Questions tagged [at.algebraic-topology]

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

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3
votes
1answer
72 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 ...
5
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1answer
111 views

Cohomology theories for spaces vs cohomology theories for spectra

It is a standard consequence of the Brown Representability Theorem for $\operatorname{Ho}(\operatorname{Top}_*)$ that the category of generalized cohomology theories for spaces (pointed CW complexes, ...
1
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0answers
41 views

Index bounded Riemannian metrics

Let $L$ be a closed simply-connected smooth manifold with a Riemannian metric $g$. We say $g$ is index bounded if the energy functional (which is assumed to be Morse/Morse-Bott) $$ E: C^k(L,g) \...
2
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0answers
200 views

Rational homotopy type of a complex algebraic variety defined over $\mathbb{Q}$

Does there exist a simply connected smooth proper complex variety that is not rationally homotopy equivalent to a simply connected smooth proper complex variety defined over $\mathbb{Q}$?
11
votes
3answers
504 views

A nontrivial principal bundle which satisfies Leray-Hirsch theorem

What is an example of a nontrivial principal bundle whose fibre space $G$, total space $P$ and base space $M$ are compact connected manifolds (the fiber $G$ is a compact Lie group) such that $$H^*(P,\...
4
votes
1answer
245 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 $$\...
10
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3answers
637 views

Smooth map homotopic to Lie group homomorphism

Let $G$ and $H$ be connected Lie groups. A Lie group homomorphism $\rho:G\to H$ is a smooth map of manifolds which is also a group homomorphism. Question: Can we find a smooth (or real-analytic) map $...
2
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0answers
80 views

Combinatorial condition for orientability on simplicial complexes

Let $K$ be a simplicial complex whose geometric realization is a topological or smooth manifold. Is it possible to restate the condition of orientability of $M$ exclusively in (combinatorial) terms of ...
1
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2answers
216 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 ...
15
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0answers
306 views
+200

Homotopy type of the affine Grassmannian and of the Beilinson-Drinfeld Grassmannian

The affine Grassmannian of a complex reductive group $G$ (for simplicity one can assume $G=GL_n$) admits the structure of a complex topological space. More precisely, the functor $$X\mapsto |X^{an}|$$ ...
0
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0answers
94 views

Understanding a step in proof of how the localization of an additive category by a subclass of morphism satisfies Ore is also additive

I started to study localization in additive and triangulated categories via a subclass of morphism which satisfies the Ore conditions by my own. Right now, I'm studying how for an additive category $...
7
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4answers
1k views

How to prove the connected sum of two closed aspherical n-manfolds (n >2) is not asperical?

The intuitive idea is that the sphere connected the two manifolds is not contractible, which implies the (n-1)th homotopy group is not zero. Another argument, which I am not totally understand, uses ...
4
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0answers
156 views

Instanton numbers for diverse gauge bundles on diverse manifolds — their relations to characteristic classes

It is standard (?) that the $SU(N)$ gauge theory has the instanton number $n$ quantized as $n \in \mathbb{Z}$ $$ n = { 1 \over 8\pi^2} \int_{\mathcal{M}_{4}} \text{tr} \left(F \wedge F\right) = {1 \...
6
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0answers
118 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 ...
4
votes
1answer
213 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 ...
1
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0answers
73 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 ...
25
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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 ...
7
votes
3answers
285 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 ...
76
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23answers
9k views

Solving algebraic problems with topology

Often, topologists reduce a problem which is - in some sense - of geometric nature, into an algebraic question that is then (partiallly) solved to give back some understanding of the original problem. ...
7
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0answers
129 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 $w_{2^k}(E)$ (...
6
votes
2answers
358 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 =...
12
votes
6answers
4k views

What is the best way to study Rational Homotopy Theory

I studied basic algebraic topology elements: fundamental group, higher homotopy groups, fibre bundles, homology groups, cohomology groups, obstruction theory, etc. I want to study Rational Homotopy ...
11
votes
2answers
457 views

What do absolute neighborhood retracts look like?

In the course of filling in my map of non-pathological topology, I'd like to understand the class of ANRs (Absolute Neighborhood Retracts) as a sort of "neighborhood" of the class of CW complexes. ...
4
votes
1answer
143 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 ...
16
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3answers
768 views

Can an oriented closed $n(\geq 2)$-dimensional manifold be smoothly embedded in $\mathbb{R}^{2n-1}$?

Can anyone provide me with an example of an orientable closed manifold $M$ of dimension $n\geq 2$, which cannot be smoothly embedded in $\mathbb R^{2n-1}$? I know these cannot exist for $n=1$, i.e. $S^...
1
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0answers
79 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 ...
5
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0answers
200 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 ...
4
votes
0answers
209 views

Enlarging a compact set in order to improve its shape

In my previous question it was established that if $X$ is a metrizable, connected, locally path connected space and $K\subset X$ is compact, then there is a Peano continuum $L\subset X$ such that $K\...
5
votes
2answers
335 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 ...
2
votes
0answers
94 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 ...
12
votes
1answer
228 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(...
2
votes
0answers
69 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....
11
votes
1answer
394 views

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

$\DeclareMathOperator{\span}{span}$ $\DeclareMathOperator{\co}{H}$ $\newcommand{\kk}{\mathbb{F}}$ $\newcommand{\qq}{\mathbb{Q}}$ $\newcommand{\zz}{\mathbb{Z}}$ $\newcommand{\rr}{\mathbb{R}}$ $\...
26
votes
4answers
799 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 ...
4
votes
0answers
164 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 ...
0
votes
0answers
72 views

Inverse image of simplex [closed]

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_{*} : ...
2
votes
0answers
122 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 ...
8
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0answers
165 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 ...
3
votes
0answers
175 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?
6
votes
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 ...
7
votes
1answer
124 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, ...
10
votes
2answers
508 views

Is limit of null-homotopic maps null-homotopic?

The question is motivated by my failed comment to this one. Let $M$ and $N$ be path connected locally compact, locally contractible metric spaces (you may assume that they are manifolds). Let $\...
4
votes
1answer
92 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 ...
5
votes
1answer
398 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 ...
5
votes
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:\...
1
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0answers
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 ...
17
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0answers
465 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 ...
7
votes
2answers
366 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\}...
0
votes
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: $$[X\times F] = [X]\times [F]?$$
4
votes
1answer
209 views

Is the derived category of local systems equivalent to the derived category of sheaves of vector spaces with local system cohomology?

Let $k$ be a field and $X$ a topological space. Write $\mathrm{Sh}(X)$ for the category of sheaves of vector spaces on $X$, and $\mathrm{Loc}(X)$ for the subcategory of local systems of finite ...

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