# Questions tagged [at.algebraic-topology]

Homotopy theory, homological algebra, algebraic treatments of manifolds.

8,198
questions

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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 ...

2
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0
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96
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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 ...

4
votes

1
answer

183
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### 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 ...

6
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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 ...

4
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2
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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 ...

6
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0
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101
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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) ...

2
votes

1
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149
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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 ...

1
vote

1
answer

110
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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'...

4
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350
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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 ...

0
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121
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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 ...

3
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0
answers

178
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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 ...

2
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0
answers

107
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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 $\...

3
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156
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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 ...

5
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180
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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 \...

7
votes

1
answer

224
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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 ...

4
votes

1
answer

321
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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, ...

3
votes

1
answer

117
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### 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,\...

3
votes

1
answer

86
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### 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 ...

5
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1
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225
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### 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}^{\...

7
votes

1
answer

406
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### 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 ...

6
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answers

137
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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 ...

2
votes

1
answer

179
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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 ...

11
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1
answer

510
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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 ...

2
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0
answers

62
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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-...

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 $...

5
votes

1
answer

99
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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 ...

3
votes

1
answer

132
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### 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 ...

2
votes

1
answer

174
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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 ...

2
votes

2
answers

204
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### 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 ...

1
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0
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84
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### 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 ...

5
votes

1
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254
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### 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 ...

3
votes

1
answer

113
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### 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 ...

3
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### 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 ...

2
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137
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### 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 ...

1
vote

1
answer

215
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### 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 ...

2
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0
answers

152
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### 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 ...

5
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0
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131
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### 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 ...

5
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0
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134
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### 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 ...

13
votes

1
answer

462
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### 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 $2$ ...

7
votes

1
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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
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### 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 ...

2
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0
answers

139
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### 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 ...

3
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0
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181
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### 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 ...

10
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1
answer

219
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### 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 ...

0
votes

1
answer

267
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### 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 $[0,1]$...

3
votes

1
answer

313
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### "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}^...

23
votes

1
answer

993
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### 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,...

4
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0
answers

232
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### 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 ...

14
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1
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832
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### 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 ...

4
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0
answers

158
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### 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 ...