# Questions tagged [at.algebraic-topology]

Homotopy, stable homotopy, homology and cohomology, homotopical algebra.

6,370
questions

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### Cellular chain complex of $G$-CW-complexes & their differentials

I not completly understand EXAMPLE 2.31 (page 19) dealing with homology of
$G$-CW-complexes. Source: http://www.ltcc.ac.uk/media/london-taught-course-centre/documents/LTCC-notes-Lecture3-2019.pdf
...

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101 views

### Realizing Stiefel-Whitney classes via vector bundles

Let $X$ be a CW complex. If $E$ is a vector bundle over $X$, then it's well-known that the Stiefel-Whitney classes $w_j(E) \in H^j(X,\mathbb F_2)$ of $E$ are determined from the classes $w_{2^k}(E)$ (...

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**1**answer

159 views

### Oddness of intersection form of surface bundle

Let $\Sigma_g$ be a Riemannian surface of genus $g$. Let $M^4$ be a surface bundle: $\Sigma_g \to M^4 \to \Sigma_h$. When $g=1$, $M^4$ is called a torus bundle.
My question: is there a torus bundle ...

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338 views

### Chern number on non-spin manifold

Let $M^4$ be an orientable closed 4-manifold and $c_1$ be the first Chern class of a complex line bundle on $M^4$. Let $b$ be the mod 2 reduction of $c_1$, ie $b=c_1$ mod 2.
We have a relation $w_2 b =...

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67 views

### Is $\beta^{*}(w_{2k-2}) = 0$ for an open orientable $2k$-manifold?

This question is motivated by the vector field question I asked recently. Panagiotis Konstantis answered this question for odd manifolds and I am trying to figure out the even case.
Let $M$ be a ...

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257 views

### Homotopy group action and equivariant cohomology theories

Many of the introductory notes on generalized equivariant cohomology theories assume that one is working over the category of $G$-spaces or $G$-spectra. However, one thing that concerns me is that the ...

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184 views

### Under what conditions are two orientation-reversing involutions of a compact surface equivalent?

Let $M$ be a compact, connected, orientable surface and $\varphi_1,\varphi_2$ be two orientation-reversing involutions (i.e., diffeomorphisms for which $\varphi^2=Id$) such that the fixed-point set ...

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72 views

### Notation question: bigraded direct sum of graded objects

In some work I'm doing I have two graded modules $M$ and $N$ (graded on $\mathbb Z$, say) and need to take, not the usual direct sum, but the bigraded sum consisting of all $M_p \oplus N_q$ (so graded ...

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139 views

### The Thom map for the Brown-Peterson cohomology

For a prime number $p$ and the Brown-Peterson spectrum $BP$, let $T:BP\to H\mathbb{Z}_{(p)}$ be the Thom map, and $T':BP\to H\mathbb{Z}_p$ be the mop $p$ reduction of $T$. Tamanoi (1) determined the ...

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217 views

### Rational homotopy invariance of algebraic $K$-theory

Suppose that $R\to S$ is a 1-connected morphism of connective structured ring spectra that induces an isomorphism on rational homotopy groups. Is the induced map of (Waldhausen) K-theory spectra
$$
K(...

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68 views

### Is there discrete Morse theory on acyclic categories?

Forman introduced discrete Morse theory on finite regular cell complexes. Minian introduced a version of discrete Morse theory for posets which generalizes Forman's original Morse theory https://arxiv....

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390 views

### When does an open manifold admit two linearly independent vector fields?

$\DeclareMathOperator{\span}{span}$
$\DeclareMathOperator{\co}{H}$
$\newcommand{\kk}{\mathbb{F}}$
$\newcommand{\qq}{\mathbb{Q}}$
$\newcommand{\zz}{\mathbb{Z}}$
$\newcommand{\rr}{\mathbb{R}}$
$\...

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120 views

### Interpreting the Bockstein lemma?

I am reading through "Cohomology Operations and Applications in Homotopy Theory" by Mosher and Tangora and I had a little bit of confusion with the Bockstein lemma. All cohomology will be ...

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197 views

### Galois action on torsion in homotopy groups not induced by homotopy equivalences

Let $V$ be a simply connected smooth projective complex variety defined over the rationals. Then for any integer $n\geq 2$ the group $\pi_n(V)$ is finitely generated abelian so profinite completion ...

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69 views

### Inverse image of simplex

Let $M=B\times S^{1}$ be the solid torus where $\partial M=X\times F= S^{1}\times S^{1}$. We consider the projection $\pi : \partial M \longrightarrow X$ which induces the simplicial map $$\pi_{*} : ...

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161 views

### Some computational results and goals of stable motivic homotopy theory of schemes

I am trying to learn ($\mathbb{P}^1$-)stable motivic ($\mathbb{A}^1$-)homotopy theory of schemes from the Cisinski-Deglise book, Triangulated Categories of Mixed Motives. In order to keep myself going ...

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173 views

### A homotopy equivalence from a variety to itself that is not homotopic to a homeomorphism

Let $V$ be a simply connected smooth projective complex variety. Can there be a homotopy equivalence $V\to V$ that is not homotopic to a homeomorphism?

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780 views

### Which stable homotopy groups are represented by parallelizable manifolds?

The Pontryagin-Thom construction allows one to identify the stable homotopy groups of spheres with bordism classes of stably normally framed manifolds. A stable framing of the stable normal bundle ...

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122 views

### Groupoid completion of a topological category vs its homotopy category?

Given a category $\mathcal{C}$ enriched in spaces, we can take the nerve (a simplicial space) and then geometric realization to get a space $B\mathcal{C}$. If we view spaces as $\infty$-groupoids, ...

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292 views

### 3-colored triangulations of the sphere $S^2$, and Sperner's Lemma

I noticed something about colored triangulations of the topological sphere $S^2$ and have a question about this.
Observation. If you triangulate the sphere $S^2$ and color the vertices with three ...

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59 views

### Constructive factorisation of null-homology map through acyclic complex

Let $f: C \rightarrow D$ be a maps of chain complexes on an idempotent complete additive category with all kernel or cokernel (or chain complexes on abelian category).
If $f$ induces a null map in ...

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91 views

### Creating an inverse system which “stratifies density”

Setting:
Let $X'$ be a dense subset of an infinite-dimensional Fréchet space $X$ and suppose that $(X_n')_{n \in \mathbb{N}}$ is a nested sequence of non-empty subsets of $X'$ satisfying
$$
\bigcup_{n ...

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162 views

### Chern-Weil theory in the cohomological Atiyah-Singer theorem

I am interested in the following piece of data appearing in the cohomological Atiyah-Singer theorem. My reference is "The index of elliptic operators. III" by Atiyah and Singer.
Let $D:\...

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362 views

### Chromatic t-structures?

Questions: Fix a prime $p$ and $n \in \mathbb N_{\geq 1}$.
Does the category $Sp_{K(n)}$ of $K(n)$-local spectra admit a nontrivial $t$-structure?
By "nontrivial", I simply mean that $\{0\}...

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92 views

### Factorizing vector fields near manifolds of singularities

Let $V: \mathbb{R}^n \to \mathbb{R}^n$ be a smooth vector field containing a smooth $k$-dimensional manifold $M$ (with $1\leq k < n$) of singularities: $V(M)=0$.
Suppose furthermore that at every ...

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161 views

### Modelling rational spaces of finite $\mathbb{Q}$-type with spaces with finite simplices in every simplicial dimension

EDIT 2
Original question below. I will award the outstanding bounty for an answer to the following question (question (2) in the OP).
Let $X$ be a Kan complex which is connected, nilpotent, and of ...

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53 views

### Fundamental class of products of spaces [migrated]

Let $M$ be smooth oriented manifold where $M=X\times F$, $X$ and $F$ smooth oriented manifolds.
We note by $[M]$ the fundamental class of $M$. Is this equality true: $$[X\times F] = [X]\times [F]?$$

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190 views

### De Rham's theorem for top-forms in manifolds with boundary

In page 79 of Bott-Tu, "Differential Forms in Algebraic Topology", they define the relative de Rham theory as follows:
Let $f:S\to M$ be a smooth map. Define the complex $\Omega^*(f)$ by
$$\...

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129 views

### chromatic minimal cell structures

If $X$ is a finite $p$-local spectrum, then the minimal number of cells needed to construct $X$ is exactly $\dim_{\mathbb F_p} H_\ast(X,\mathbb F_p)$. Is there an analogous result in the $K(n)$-local ...

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120 views

### To what extent is a vector bundle on a smooth manifold determined by its restriction to the complement of a closed smooth submanifold?

The question is a follow-up to this one.
Let $N\subset M$ be a closed smooth submanifold of codimension $k\geq 1$ of a smooth manifold $M$. Let $E_1\rightarrow M$ and $E_2\rightarrow M$ be two vector ...

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1k views

### Surprising properties of closed planar curves

In https://arxiv.org/abs/2002.05422 I proved with elementary topological methods that a smooth planar curve with total turning number a non-zero integer multiple of $2\pi$ (the tangent fully turns a ...

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394 views

### To what extent is a vector bundle on a manifold with boundary determined by its restriction to the interior?

Let $M$ be a manifold with boundary $\partial M$ and interior $M_0$. Let $E\rightarrow M_0$ be a fixed vector bundle. How many extensions of $E$ to a vector bundle $E'\rightarrow M$ are there, up to ...

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93 views

### The homotopy type of the simplicial space obtained by free adding degeneracies to a semi-simplicial space

Let $\text{sTop},\text{ssTop}$ denote the categories of simplicial, semi-simplicial spaces respectively. There is a functor $E:\text{ssTop}\rightarrow \text{sTop}$ that is left adjoint to the ...

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451 views

### If $A, B$ are abelian groups such that $\mathrm{Hom}(A, G) \cong \mathrm{Hom}(B, G)$ for all abelian groups $G$, must $A$ and $B$ be isomorphic?

The question is in the title. If the isomorphism $\mathrm{Hom}(A, G) \cong \mathrm{Hom}(B, G)$ is natural in $G$ then this is just the Yoneda Lemma. If $A$ and $B$ are finitely generated this is also ...

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131 views

### Using the Serre spectral sequence - moving between $\mathbb{Z}/2$ and $\mathbb{Z}$ information

I am trying to understand the computation of $\pi_5(S^3)$ and $\pi_6(S^3)$ using the Serre spectral sequence. I know already that $\pi_5(S^3)$ is only 2-torsion and $\pi_6(S^3)$ is 2-torsion together ...

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386 views

### Why does this construction give a (homotopy-invariant) suspension (resp. homotopy cofiber) in an arbitrary pointed model category?

In their text Foundations of Stable Homotopy Theory, Barnes and Roitzheim define the suspension of a cofibrant object X of a pointed model category to be the pushout of the diagram $*\leftarrow X\...

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92 views

### Examples of commutative ring spectra with graded-artinian coefficients?

Question: What are some ring spectra $E$ satisfying the following conditions?
The coefficients $E_\ast = \pi_\ast(E)$ are graded-commutative;
There is a Kunneth spectral sequence $E_\ast(X) \otimes_{...

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160 views

### Does cohomology ring determine a compact symmetric space?

Suppose that $M_1, M_2$ are compact connected symmetric spaces with isomorphic integer cohomology rings. Does it follow that $M_1$ is diffeomorphic to $M_2$?
The only result I am aware of is this ...

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152 views

### Bousfield $p$-completion on spectra

Bousfield p-completion on spaces is a functor $(-)^{\wedge p}$ whose main property is that a map $f:X\rightarrow Y$ induces an isomorphism $f_{\ast}:H_\ast(X,\mathbb{F}_{p})\rightarrow H_\ast(Y,\...

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67 views

### Binary and n-ary topological spaces

I am interested in various generalizations of the notion of topological space; also in topologies placed in untypical frameworks, i.e. intuitionistic topological spaces or nano topological spaces. ...

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325 views

### Two definitions of power operations — how do they relate?

Below are two different stories about power operations for $\mathbb{E}_\infty$-ring spectra, and I am struggling to see how they relate. In the following we let $R$ be an $\mathbb{E}_\infty$-ring ...

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57 views

### relationship between “linear approximation” to immersions and formal immersions

I'm reading these notes
Here, I am regarding $\mathrm{Imm}(-,N)$ as a presheaf on the open sets of some manifold $M$
If we take $\mathrm{Imm}^f(-,N)$ to be the sheaf of formal immersion (an element ...

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60 views

### Multi-simplicial generalization of $\Gamma$-spaces

Is there a generalization of Segal's theorem that the inclusion of $X_1$ into $\Omega|X_*|$ is a weak equivalence for a $\Gamma$-space $X_*$ if $X_1$ is group like? Specifically, I am looking for a ...

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290 views

### Morphism with connected fibers induce surjection on fundamental groups?

Let $X,Y$ be path-connected finite CW complexes with base points $x_0,y_0$, let $f\colon X\to Y$ be a surjective continuous map, such that for every $y\in Y$, the fiber $f^{-1}(y)$ is path connected. ...

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117 views

### Dress' construction and Serre spectral sequence

Currently, I am reading Serre spectral sequence, given below, using Dress' construction.
Let $f:E\to B$ be a Serre fibration. Then, there is a first quadrant
spectral sequence $\big\{E^r,d^r\}_{...

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465 views

### Can triangulations (or some related combinatorial structure) distinguish smooth structures on $RP^4$?

There are exotic versions of $RP^4$, constructed by Cappell-Shaneson, which are homeomorphic but not diffeomorphic to the standard $RP^4$. One way to distinguish them is via the $\eta$ invariant of $...

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79 views

### Decomposing a compact connected Lie group

I want to prove the following. Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak g$ and center $Z_G.$ It is not hard to prove that $\mathfrak g$ is reductive. Therefore, we can ...

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112 views

### An alternative proof of Künneth spectral sequence, independent of Künneth formula for homology

I am currently reading Künneth spectral sequence, which is given below.
Let $R$ be a ring and A$=\big\{A_n,d_n:A_n\longrightarrow A_{n-1}\big|d_{n-1}\circ d_n=0\big\}_{n\in \Bbb Z}$ be a chain ...

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**1**answer

285 views

### Examples of non-zero negative Steenrod operations

In JP May's paper A general algebraic approach to Steenrod operations, Steenrod operations are constructed in wide generality. In this context, it is not necessarily true that negative Steenrod ...

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52 views

### Structure of boundary labelling in Sperner‘s Lemma

Consider a triangulated polygon in the 2-dimensional plane, where each vertex is labelled green, blue, or orange. Sperner's Lemma asserts that a fully-colored triangle exists in the triangulation, if ...