Questions tagged [at.algebraic-topology]

Homotopy theory, homological algebra, algebraic treatments of manifolds.

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5 votes
5 answers
888 views

Two arcs in the complement of a disc must intersect?

Let $D=\{z\in \mathbb C:|z|\leq 1\}$ be the unit disc in the complex plane, with interior $U=\{z\in \mathbb C:|z|<1\}$. Let $A\subset \mathbb C\setminus U$ be an arc intersecting $D$ only at its ...
3 votes
0 answers
210 views

Classifying spaces beyond CW complexes

We know that for a reasonable topological group $G$ (say a compact Lie group) admits a classifying space for $G$-bundles within the category of countable CW complexes. That means, there is a space $BG$...
4 votes
2 answers
279 views

On a generalized homotopy transfer theorem

In the book of Loday and Vallette "Algebraic Operads" a necessary condition for the Homotopy Transfer Theorem is that the starting operad is Koszul. I am interested in a generalization of ...
13 votes
3 answers
825 views

Are negatively pinched manifold locally conformally flat?

One knows that hyperbolic manifolds are locally conformally flat. How about those negatively pinched manifolds, i.e. the sectional curvature $K$ satisfy: $$ -\Lambda \le K \le -\lambda$$ for $\Lambda&...
3 votes
2 answers
412 views

A question on the manifold $ \{n\otimes n-m\otimes m:n,m\in S^2,(n,m)=0\} $

Consider a manifold $ N $ defined as follows $$ N=\{n\otimes n-m\otimes m:n,m\in S^2,\quad(n,m)=0\}\subset M^{3\times 3}, $$ where $ S^2 $ denotes the two dimensional sphere, $ (\cdot,\cdot) $ ...
2 votes
0 answers
142 views

Do the nearby cycle and Beilinson's vanishing cycle functors commute?

Let $X$ be a complex algebraic variety with a pair of regular functions $f_1,f_2$. To these functions we can associate various functors: the nearby cycles functor $\Psi_{f_i}$, the vanishing cycles ...
23 votes
2 answers
3k views

Calculating Mayer-Vietoris efficiently

This is a question whose motivation and framing seem to involve a lot of topology, but which I suspect comes down to some simple and standard combinatorics that's probably recorded in a book somewhere....
23 votes
3 answers
2k views

What are some toy models for the stable homotopy groups of spheres?

The graded ring $\pi_\ast^s$ of stable homotopy groups of spheres is a horrible ring. It is non-Noetherian, and nilpotent torsion outside of degree zero. Question: What are some "toy models" ...
1 vote
0 answers
141 views

Minimal first Pontryagin class $p_1=1$?

From Hirzbuch theorem, the signature of 4-manifold $\sigma = p_1/3$ with the first Pontryagin class $p_1$. I know that the $\sigma=p_1/3 =1$ so $p_1=3$ for complex projective space $\mathbb{P}^2$. Is ...
39 votes
1 answer
6k views

Classification of surfaces and the TOP, DIFF and PL categories for manifolds

A surface is simply a 2-manifold. The classification theorem for compact connected surfaces (with boundary) is commonly regarded in the categories TOP, DIFF and PL. Well known proofs (e.g. via ...
4 votes
0 answers
91 views

What is the Goldie dimension of the ring of stable stems?

Let $p$ be a prime, and let $\pi_\ast^{(p)}$ be the ring of stable homotopy groups of spheres localized at the prime $p$. This is a nonnegatively-graded-commutative ring with $\mathbb Z_{(p)}$ in ...
2 votes
0 answers
107 views

Extension of isotopies

In what follows $M$ will be a manifold (without boundary, for simplicity) and $C\subseteq M$ will denote a compact subset. In the paper Deformations of spaces of imbeddings Edwards and Kirby prove the ...
6 votes
1 answer
224 views

Relationship between the holonomy pseudogroup and holonomy homomorphism (foliation)

This question is surely a duplication of https://math.stackexchange.com/questions/4343635/relationship-between-the-holonomy-pseudogroup-and-holonomy-homomorphism-foliati , however, I got no replies. ...
1 vote
0 answers
112 views

A question about cohomology with local coefficient

Let's consider the next theorem. Theorem [The cohomology Leray-Serre Spectral sequence] Let $R$ be a commutative ring with unit. Given a fibration $F\hookrightarrow E\overset{p}{% \rightarrow }B$, ...
1 vote
1 answer
138 views

Lie group framing and framed bordism

What is the definition of Lie group framing, in simple terms? Is the Lie group framing of spheres a particular type of Lie group framing? (How special is the Lie group framing of spheres differed ...
3 votes
1 answer
158 views

Pontryagin product on the homology of cyclic groups

Consider the cyclic group $C_{p^N}$ of order $p^N$, and let $k$ be a field of characteristic $p$. I would like to know what the algebra structure on the homology $H_*(C_{p^N};k)$ induced by the ...
2 votes
0 answers
62 views

Framed bordism and string bordism in 3-dimensions vs topological modular form

In simple colloquial terms, how are the framed bordism and string bordism in 3-dimensions related to the study of the theory of topological modular form TMF? I want to know some simple derivable ...
1 vote
2 answers
151 views

Reference for choosing a path lifting function?

I recall having seen discussion of a Hurewicz or Serre fibration equipped with a chosen path lifting function. Citation??
34 votes
6 answers
4k views

Why study finite topological spaces?

In rereading Thurston's essay On Proof and Progress in Mathematics I ran across this passage: … this means that some concepts that I use freely and naturally in my personal thinking are foreign to ...
3 votes
2 answers
219 views

Is the free algebra functor over an $\infty$-operad symmetric monoidal?

Suppose $F: \mathcal{O}^\otimes \to \mathcal{P}^\otimes$ is a map of $\infty$-operads, and $\mathcal{C}$ is a symmetric monoidal $\infty$-category that admits small colimits, such that the tensor ...
2 votes
2 answers
160 views

$String/CP^{\infty}=Spin$ or a correction to this quotient group relation

We know that there is a fiber sequence: $$ ... \to B^3 Z \to B String \to B Spin \to B^2 Z \to ... $$ Is this fiber sequence induced from a short exact sequence? If so, is that $$ 1 \to B^2 Z = B S^...
1 vote
0 answers
86 views

A question about the Conner Conjecture

In some sources, Conner conjecture is expressed as follows: Theorem [Conner Conjecture] Let $G$ be a compact Lie group, and let $X$ have the homotopy type of a finite dimensional $G$-CW complex with ...
9 votes
2 answers
571 views

Generalization of the sphere theorem in dimension at least 4

In 1956, Papakyriakopoulos proved Dehn's lemma, loop theorem and the sphere theorem. The proofs are based on a clever technique called "tower construction". Later, Whitehead, Shaprio, ...
1 vote
1 answer
85 views

Simplicial cochain representing the pullback of a class Poincaré dual of a submanifold

Let $K$ be a simplicial complex of dimension $n$, $M$ be a topological manifold, and $f \colon |K| \to M$ be a continuous map. Let $X$ be an embedded manifold in $M$ of codimension $n$, such that $f(|...
0 votes
0 answers
68 views

Topological transversality by dimension

We know that to achieve transverality in the topological category, for example to make a continuous map into a manifold transverse to a topological submanifold, we need the existence of micro normal ...
2 votes
1 answer
142 views

string bordism group and framed bordism group for $d \leq 6$ and $d \geq 7$

Why do the string bordism group and the framed bordism group coincide the same in dimensions lower than 7 ($d = 0,1,2,3,4,5, 6$)? Why do the string bordism group and the framed bordism group differ ...
3 votes
1 answer
250 views

Do objects in the derived category behave stackily?

It is well known that derived categories (I'm particularly thinking of constructible derived categories and derived categories of D-modules) don't form a stack. In particular given morphisms in the ...
1 vote
0 answers
125 views

Determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower?

What and how to determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower? Namely, how do we know $$ K(Z_2,1)?, \quad K(Z_2,2)?, \quad K(Z,4)? $$ Naively -- in each step ...
97 votes
10 answers
13k views

equivalence of Grothendieck-style versus Cech-style sheaf cohomology

Given a topological space $X$, we can define the sheaf cohomology of $X$ in I. the Grothendieck style (as the right derived functor of the global sections functor $\Gamma(X,-)$) or II. the Čech ...
2 votes
0 answers
128 views

Tensor product of objectwise weak homotopy equivalences of $\mathcal{M}$-spaces

I consider the enriched category $[\mathcal{M}^{op},\mathrm{Top}]$ of enriched functors (I call them $\mathcal{M}$-spaces) from the enriched small category $\mathcal{M}^{op}$ to the enriched category $...
1 vote
0 answers
216 views

Examples of when $X$ is homotopy equivalent to $X\times X$

I was thinking about this question the other day: When is a topological space $X$ homotopy equivalent to $X\times X$ (with the product topology)? This is essentially a cross-post of this MSE question.....
3 votes
1 answer
187 views

On infinity-morphisms between algebras over algebraic operads

I posted this question in the "Mathematics" stack exchange, but it hasn't got much attention... I hope it will get more here. Let $P$ be a Koszul operad. In the book of Loday-Vallette "...
2 votes
0 answers
89 views

Explicit CW-complex replacement of the space of reparametrization maps

Let $P$ be the space of nondecreasing surjective maps from $[0,1]$ to itself equipped with the compact-open topology: $P$ is contractible. There exists a trivial fibration $P^{cof} \to P$ from a CW-...
8 votes
1 answer
335 views

Telescopic localisation of Eilenberg-MacLane spaces

Fix a prime $p$ and an integer $n>0$. Let $K$ be the corresponding Morava $K$-theory spectrum, and let $T$ be the telescope of a $v_n$-self map of a finite spectrum of type $n$, and let $X$ be the ...
0 votes
0 answers
83 views

What happens if I take a doubly-free simplicial abelian group?

Suppose that I have a simplicial set $X_\bullet$. I can take the free abelian group generated by $X_\bullet$, $\mathbb{Z}X_\bullet$. But then I can forget that this has an abelian group structure, ...
0 votes
0 answers
100 views

Basis of Lambda algebra for a programmer

First of all, I'm not a specialist in alg. top., but I try to apply computational math to it, so if I'm wrong in something you'd be doing a better thing explaining it to me instead of blaming me :) ...
5 votes
1 answer
553 views

Intersection cohomology and Poincaré duality

When trying to learn about perverse sheaves I hand-wavingly thought that intersection cohomology is the ‘minimal’ way of fixing the failure of Poincaré duality. But I am very aware that it is risky to ...
4 votes
0 answers
162 views

How to think about Beilinson's gluing data?

Let $X$ be a complex manifold, $D$ a divisor (that is globally the zero locus of a function) and $U$ its complement. Recall Beilinson's "how to glue perverse sheaves": Given a perverse ...
2 votes
1 answer
221 views

Comparing the exit path category and the nerve of a stratified space

Let $P$ be a finite poset, and $X$ be a topological space stratified by $P$, in the sense that $X$ is equipped with a continuous map $X \to P$ in the Alexandroff topology (or equivalently with a ...
2 votes
1 answer
226 views

Derived category of local systems of finite type on a $K(\pi,1)$ space: an explicit counterexample

Let $X$ be a nice enough topological space. I am mostly interested in smooth complex algebraic varieties. One may ask whether the bounded derived category of the category $\mathrm{Loc}(X)$ of local ...
25 votes
1 answer
1k views

What can we say about the Cartesian product of a manifold with its exotic copy?

Let $M$ be a smooth oriented manifold, and let $M^E$ be an exotic copy, i.e homeomorphic but not diffeomorphic to $M$. Is it true that $M\times M$ is diffeomorphic to $M\times M^E$? I am ...
2 votes
0 answers
165 views

How a circle $S^1$ acts on the Cayley plane $OP^2$ with exactly three fixed points?

The complex projective $CP^2$, the quaternionic projective space $HP^2$, and the octonionic projective space $OP^2$ each admit a circle action with $3$ fixed points. The circle action on $HP^2$ can be ...
8 votes
2 answers
685 views

Pullbacks of classifying spaces

In what follows all the groups will be discrete, not necessarly finite. Let $f:G\to H$ be a morphism of groups and $H'\to H$ be the inclusion of a subgroup. It seems to me (but correct me if I am ...
4 votes
2 answers
364 views

Classifying space of a non-discrete group and relationship between group homology and topological homology of Lie groups

I have a very soft question which might be very standard in textbooks or literature but I haven't seen it. To a fixed group $G$ we may attach different topologies to make it different topological ...
13 votes
1 answer
336 views

Is $KU\otimes S^1_+$ isomorphic to $F(S^1_+,KU)$ as $E_\infty$ rings?

There are various ways to construct $KU$ as an $E_\infty$ ring spectrum; I will take that as given. Using this, we can make $KU\otimes G_+$ into an $E_\infty$ ring for any commutative topological ...
2 votes
0 answers
95 views

Unordered configuration space with non-distinct points

Consider a topological space $X$, a natural number $n>0$ and the quotient topological space $Q_n(X)$ of $X^n$ by the equivalence relation : $x\sim y$ if and only if there is a permutation $\sigma$ ...
2 votes
1 answer
208 views

Projective objects in chain complexes of an abelian category: Further question

Yes, I see there are other Q&A's on this, for instance here: Projective objects in the category of chain complexes I am wondering why a level-wise projective chain complex $P$ which is split ...
14 votes
1 answer
547 views

Mednykh's formula for 2d manifold with boundaries? How to count principal G-bundles with prescribed holonomies?

Let S be a closed, orientable 2d manifold and G a finite group. Since a principal G-bundle over S is specified by maps $\phi : \pi_1(S) \rightarrow G$ modulo the adjoint action by G, the way to count ...
6 votes
2 answers
619 views

Removing a submanifold from a closed manifold

Let $M$ be a simply-connected closed manifold. Can we find a closed submanifold $N \subsetneq M$ such that $M\backslash N$ is simply-connected and has finite second homotopy group?
2 votes
0 answers
58 views

Module structure of $\Omega_*(Z_p)$

In Conner-Floyd's book, Differentiable Periodic Maps in (46.1), for p an odd prime and $k=1,2,\dots$, it is posted the identities: $$p\alpha_{2k+1}+[M^4]\alpha_{2k-3}+[M^8]\alpha_{2k-7}+\dots=0$$ in $\...

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