is unrelated to this question). After having read most of Kock's book on the equivalence between 2D ...
Santiago Pareja Pérez's user avatar
2 votes
0 answers
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cocycle datum for principal $G$-bundle over base space Delta set

Let $X$ be topological realization of a (finite) Delta set, $G$ a finite group and $p: P \to X$ a principal $G$-bundle. Let's recall the standard fact that more generally any numerable principal G-...
JackYo's user avatar
  • 541
12 votes
1 answer
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Who proved the motivic 6-functor formalism?

In the recent beautiful talk "Motives and ring stacks" Peter Scholze states the theorem saying that there exists an initial 6-functor formalism on $\mathit{Sch}_\mathbb{Z}$ such that when $...
Ola Sande's user avatar
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5 votes
1 answer
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Minimal cell structures in combinatorial model categories

I recently rediscovered a classical theorem from Hatcher which states that simply-connected CW complexes have a 'minimal' cell structure, where the cells correspond to spheres and disks indexed by the ...
kelly maggs's user avatar
3 votes
1 answer
132 views

Can a phantom map have finite cofiber?

Let $f : X \to Y$ be a nonzero phantom map between spectra. Can the cofiber of $f$ be a finite spectrum? Recall that a map $f$ is said to be phantom if $f \circ i = 0$ whenever $i : F \to X$ is a map ...
Tim Campion's user avatar
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combinatorical description of classifying map for principal $G$-bundle over Delta set

Let $X$ be topological realization of a (finite) Delta set, $G$ a finite group and $p: P \to X$ a principal $G$-bundle. It's standard that the isom' classes of such principal $G$-bundles are ...
JackYo's user avatar
  • 541
2 votes
2 answers
204 views

Is the mapping cylinder a replacement for morphism by cofibration in model categories?

Let $M$ be a model category, consider a very good cylinder object $X \coprod X \to X \times I \overset{\operatorname{pr}}{\to} X$ (here $X \times I$ is just a notation, no object $I$ is implied), that ...
Arshak Aivazian's user avatar
1 vote
0 answers
84 views

Stable equivalence and stability theorem of vector bundles

I am going through this paper by Tanaka. In the proof of Proposition 3.2(1) given below The author says that by the stability theorem as $\dim (B)\le m$ we have $\alpha\oplus1\cong m\oplus1$. But I ...
Devendra Singh Rana's user avatar
5 votes
1 answer
254 views

Are Euclidean spaces $\Delta$-generated?

From the definition of $\Delta$-generated it seems like $\mathbb R$ should be $\Delta$-generated, as $\mathbb R$ is final with respect to all continuous maps $\mathbb R^n \to \mathbb R$. However, the ...
William B.'s user avatar
3 votes
1 answer
113 views

Derived flat bundles

I am looking for a notion of derived flat bundles over a surface $X$. Flat vector bundles may be thought of in terms of surface representations $\pi_1(X)\rightarrow\text{GL}(V)$. Is there a notion of ...
user521599's user avatar
3 votes
0 answers
46 views

Natural morphisms between stable unitary, orthogonal, and (compact) symplectic groups

I am a physicist knowing a bit of algebraic topology, and trying to answer the following question. This is perhaps not appropriate as a question on MO, in which case I apologize. I posted this ...
Hyeongmuk LIM's user avatar
2 votes
0 answers
137 views

Local-to-global philosophy for crossed modules

In the survey Groupoids and crossed objects in algebraic topology Ronald Brown made after Corollary 5.17 (p 30) an very interesting remark I not fully understand. He stated that this Corollary 5.17 ...
user267839's user avatar
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1 vote
1 answer
215 views

Symmetric-monoidal-associative smash product up to homotopy

I am thinking about sequential spectra. I am trying to figure out if the smash product here is symmetric monoidal associative up to homotopy. See definition 3.16 in the above. Recall that a sequential ...
Ronald J. Zallman's user avatar
2 votes
0 answers
152 views

Classification of bundles with fixed total space

I am aware of classification theorems for principal bundles, vector bundles, and covering spaces $\pi:E\to B$ over a fixed base space $B$. Principal and vector bundles over $B$ are classified by ...
Matthew Kvalheim's user avatar
5 votes
0 answers
131 views

Representing some odd multiples of integral homology classes by embedded submanifolds

Consider an $m$-dimensional compact closed orientable smooth manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]$ on $M$, with $1 \le n \le m-1$. Then does there exist an odd integer ...
Zhenhua Liu's user avatar
5 votes
0 answers
134 views

Group homology of $\mathrm{GL}_2(\mathbb{R})$ with real coefficients

What is known about the group homology of $\mathrm{GL}_2(\mathbb{R})$ with real coefficients and what are strategies to compute it (or at least some groups for low degrees)? Here I want to consider ...
ThorbenK's user avatar
  • 1,115
13 votes
1 answer
462 views

Impossibility of realizing codimension 1 homology classes by embedded non-orientable hypersurfaces

Suppose we have an $n+1$-dimensional compact closed oriented manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]\in H_n(M,\mathbb{Z})$ on $M.$ Then is it true that $[\Sigma]$ mod $2$ ...
Zhenhua Liu's user avatar
7 votes
1 answer
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Which revolutions in topology and geometry can we expect in the next 20 years? [closed]

In my limited perspective on the history of mathematics, I can name at least two big revolutions in Topology and Geometry (broadly construed): the introduction of Schemes in Algebraic Geometry, and ...
1 vote
1 answer
207 views

On the definition of a derived $A_\infty$-category

Let $\mathcal{A}$ be an $A_\infty$-category. The derived $A_\infty$-category is defined to be the 0th cohomology category of the category of twisted complexes of $\mathcal{A}$. I have troubles ...
warzasch's user avatar
  • 149
2 votes
0 answers
139 views

Connection on relative topological periodic cyclic homology

I have been looking Bhatt-Morrow-Scholze's paper: https://arxiv.org/pdf/1802.03261.pdf and came to a naive question. Let $C$ be a dg-category (with assumptions?) over $\mathbb{F}_p[[z]]$ and view this ...
Daniel Pomerleano's user avatar
3 votes
0 answers
181 views

Reference for a folklore theorem about h-cobordisms

I've seen referenced here that if $M$ and $N$ are closed topological $n$-manifolds and $f: \mathbb{R}\times M \to \mathbb{R}\times N$ is a homeomorphism, then $M$ and $N$ are h-cobordant. I know that ...
nick5435's user avatar
10 votes
1 answer
219 views

Classifying space of centralizer

$\DeclareMathOperator\Map{Map}\newcommand{\B}{\mathrm{B}}\newcommand{\h}{\mathrm{h}}$Let $f:G\to H$ be a morphism of topological groups and let $$H^{\h G}:=\Map_G(\mathrm{E}G, H)$$ be the homotopy ...
Thomas's user avatar
  • 103
0 votes
1 answer
267 views

what is this simple topological space?

Take $M_p$ the mapping cylinder (MC) of the $p=3$-fold cover of $S^1$, $M_q$ the MC of the $q=2$-fold cover of $S^1$, where for both, the identification of the MC is done on the side $\{0\}$ of $[0,1]$...
Virgile Guemard's user avatar
3 votes
1 answer
313 views

"Totally real" linear transformations

Identify $\mathbb{C}^n$ with $\mathbb{R}^{2n}$ via the equality $$(z_1, z_2, \ldots, z_n)=(x_1, \ldots, x_n, y_1, \ldots , y_n)$$ Where $z_j=x_j + iy_j$. We call a linear invertible map $A: \mathbb{R}^...
user avatar
23 votes
1 answer
993 views

What topological principle is at work here?

[I'm cross-posting this from MSE. I initially asked there 10 days ago, and the question was well-received, but left unanswered.] My question is inspired by a problem I discovered in Putnam and Beyond,...
Yly's user avatar
  • 936
4 votes
0 answers
232 views

Homotopy group of maps into S^3 using its Lie group multiplication to define the group operation

The Bruschlinsky group of maps of a space X into S1 up to homotopy, using the multiplication on S1, is well-known to equal the first cohomology group of X (at least assuming X is a reasonably nice ...
Daniel Asimov's user avatar
14 votes
1 answer
832 views

What is $\pi_{23}(S^2)$?

The first $22$ homotopy groups of the $2$-sphere were worked out by Toda in 1962, but I cannot find any results extending that to any higher homotopy groups of $S^2$. Are any more of these groups ...
Joe Shipman's user avatar
4 votes
0 answers
158 views

Possible Euler characteristics of manifolds with tangential structures

Let $p:B\to BO$ be a fibration. We say that a manifold has a $B$-structure if its stable tangent bundle lifts to $B$. I am interested in the question of whether there exists, for a given even ...
Simona Vesela's user avatar

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