Questions tagged [at.algebraic-topology]

Homotopy theory, homological algebra, algebraic treatments of manifolds.

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2 votes
1 answer
141 views

Are all Torus Links in fact Lorenz links or not?

I'm currently trying to work through the material on Lorenz knots in the literature and there seems to be conflicting information. On p. 66, in the Birman-Williams' paper Knotted Periodic Orbits in ...
6 votes
3 answers
776 views

Cylinders dividing $\mathbb{R}^{3}$

Consider $n$ affine copies of a compact cylinder, say $S^{1}\times [-3,3]$ with top and botom, sitting inside $\mathbb{R}^{3}$. For each $n$ we may ask ourselves how to arrange the $n$ cylinders so ...
18 votes
1 answer
559 views

A search for a sequence of $6$-manifolds

How to construct closed, orientable, smooth, simply-connected $6$-manifolds such that $H^{*}(M,\mathbb{Z}) \cong \mathbb{Z}[a]/(a^{4})$ (Where $a$ is a generator of degree 2) satisfying $p_{1}(M) = n ...
1 vote
1 answer
562 views

Bijection homotopy class of maps and homomorphisms of fundamental group

I was told there was a bijection between $[X;BG]$, the set of homotopy types of maps from a topological space $X$ to the classifying space $BG$, and the set of group homomorphisms $Hom(\pi_1 (X), G)$. ...
8 votes
2 answers
530 views

Cohomology theories as colimits

I am looking for examples of cohomology theories that can be written as (filtered, or another nice class of) colimits of "simpler" functors, i.e. which $\{h^n : {\bf Top}^2 \to {\bf Ab}\}_n$ are such ...
4 votes
0 answers
97 views

What is the correct generalization of "sigma-free" to props?

This is a question about props, a generalization of operads (used to model operations with several inputs and several outputs). By forgetting the composition structure of an operad one obtains a so ...
49 votes
5 answers
8k views

Fundamental group as topological group

Background Let $(X,x)$ be a pointed topological space. Then the fundamental group $\pi_1(X,x)$ becomes a topological space: Endow the set of maps $S^1 \to X$ with the compact-open topology, endow the ...
2 votes
0 answers
2k views

How can one define "punctured torus" in Homotopy Type Theory? Is its fundamental group the free product of the integers with themselves?

Questions. Has the beautiful old idea, in part already known to Gauß, of a punctured torus surface (take, if you will, the classical set-theoretic definition as the meaning of the latter three words)...
5 votes
2 answers
298 views

3-folds with "simple" Betti numbers and positive Kodaira dimension

I am interested to know an example of a simply connected smooth projective 3-fold $X$ (over $\mathbb{C}$) satisfying the following two constraints: $X$ has the same Betti numbers as $\mathbb{C}\...
6 votes
1 answer
321 views

Minimal model (resolution) for a specific colored operad

We know that for the operad $As:=\mathcal{F}(\mu)/(\mu\circ_1\mu-\mu\circ_2\mu)$, its minimal model is the free operad $\mathcal{F}(E)$ where $E=\mathbb{k}\langle\mu_2,\mu_3,\dots,\mu_n,\dots\rangle$ ...
48 votes
6 answers
7k views

Total spaces of $TS^2$ and $S^2 \times R^2$ not homeomorphic

Hello, I'm looking for an invariant to distinguish the homeomorphism types of homotopy equivalent spaces. Specifically, how does one show that the total spaces of the tangent bundle to $S^2$ and the ...
8 votes
1 answer
1k views

semi flat connections

Let $L\to V$ be complex line bundle and $F_{t}:V\to V$, $t\in [0,1]$, be a loop of diffeomorphisms, $F_0=F_1=$ identity. For every $x\in V$, we get a loop $\gamma_x(t)=\{F_t(x)\}$ whose class in $\...
7 votes
1 answer
539 views

Continuous maps $f:S^n \to \mathbb{C}P^m$ with $f(x)\perp f(-x) $

Question 1: What is a complete classification of all positive integers $m,n$ with the following property: There is a continuous map $f:S^n \to \mathbb{C}P^m$ such that $f$ maps antipodal ...
9 votes
1 answer
145 views

Minimal approximations of surfaces by convex polygons

Suppose you want a collection of convex polygons in $3$-space such that, when you glue them together edge-to-edge, you obtain an orientable surface of genus $g$. What is the fewest number of polygons ...
3 votes
1 answer
660 views

Higher Euler characteristics (possible generalizations)

Let $X$ be projective and Gorenstein (over $\mathbb{C}$), of dimension $n$, then $\chi(\mathcal{O}_X)=(-1)^n\chi(\omega_X)$. Hence a "generalization": $\chi(\omega^{\otimes k}_X)$. I'd like ...
29 votes
4 answers
2k views

Geometric Interpretation of the Lower Central Series for the Fundamental Group?

For any group G we can form the lower central series of normal subgroups by taking $G_0 = G$, $G_1 = [G,G]$, $G_{i+1} = [G,G_i]$. We can check this gives a normal chain $G_0 > G_1 > ... > ...
4 votes
0 answers
208 views

Two definitions of central extensions of simplicial groups

This is a cross-post from MSE. Let $\overline W$ be a classifying space functor on $\mathrm{sGrp}$ with $G$ be a corresponding left adjoint (Kan's loop group). Def 1 : a sequence of maps $A\to E\...
24 votes
1 answer
1k views

Finite-order self-homeomorphisms of $\mathbf{R}^n$

Consider the $n$-dimensional euclidean space $\mathbf{R}^n$. A self-homeomorphism $\phi:\mathbf{R}^n\to \mathbf{R}^n$ is said to be of finite order if $\phi^m = \mathrm{id}_{\mathbf{R}^n}$ for some ...
3 votes
1 answer
347 views

Is Leray's theorem on commutative Hopf algebras proven in Milnor-Moore?

Question 1. Is a correct proof of Leray's theorem (the one that says that a connected graded Hopf algebra $H$ over a field of characteristic $0$ is isomorphic as an algebra to the symmetric ...
8 votes
1 answer
685 views

Topological fraction rings and fields

Linked to this question and as a sequel to my answer of it. Let $R$ be a topological (commutative, unital) ring and set $S$ be a submonoid of $(R,\times,1_R)$. Let $$ s_{frac}\ :\ R\times S\to S^{-...
9 votes
0 answers
647 views

Motivic Galois theory and Betti realizations?

Why Motivic Galois groups are defined with Betti realizations? (In fact Absolute Galois groups can be defined in this way (with Betti realizations), why they are so related?).
0 votes
1 answer
100 views

Continuous orthogonal preserving maps between projective space

Is there a continuous map $f:\mathbb{C}P^n \to \mathbb{C}P^m$, for some $n>m$ which preserve orthogonality?Namely $x\perp y \implies f(x) \perp f(y) $? If yes, are there two non homotopic ...
122 votes
12 answers
11k views

Spectral sequences: opening the black box slowly with an example

My friend and I are attempting to learn about spectral sequences at the moment, and we've noticed a common theme in books about spectral sequences: no one seems to like talking about differentials. ...
5 votes
1 answer
233 views

Equivariant cohomology ring is an integer domain

Let $G$ be a connected compact Lie group and let $V$ be a complex $G$-representation. Denote by $\mathbb{P}(V)$ the projectivization of the vector space $V$. I would like to ask a couple of questions ...
2 votes
1 answer
224 views

A line bundle on the wedge sum of spheres associated to a polynomial $P(z)\in \mathbb{C}[z]$

Assume that $P\in \mathbb{C}[z]$ is a polynomial of degree $n$ with $n$ distinct roots $z_1,z_2,\ldots,z_n$. We identify $\mathbb{R}^3$ with $\mathbb{C}\times \mathbb{R}$. Put $a_i=(z_i,0)$. Then ...
0 votes
1 answer
629 views

A possible proof of the Borsuk Ulam theorem without "Homology-Cohomology"

Assume that $n>1$. The configuration space of $S^n$ is defined as follows $$M_n=\{(x,y)\in S^n\times S^n\mid x \neq y\}$$ We have two questions: 1.Is there a continuous function $f:M_n ...
12 votes
1 answer
216 views

A variant of $\ell^2$-cochains

Suppose $X$ is an infinite countable CW complex which satisfies the following property: for all $k$-cells $e$, the number of $(k+1)$-cells incident to $e$ is at most $c_k$, where the latter is some ...
3 votes
0 answers
232 views

Topological space with non-finitely-generated second homotopy group [closed]

Let $M$ be a connected, simply connected topological space, and $\pi_2(M)$ its second homotopy group. I am searching for some work/references/papers/... done about such a space with $\pi_2(M)$ which ...
3 votes
0 answers
223 views

How can I find the differential in the Serre spectral sequence for this sphere fibration?

Consider the assocaited sphere bundle $$S(E) \to \mathbb{P}^n$$ for the vector bundle $\mathcal{O}(k)\oplus \mathcal{O}(l) \to \mathbb{P}^n$. Is there a way to determine the differentials $$ d_4^{p,m}:...
9 votes
2 answers
694 views

Section of the homology functor on spectra

Consider the (reduced) homology functor $H_*$ from the category of spectra to the category of graded Abelian groups. I wanted to know whether there is a "section" of this functor, i.e., a functor $F$ ...
14 votes
3 answers
2k views

What are Homotopy rings good for?

In his paper, Note on quasi-Lie rings, P. J. Hilton defines the (non-associative) Homotopy ring of a pointed space $X$ as$$\bigoplus_{n>1}\pi_n(X)$$ where the Whitehead product $\pi_m(X)\times\pi_n(...
13 votes
1 answer
373 views

Homotopy groups of $MO(2)$

Have there been any computations of the higher homotopy groups of $MO(2)$, the Thom space of the universal $O(2)$-bundle? Thom himself noted in his landmark 1954 paper that $$ \pi_1(MO(2))=0,\quad \...
2 votes
0 answers
134 views

Fibers of double suspension and fundamental class maps

We can take a fundamental class $S^3 \rightarrow K(\mathbb{Z}, 3)$ and consider homotopy theoretic fiber of this map $\bar{S}^3$. Then we obtain the following homotopy fiber sequence $\Omega^2\bar{S}^...
4 votes
0 answers
85 views

On decidability of a homeomorphism with a prescribed pushforward

This is a refinement of my older question A homeomorphism with a prescribed action on the fundamental group - decidable or not? The problem under considreation is the following. Let $M,N$ be two ...
11 votes
3 answers
2k views

Determining homotopy classes [T^2, RP^2]

So I've been interested in computing homotopy classes of maps $T^2=S^1\times S^1$ to $\mathbb{R}\mathbb{P}^2$. So first, we can decompose $T^2$ into a cell complex with one zero cell, $S^1\vee S^1$ ...
15 votes
2 answers
899 views

Semidirect product decomposition of the Borromean rings group

Let $X=S^3\setminus B$ be the link complement of the Borromean rings.     (source) Then $G=\pi_1(X)$ has a presentation of the form $$ G = \langle \; a,b,c \mid [a,[b^{-1},c]],\; ...
6 votes
0 answers
188 views

Existence of complementary pairs of foliations on spheres

Let $M$ be an $n$-manifold, $0\leq k\leq n$. We define a $(k,n-k)$-bifoliation on $M$ to be a pair $(\mathscr{E},\mathscr{F})$ consisting of ($C^\infty$ nonsingular) foliations $\mathscr{E},\mathscr{F}...
9 votes
1 answer
272 views

A homeomorphism with a prescribed action on the fundamental group - decidable or not?

I am curious if the following topological problem is decidable. Let $M,N$ be two closed manifolds. Given a group isomorphism $p: \pi_1(M)\to \pi_1(N)$, is there a homeomorphism $\phi: M\to N$ such ...
26 votes
1 answer
2k views

A refinement of Serre's finiteness theorem on unstable homotopy groups of spheres

Serre's finiteness theorem says if $n$ is an odd integer, then $\pi_{2n+1}(S^{n + 1})$ is the direct sum of $\mathbb{Z}$ and a finite group. By looking at the table of homotopy groups, say on ...
7 votes
1 answer
491 views

Homotopy groups of even-dimensional spheres

I need to understand the structures of $\pi_{4n}(S^{2n})$ and $\pi_{r}(S^{2n};\mathbb{Z}_k) (r\geq 4n-1)$, where the latter group is the homotopy group with coefficient defined as $[P^r(k), S^{2n}]$. ...
1 vote
1 answer
114 views

On graph imbedding genus clarification

Given a graph the minimum genus $g$ is the minimum number of handles needed so that there an imbedding of the graph on the surface with no edge crossings. If the graph is of genus $g$ then is there ...
17 votes
3 answers
1k views

Homology generated by lifts of simple curves

Let $\Sigma$ be a compact connected oriented surface and $p:\tilde{\Sigma}\to\Sigma$ a finite regular cover. Consider the set $\Gamma$ of simple closed curves on $\tilde{\Sigma}$ obtained as a ...
4 votes
0 answers
452 views

Are homotopy equivalent submanifolds also cobordant?

Let $M$ be a manifold, and let $A$ and $B$ be two submanifolds of $M$ which are diffeomorphic to each other. I'll say $A$ and $B$ are homotopy equivalent in $M$ if there is a $C^1$ function $f:A \...
4 votes
1 answer
256 views

Invariance of Khovanov homology under first Reidemester move

I am studying Khovanov homology from five lectures on Khovanov homology and I want to try to show Khovanov homology is invariant under first Reidemester move but I cannot understand how we can write ...
11 votes
1 answer
1k views

Necessary conditions for cofibrancy in global projective model structure on simplicial presheaves

Consider the global projective model category of simplicial presheaves on some category (the category of smooth manifolds is particularly interesting to me). In Section 9.1 of Dugger's paper “...
3 votes
0 answers
173 views

"Extending scalars" for (motivic) ring spectra and for modules over them: are the corresponding Moore spectra highly structured ring objects?

Let $S$ be a (motivic symmetric) ring spectrum (more generally, one can possibly consider a commutative ring object in any symmetric stable model category); let $R$ be an associative commutative ...
4 votes
0 answers
166 views

Bott Periodicity: Morse and K-Theory [duplicate]

Is there an easy way to see the equivalence of the two statements of Bott periodicity via K-Theory and the original Morse Theory proof? So, $$BU \times \mathbb{Z} \simeq \Omega^2BU$$ and $$K(X)\...
7 votes
1 answer
520 views

Naive equivariant transfer

Given a group $G$, a $\mathbb Z$-graded cohomology theory $E^*_G$, and a $n$-sheeted covering $p\colon X \to Y$, I would like a transfer map $$p_!\colon E^*_G X \to E^*_G Y$$ satisfying $$\require{...
9 votes
0 answers
201 views

Eilenberg-Moore spectral sequence in etale cohomology?

Let $X,Y \rightarrow S$ be schemes over an algebraically closed field $k$. (Actually I'm interested in the case where they are stacks, but I'll ignore that for now.) The vague form of my question is: ...
14 votes
1 answer
872 views

Contractible topological groups

Does there exist a Hausdorff topological group which is contractible and of finite covering dimension but which is not homeomorphic to $\mathbb{R}^n$ for some $n$?

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