Questions tagged [at.algebraic-topology]

Homotopy theory, homological algebra, algebraic treatments of manifolds.

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4
votes
1answer
196 views

Support of torsion in the Borel–Moore homology

Given a complex quasi-projective variety $X$, let $\alpha$ be an element of the Borel–Moore homology $H_i^\text{BM}(X)$ such that it can be killed by a prime $p$. Under what conditions one can say ...
31
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2answers
1k views

Can the nth projective space be covered by n charts?

That is, is there an open cover of $\mathbb{R}P^n$ by $n$ sets homeomorphic to $\mathbb{R}^n$? I came up with this question a few years ago and I´ve thought about it from time to time, but I haven´t ...
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0answers
22 views

Homotopic upper-bound for ``tesselation number'' covering closed complete Riemannian manifolds

Suppose that I'm given a $d$-dimensional closed and connected Riemannian manifold $(M,g)$. Then there exist $(K_n,\phi_n)_{n=1}^N$ such that $\phi_n:K_n \rightarrow \Delta_{d}$ is a homomorphism from ...
3
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1answer
431 views

Why is a simply connected homology sphere a topological sphere?

I post this for a friend who currently doesn’t have access to this site. It is about an implication in the last paragraph of the following paper: KATSUHIRO SHIOHAMA and HONGWEI XU, The topological ...
9
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2answers
830 views

Algebraic topology and homotopy theory with simplicial sets instead of topological spaces

To quote Kerodon: In fact, it is possible to develop the theory of algebraic topology in entirely combinatorial terms, using simplicial sets as surrogates for topological spaces. A similar quote can ...
3
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0answers
166 views

Weakening of weak Lefschetz theorem

Is there some sort of general condition that implies for a closed immersion of projective complex varieties $i:Z\hookrightarrow X$, the map on the $n$-th homology sends non $p$-divisible elements to ...
3
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0answers
129 views

Homological stability of Chow varieties

Given a connected component $C$ of the degree $d$ Chow variety of $r$ cycles on $C_{d,r}(X)$ ($X$ is smooth projective variety over $\mathbb{C}$), let $C'$ be another connected component of $C_{d',r}(...
0
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1answer
140 views

Why does $X_0\times S^1\simeq X-X_0$? [closed]

Let $X$ be an $n$-dimensional connected smooth manifold, and let $X_0$ be an embedded $(n-2)$-dimensional compact submanifold of $X$ with the trivial normal bundle. How do we get inclusion? $$X_0\...
5
votes
1answer
224 views

The contravariant mapping space represented by a homotopical classifying space (e.g. BG)

In classical homotopy theory, there are a number of spaces which are important because they represent an interesting functor on $\operatorname{Ho(Top)}$; for example, $K(G,n)$ represents singular ...
6
votes
1answer
210 views

Orbifolds are Thom-Mather stratified spaces

Where can I find a proof of (or if it is even true) that an (effective) orbifold is a Thom-Mather stratified space? edit: after some search, I found the proof should be contained in either GIBSON, C....
5
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0answers
160 views

Factorization homology and topological conformal field theories

My question concerns some of the results of Costello's "Topological conformal field theories and Calabi-Yau categories" and how they are related/ can be rederived via the description of (...
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0answers
176 views

Does a ring homomorphism induce a morphism in local cohomology?

Let $\rho:R\longrightarrow S$ be a homomorphism of Noetherian rings and, for the ideals $I\subset R$ and $J\subset S$, we have $\rho(I)\subseteq J$. Does this induce a morphism in local cohomology ...
1
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0answers
108 views

Homotopy of sheaves

On a certain topological space $X$ I want to think about sheaves up to homotopy, i.e., homotopies in the space of sheaves over $X$, and then see what homotopy classes of sheaves I get. Is there a good ...
8
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0answers
138 views

Operads and Inverses

One slightly displeasing thing about the theory of operads is that they are unable to encode the structure of inverses; e.g. the recognition principle for $n$-fold loop spaces says "there is an ...
4
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0answers
97 views

Submanifolds of $4$-manifolds and their intersections

Suppose we have two (oriented) submanifolds $A,B$ of an oriented $4$-dimensional manifold $M$, that intersect transversally. Looking at the standard references for $4$-manifolds, I couldn't find a ...
6
votes
2answers
209 views

If $G$ is a topological group that contains a torsion element, then the classifying space $BG$ is infinite-dimensional?

We know that if $G$ is a topological group that contains a torsion element and $G$ satisfies additional conditions such as $G$ discrete or $G$ finite-dimensional, then the classifying space $BG$ is ...
3
votes
1answer
143 views

Kan Complexes, proof of extension of a map to a product

I'm reading a book about Kan Complexes in Simplicial Homotopy Theory (Curtis), and came across this theorem at the beginning : if $f : \Delta[n] \times \Lambda^k[m] \to K$ is a simplicial map and $K$ ...
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0answers
163 views

are acyclic fibrations of nice spaces absolute extensors for perfectly normal spaces?

A space $Y$ is called an absolute extensor for normal spaces (also sometimes solid) if, for any normal space $X$, closed subset $A$ of $X$, and map $f:A\to Y$, there exists a map $f′:X\to Y$ such that ...
4
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0answers
153 views

Eilenberg–Moore equivalences for $C_*(\Omega M)$

Let $M$ be a nice connected topological space (I'm actually interested in manifolds) with base point $p$ and let $\pi: E \to M$ be a fibration. Then chains on the fiber $F$ at $p$, $C_*(F)$, become a ...
10
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2answers
2k views

Why doesn't local cohomology seem to be used as much in algebraic geometry?

In algebraic topology, relative (co)homology is very useful. For example, we have a long exact sequence which is often helpful for lots of calculations. In algebraic geometry, we have local cohomology,...
8
votes
1answer
205 views

Homotopy type of the Hausdorff metric

Recall that if we have a metric space $X$ then we can consider the set of its nonempty compact subsets and equip this with a metric called the Hausdorff distance. Denote the resulting metric space $\...
6
votes
2answers
276 views

Classifying space $\text{BU}(n)$ from the differential-geometric point of view?

The classifying space of a topological group $G$ is the quotient of $EG$ (a topological space with vanishing homotopy groups) by a proper free action of $G$. The standard notation for the classifying ...
6
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0answers
180 views

Topological vs smooth (equivariant) bordism group

In Remark 1.31 of this work, it is claimed that "Standard arguments in Pontryagin-Thom theory imply that the relevant smooth and topological G bordism groups are isomorphic." The adjective &...
0
votes
1answer
122 views

Künneth formula and induced map in homologies

Let $X,Y,Z$ be smooth connected manifolds and $f \colon X \times Y \rightarrow Z$ a smooth map. Suppose that we have $H_{*}(X \times Y; \mathbb{Z})$ is isomorphic to $\bigoplus_{p+q=*}(H_{p}(X; \...
7
votes
2answers
825 views

Does the isomorphic of the fundamental groups imply the existence of a mapping inducing an isomorphism?

A pair of continuous mappings $f \colon X \to Y$ and $g \colon Y \to X$ is called $\pi_1$-equivalence if they induce mutually inverse isomorphisms of fundamental groups. Spaces are called $\pi_1$-...
4
votes
2answers
337 views

Can the loops in the definition of the fundamental group be considered injective?

Let $\mathrm{С}$ be some class of topological spaces that includes at least all subspaces of $\mathbb{R}^n $. Further we are in the category $\mathrm{С}_{*}$ (the category of point spaces; all ...
11
votes
1answer
384 views

Asking whether there is a compact Lie group containing affine symplectic group

The affine symplectic group is interesting and important in physics. However, the Lie group is noncompact. In order to have some good properties (Basically, we need some good behavior of Haar measure) ...
9
votes
1answer
559 views

"Nice" way to compute the signature of a toric manifold?

Is there a "nice" way to compute the signature of a smooth toric manifold of even complex dimension in terms of the moment polytope? By signature I mean in the sense of topology (see https://...
1
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0answers
103 views

Lifting theorem for finite spaces: replacing perfect normality by normality

In the Lifting theorem for finite spaces (Thm. 3.5, Eric Wofsey, quoted below), can one relax the condition "$A$ is a closed subset of a perfectly normal $X$" to "$A\to X$ has the right ...
4
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0answers
90 views

Equivariant phantom maps

In the stable homotopy category a map of spectra $f\colon X \rightarrow Y$ is called phantom is the induced map between the associated homology theories $X_* \rightarrow Y_*$ is zero, it is know that ...
5
votes
0answers
109 views

Higher homotopy groups of an orbifold

Given an orbifold $\mathcal{O}$, I have seen many ways to define the orbifold fundamental group: Thinking of $\mathcal{O}$ as a groupoid $\mathcal{G}$, $\pi_1^{orb}(\mathcal{O})$ can be defined as ...
3
votes
0answers
170 views

Cohomology ring $H^*(BG,\mathbb{Z}_2)$ for $G=\mathbb{Z}_2\ltimes B^2\mathbb{Z}^2$

$$ \newcommand{\Z}{\mathbb{Z}} $$ Consider the group $G=\Z_2\ltimes B^2\Z^2$ where $\Z_2$ acts on $\Z^2$ by interchanging the two factors of $\Z^2$, hence $\Z_2$ also acts on the Eilenberg-MacLane ...
6
votes
0answers
173 views

The role of spectral Lie algebras and twisting for operads in spectra

In the theory of differential graded (co)operads, the notion of twisting is ubiquitous. The fundamental notion is the twisting map from a cooperad $C$ to an operad $P$. It is defined as a Maurer-...
2
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0answers
150 views

Fibering cobordant to projectivization of a vector bundle

I was going through this Stong's paper, I am stuck in the proof of the proposition 8.4 (given below) I understand the proof till he derives the expression for the Steenrod square operation of the ...
2
votes
1answer
332 views

The nerve of the Ising category

A semi-simplicial complex (see Moore's 1958 paper: Semi-simplicial complexes and Postnikov systems, available at http://www-math.mit.edu/~hrm/kansem/moore-semi-simplicial-complexes.pdf, cf MathSciNet) ...
3
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0answers
209 views

CW structure on $\mathrm{PU}(3)$/Heisenberg group

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\PU{PU}$Consider the quotient space $\PU(3)/H=\SU(3)/G_{81}$ where $H$ is the Heisenberg group of order 27 $G_{81}$ is the No. 9 group of order 81 (...
0
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0answers
94 views

Spaces of $n$-dimensional topological spaces whose fundamental group is given

If we fix a group $G$ and a dimension $n$, we can ask which $n$-dimensional locally path-connected$^*$ (or otherwise sufficiently nice) topological spaces $X$ have $\pi_1(X) \cong G$. Would these ...
-1
votes
1answer
150 views

If $H_i(U_j)=0$ for infinitely many $j$ then $H_i(X)=0$ [closed]

Let $X$ be a topological space and $U_i$ open subsets. If $U_i\subset U_{i+1}$ and $\bigcup^{\infty}_{i=1}U_i=X$. How can I prove that if for infinitely many $j$, the $i$-th homology vanishes $H_i(U_j)...
7
votes
0answers
244 views

Topological operations corresponding to abelianization of the fundamental group

$\newcommand{\ab}{\mathrm{ab}}$Suppose we have a topological space $X$ with a non-abelian fundamental group $\pi_1(X)$. We'd like to perform a sequence of topological operations on $X$ (for example, ...
4
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0answers
259 views

Finite subgroup of $\mathrm{SO}(4)$ which acts freely on $\mathbb{S}^3$

Let $\Gamma$ be a finite subgroup of $\mathrm{SO}(4)$ acting freely on $\mathbb{S}^3$. It is known that all such $\Gamma$ can be classified. Is there any characterization of $\Gamma$ such that $\Gamma$...
3
votes
0answers
139 views

Finite generation of algebraic $K$-theory with finite coefficients

Given a smooth connected complex quasi-projective variety $X$, is it possible that $K_i(X, \mathbb{Z}/l)$ to be infinitely generated for $i>0$? I think Quillen-Lichtenbaum implies that above the ...
3
votes
1answer
146 views

Parameterizing the space of convex quadrilaterals

If $P=\mathbb{R}^2$ is the plane, is there a continuous surjection from $P^4$ to the space of convex quadrilaterals? Specifically, I'm looking for a continuous $f:P^4\to P^4$ such that: [convexity] ...
7
votes
2answers
251 views

Indexing categories of derivators

It is not clear to me the role of the domain and target in the definition of prederivators. For instance, the classical references put the domain as $\mathit{Dia}$, others as $\mathit{Cat}$ itself. ...
5
votes
1answer
259 views

Set of proper homotopy classes of arcs in a manifold

Let $M^n$ be an $n$-manifold with nonempty boundary and let $\partial_0 M$ be a specific connected component of $\partial M$. I am interested in the set of continuous maps $f : [0,1] \to M$ such that $...
4
votes
2answers
430 views

How much do characteristic classes fail to characterize bundles?

Given a group $G$, let $E \to B$ be a principal $G$-bundle. It is well-known that when $B$ is a nice enough topological space (e.g. CW-complex), such a thing corresponds to a connected component of $...
6
votes
1answer
257 views

Do all spaces doubly covered by $S^{2n}$ have the homeomorphism type of $\mathbb{P}^{2n}_{\mathbb{R}}$?

For reference, my motivation: It's of interest to classify free actions of groups on spheres of positive even dimension. Establishing such a classification up to homotopy is not too difficult: Every ...
4
votes
1answer
140 views

Landweber-exactness type theorems for arbitrary morphisms of $E_\infty$-ring spectra

This question comes out of a join of this question and this other question of mine, so please remove it as a duplicate if you feel it is so (to my eyes it puts the two previous ones in a context of ...
3
votes
0answers
178 views

Phantom map in homotopy and homology exists?

Let $f:X\rightarrow Y$ be a map between finite spectrum such that $\pi_{\ast} (f)=0$. (stable homotopy groups) $H_{\ast}(f)=0$. (Homology with integer coefficients) Does it imply that $f$ is null ...
15
votes
1answer
355 views

On diagrams in model categories and rectification

For a model category $C$, I'm denoting $h_\infty(C)$ the associated $\infty$-category (for example its Dwyer-Kan localization at weak equivalences, or if $C$ is simplicial the simplicial nerve of the ...
4
votes
1answer
145 views

Cobordism class of projectivization of a bundle

I was reading the book "Differentiable Periodic Maps" by P.E. Conner (1979). I am stuck at the following problem given at the end of section 21: Let $\xi\to V^n$ be a $k$-plane bundle over a ...

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