Questions tagged [at.algebraic-topology]

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

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45 views

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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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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1answer
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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2answers
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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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 ...
7
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3answers
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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2answers
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 ...
4
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1answer
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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1answer
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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1answer
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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0answers
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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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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4answers
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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1answer
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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2answers
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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1answer
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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0answers
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:\...
7
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2answers
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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0answers
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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0answers
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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1answer
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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5answers
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 ...
5
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1answer
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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0answers
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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3answers
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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0answers
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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0answers
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. ...
6
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1answer
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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0answers
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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2answers
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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0answers
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\}_{...
8
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3answers
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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0answers
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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0answers
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 ...
5
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1answer
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 ...

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