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Questions tagged [at.algebraic-topology]

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

6
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0answers
124 views

Survey on homological stability

Background: homological stability is the phenomenon that for many natural families of groups $G_0 \to G_1 \to G_2 \to \dots$, the group homology $H_i(G_n)$ stabilizes for $n \gg i$. This is e.g. the ...
2
votes
0answers
130 views

Deriving category of quadratic functors

Recall that a fuctor $T: \cal C \to \cal A$ from pointed small $\cal C$ with coproducts to additive and Karoubian — or, even better, abelian $\cal A$ is called quadratic if kernel of sum of obvious ...
9
votes
2answers
199 views

Is the poset of affine subspaces of a vector space highly connected?

The question is in the title. Fix a field $k$. Let $P_n$ be the poset of proper nonempty affine subspaces of $k^n$ under inclusion. The geometric realization $|P_n|$ is $n$-dimensional. Is it $(n-...
4
votes
1answer
122 views

Representing simplicial homotopy classes cubically?

Let $(X,x_0)$ be a pointed simplicial set. Assume if you like that $X$ is the nerve of a category but do not assume that $X$ is a Kan complex. Because $Ex^\infty X$ is a Kan complex, every homotopy ...
5
votes
0answers
100 views

symplectic sum of two copies of $Bl_{p}(\mathbb{CP}^{2})$

Let $M^{4}= \mathbb{P}(\mathcal{O} \oplus \mathcal{O}(1))$ and $\omega$ the symplectic form on $M^{4}$ given by the anticanonical polarisation. Suppose we form a symplectic (Gompf) sum of two copies ...
8
votes
1answer
464 views

Last Results in Chromatic Homotopy Theory

I started a PhD in Chromatic Homotopy three years ago, but I had to quit it due to personal reasons after one year. Last week I was looking at all my notes from that period and I was wondering where ...
2
votes
1answer
147 views

Rationalization of topological groups and degree maps

Suppose $G$ a finitely generated nilpotent topological group and we consider its rationalization $G_\mathbb{Q}$. This space may fail to be a topological group, but it's always a group-like H-space. ...
3
votes
0answers
62 views

Homotopy type of hyperplane arrangements intersected with real subspaces

The homotopy type, and especially the higher homotopy groups of complement of hyperplane arrangements in $\mathbb{C}^n$ has been extensively studied, for example Falk and Randell - On the homotopy ...
7
votes
0answers
113 views

Relationship between the p-radical subgroups and the parabolics in a BN-pair generality

A theorem of Quillen says that if $G$ is a finite Chevalley group over characteristic $p$, then the poset $\mathcal{A}_p(G)$ of nontrivial elementary abelian subgroups of $G$ is homotopy equivalent (I ...
7
votes
1answer
246 views

Maps into a Postnikov tower of the sphere

Suppose I have a CW complex $Y$ of dimension $n+2$ and let $X_{n+2}$ be the third non-trivial Postnikov stage of $S^n$ (i.e. there is a map $S^n \to X_{n+2}$ which is an $(n+2)$-equivalence). We ...
2
votes
0answers
83 views

On the Puppe sequence for the cofibration induced by the inclusion of skeleta

Let $X$ be a finite CW complex and denote by $X_k$ the $k$-skeleton of $X$. We have the natural cofibration $$ X_k \to X \to C(i_k) $$ where $i_k\colon X_k \to X$ is the inclusion. The Puppe ...
2
votes
0answers
87 views

Space of non-vanshing sections path-connected?

Let $M$ be a path connected smooth manifold and $E$ be a vectorbundle over $M$ of rank at least two. My question is: Under which conditions is the space of global non-vanishing sections path connected?...
5
votes
0answers
86 views

Is there any work in topological data analysis on something like “Voronoi complexes”?

Given a finite set $X \subset \mathbb{R}^n$, we can of course construct the corresponding Čech or Vietoris-Rips filtration. At each level of this filtration the scale parameter is fixed and unrelated ...
2
votes
0answers
119 views

First cube theorem for homotopy cartesian squares?

Does the following statement, similar to the first cube theorem (http://dx.doi.org/10.4153/CJM-1976-029-0) hold?: -The left-hand and the rear face of a cube are homotopy cartesian (i. e. the left ...
7
votes
1answer
160 views

If a loopspace admits space-level power operations, is is a higher loopspace?

Let $G$ be a group. Suppose that the map $G \to G$, $g \mapsto g^r$ is a group homomorphism for every $r \in \mathbb N$. Then $G$ is abelian. Is this also true homotopy-theoretically? (In the ...
1
vote
0answers
102 views

Lift up characteristic class to chain complex

In derived category, there is a slogan, "cohomology is bad, chain complex is good". In the theory of characteristic classes, we could associate a vector bundle to cohomology classes of the base space. ...
4
votes
0answers
48 views

Differences among various index theories in critical point theory

Index theories help characterize critical points of functionals having certain symmetries. What are the differences (regarding problems they can be applied to) between for example these ones? the ...
14
votes
0answers
267 views

How does quotienting by a finite subgroup act on the framed-cobordism class of a group manifold?

Let $G$ be a connected simple connected compact Lie group, and $\Gamma \subset G$ a finite subgroup. Then (the underlying manifold of) $G$ can be framed by right-invariant vector fields, and this ...
5
votes
1answer
186 views

Is a $G$-cell complex always a $G$-CW complex?

I recall vaguely once reading that a cell complex—constructed like a CW-complex but without assuming the cells are appended in order of increasing dimension —"is" actually a CW-complex. I ...
2
votes
0answers
146 views

Splittings in the difference bundle construction of Atiyah-Hirzebruch

I'm reading the construction of difference bundle(generalized) in the paper of Atiyah-Hirzebruch(Analytic cycles on complex manifolds) There is one thing I cannot understand. The followings are in ...
6
votes
4answers
573 views

Homology sphere with $\mathbb{R}^3$ as the universal cover

Question. Is there a $3$-dimensional integer homology sphere whose universal cover is $\mathbb{R}^3$? I believe the answer is in the positive and I am looking for (precise) references. If not in ...
13
votes
2answers
3k views

Mistakes in Bredon's book “Topology and Geometry”?

I am preparing the notes for a course in Algebraic Topology, so I decided to borrow some of the material from the classical (and wonderful) book by G. Bredon Topology and Geometry. Looking at the ...
10
votes
0answers
174 views

Vertex algebras and factorization algebras

It is often said that vertex algebras are a special case of factorization algebras. In particular, in their book "FAs in QFT" Costello/Gwilliam construct a functor from a certain class of 2d "...
5
votes
0answers
156 views

making the group completion in homology sense unique via the plus construction

A paper by Mcduff and Segal justifies the following definition: A map of h-spaces $X \to Y$ is a group completion if the map is a localization on homology. In the paper they prove that when $X$ is a ...
3
votes
0answers
126 views

Cohomological indices VS cup-length, Lusternik-Schnirelmann category, betti sum in critical point set theory

Let $M$ be a (closed) manifold acted on by a compact Lie group $G$. For any characteristic class $\alpha \in H^*(BG)$, one can define a so-called cohomological index of $M$ as follows: $$\text{ind}_{\...
4
votes
0answers
595 views

How should one approach reading Spectral Algebraic Geometry by Lurie?

A question posed at the nForum asked for a roadmap to learn Lurie's Higher Topos Theory. This MathOverflow question asked for a roadmap to Lurie's Higher Algebra. Still another question asked for a ...
2
votes
1answer
277 views

Is there a theorem showing that de Rham homology is isomorphic to singular homology?

The only exposition of de Rham homology I've found is an appendix to Uranga and Ibanezs book on String Phenomenology. It was brief and gave only basic outline of how to construct this homology. Now ...
11
votes
1answer
880 views

How should one approach reading Higher Algebra by Lurie?

A question posed at the nForum asked for a roadmap to learn Lurie's Higher Topos Theory, including helpful sources other than HTT itself (to read along it) and information about which parts of HTT ...
8
votes
0answers
180 views

Generalize Wu formula to general Bockstein homomorphisms

The classical Wu formula claims that $$Sq^1(x_{d-1})=w_1(TM)\cup x_{d-1}$$ on a $d$-manifold $M$, where $x_{d-1}\in H^{d-1}(M,\mathbb{Z}_2)$. I wonder whether there is a generalization of the ...
11
votes
1answer
314 views

Finite complexes which are not Thom spectra

I'll be working in the stable world. It's an easy observation that any 2-cell complex (over the sphere) with bottom cell in dimension zero is a Thom spectrum: any such complex is the cofiber of some ...
14
votes
1answer
286 views

What is the value of $[S^3/G] \in \pi_3(Sphere)$ for a finite subgroup $G \subset SU(2)$?

Let $G\subset \mathrm{SU}(2)$ be a finite group. (These are famously classified through the McKay correspondence.) The Lie group framing of $\mathrm{SU}(2) = S^3$ descends to the quotient manifold $S^...
4
votes
0answers
123 views

Do the ternary braid groups arise in algebraic topology?

Let $TB_{n}$ be the group defined by the presentation with generators $t_{1},...,t_{n-2}$ and relations $t_{i}t_{i+1}t_{i+2}t_{i}=t_{i+2}t_{i}t_{i+1}t_{i+2}$ and $t_{i}t_{j}=t_{j}t_{i}$ whenever $|i-j|...
10
votes
1answer
330 views

Who is credited with the creation/invention of the cup product?

Who is credited with the creation/invention of the cup product? Wikipedia gives credit to several but I wasn't able to confirm.
11
votes
1answer
223 views

Which maps of simplicial sets geometrically realize to fibrations?

If $f:X\to Y$ is a Kan fibration of simplicial sets, then its geometric realization $|f| : |X|\to |Y|$ is (in some suitable convenient category of topological spaces, like compactly generated ones) a ...
-2
votes
1answer
74 views

Alternating property of H_2(T, Z)

Let us consider the torus $T = S^1_X \times S^1_Y$, where the former $S^1_X$ has the coordinate $X$, whereas the latter $S^1_Y$ has $Y$. We have an alternate property $dX \wedge dY = - dY \wedge dX \...
9
votes
1answer
380 views

Power operations from a Tate construction

In an action-packed three pages of Lurie's DAG-XIII: Rational and p-adic Homotopy Theory, section 2.2: Power Operations on $\mathbb{E}_{\infty}$-algebras, one finds a construction of the power ...
6
votes
0answers
116 views

Gluing $n$-homotopy equivalences

Let $f:X \rightarrow Y$ be a map between simplicial complexes. Let $C$ and $C'$ be subcomplexes of $Y$ such that $Y = C \cup C'$. Define $D = f^{-1}(C)$ and $D'=f^{-1}(C')$, so $D \cap D' = f^{-1}(C ...
4
votes
0answers
107 views

Different definitions of a structure on principal bundles

Let $P\to B$ be a principal $G$-bundle and $\psi:H\to G$ a homomorphism of topological groups. A $\psi$-structure for $P$ can be defined in two different ways. I am trying to prove their equivalence. ...
-1
votes
1answer
151 views

Alternate property of H^2(T, Z) [closed]

Let us take $T = S^1_X \times S^1_Y$, which is a torus, where the former $S^1_X$ has the coordinate $X$, whereas the latter $S^1_Y$ has $Y$. If we consider the generator $dX \wedge dY \in H^2(T, {\Bbb ...
5
votes
1answer
160 views

Does profinite completion commute with mapping spaces?

Does there exist a prime number $p$ and a smooth complex projective variety $X$ such that $F_{\infty p}\mathrm{Map}(B\mathbb{Z}/p\mathbb{Z}, X)$ is not weakly homotopy equivalent to $\mathrm{Map}(B\...
0
votes
1answer
99 views

On constructing free action of the cyclic group $\Bbb Z/p \Bbb Z$ on $\prod_i S^n$($n$ is odd) which is not conjugate to the usual action.

Let $ p$ be an odd prime. Can we construct a free action of the cyclic group $\Bbb Z/p\Bbb Z$ on $S^n \times \cdots \times S^n$($n$ is odd), which is not conjugate to the free action given by ...
4
votes
1answer
273 views

Splitting of $H\mathbb{Z}$-module spectra

It is classical result of Adams that every $H\mathbb{Z}$-module spectra splits as a wedge of Eilenberg-MacLane spectra. Let me briefly recall what he writes about the proof. Let $M$ be an $H\mathbb{Z}...
5
votes
1answer
143 views

Does the cyclic group $\Bbb Z/4 \Bbb Z$ acts freely on $S^{2k} \times \Bbb CP^n$?

I was wondering whether the cyclic group $\mathbb Z/4\Bbb Z$ acts freely on $S^{2k} \times \Bbb CP^n$ where $n>1$? It seems to me that it does not act freely. In case it acts freely then the ...
16
votes
2answers
631 views

revisiting $THH(\mathbb{F}_p)$

Reading through Bhatt-Morrow-Scholze's "Topological Hochschild Homology and Integral p-adic Hodge Theory" I encountered the following statement. We use only “formal” properties of THH throughout ...
10
votes
1answer
412 views

Serre spectral sequence for de Rham cohomology

Suppose we a given a fibration of manifolds $p\colon E\to M$ with a path connected fiber $F$ and simply connected $M$, then we have the Serre spectral sequence with $$ E_2^{p,q} = H^p(M,\underline{H^...
5
votes
1answer
242 views

Conversion formula between “generalized” Stiefel-Whitney class of real vector bundles: O(n) and SO(n)

$O(n)$ is an extension of $\mathbb{Z}_2$ by $SO(n)$, $$1\to SO(n) \to O(n)\to \mathbb{Z}_2 \to 1.$$ Below we denote the Stiefel-Whitney class of real vector bundle $V_G$ of the group $G$ as: $$ w_j(...
0
votes
0answers
29 views

dual and intersection of a simplex

In a triangulation $\Gamma$ of a (oriented) 2-manifold, consider a 2-simplex labeled by ($123$), where $1,2,3$ denote the order of vertices. Consider the dual $\Gamma^*$ of $\Gamma$, and then denote ...
3
votes
0answers
128 views

Do profinite completion and homotopy fixed points commute?

Let $X$ be a separated integral normal scheme of finite type over $\mathbb{C}$. It is my understanding that $\mathbb{Z}/2$ acts on the homotopy type of $X(\mathbb{C})$ and its Sullivan 2-profinite ...
7
votes
1answer
134 views

Set of Jones polynomials as the knot varies

Is a characterization known for the set of Laurent polynomials arising as the Jones polynomial of some knot? More generally, is such a characterization known for any of the famous knot polynomials?
2
votes
0answers
113 views

$E_\infty$-algebras and Tor-unital rings

Recall that a non-unital ring $R$ is called Tor-unital if $Tor^1_{R_+}(\mathbb Z,\mathbb Z) \cong 0$ where $R_+$ is the unitalization of $R$. See e.g. https://arxiv.org/pdf/1610.04998.pdf. If $R$ is ...