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Questions tagged [at.algebraic-topology]

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

4
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0answers
194 views

Comparing real topological K-theory and algebraic K-theory

Does there exist an integer $0<i<8$ with the following property: for any commutative unital ring $R$, there exists a compact Hausdorff space $X$ such that $KO^i(X)\approx K^i_{alg}(R)$? P.S.: ...
5
votes
0answers
124 views

Generalizing the formula between Wu class and the Steenrod square

I know that on the tangent bundle of $M^d$, the corresponding Wu class and the Steenrod square satisfy $$ Sq^{d-j}(x_j)=u_{d-j} x_j, \text{ for any } x_j \in H^j(M^d;\mathbb Z_2) . \tag{eq.1}$$ ...
1
vote
0answers
50 views

$n$-point map $S$ from $k$-manifold $X$ to $\mathbb{R}^k$

It is known that for every continuous map $M:S^1\to \mathbb{R}$ there are infinitely many double points. Also by the Borsuk-Ulam theorem, this is true for each continuous map $N:S^n\to \mathbb{R}^n$, $...
6
votes
1answer
276 views

A question about Poincare duality

Let $k$ be a field. Let $C$ be a small category and assume that for every $i\geq 0$ we have a functor $H^i:C\rightarrow FinDimVect_k$. Assume that there is a function $dim:Obj(C)\rightarrow \mathbb{Z}...
2
votes
1answer
108 views

Leray-Serre spectral sequence for projective bundles

Let $\mathcal{E} \rightarrow X$ be a complex vector bundle of rank $r+1$ and let $F=\mathbb{P}^r \rightarrow E = \mathbb{P}\mathcal{E}\rightarrow X$ be the associated projective bundle. We know that ...
5
votes
1answer
276 views

Are framed manifolds cubulatable?

Let's say an $n$-manifold is cubulated if it is glued out of cubes $[0,1]^n$ in a way that looks locally like the standard cubulation of $\mathbb R^n$. For instance, the face $[0,1]^{k-1} \times \{1\} ...
4
votes
0answers
95 views

Twisted spin-bordism invariant and a possible Postnikov square from $d=2$ to $d=5$

This is a follow up more advanced question following Twisted spin bordism invariants in 5 dimensions. We follow the definitions in the earlier post. I had discussed my computation of $$ \Omega_5^{...
3
votes
1answer
133 views

Cofibrations of functors

Let $\cal M$ and $\cal N$ be model categories, $S,T:\cal M\to N$ functors, and $\alpha:S\to T$ a natural transformation. Say that $\alpha$ is a <blank> cofibration if for any cofibration $i:A\to B$...
7
votes
1answer
181 views

Twisted spin bordism invariants in 5 dimensions

[Note]: My question will be a bit long. So, first, thank you for your careful reading, generous comments, helps and answers, in advance! The spin $G$-bordism invariant can be twisted in the way that ...
5
votes
0answers
60 views

Alexander-Whitney for cyclic objects

What is known about the extension of the AW map from simplicial to cyclic Abelian groups? Homological perturbation theory implies there is an A infinity-like sequence of maps, but is it known ...
7
votes
1answer
204 views

Example of a finite group $G$ with low dimensional cohomology not generated by Stiefel-Whitney classes of flat vector bundles over $BG$

In Stiefel-Whitney classes of real representations of finite groups, J. Algebra 126 (1989), no. 2, 327–347, Gunawardena, Kahn and Thomas dealt with the question, whether the cohomology ring $H^...
3
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0answers
151 views

Is there a natural homotopy inverse to the map $∣Sing(X)∣\rightarrow X$

Let $Sing:C \rightarrow SSet$ be the functor sending a CW complex to its singular complex (a simplicial set). Let $∣-∣: SSet \rightarrow C$ be the geometric realization functor. For every $X$ we ...
11
votes
0answers
485 views

How do topological automorphic forms fit into homotopy theory and what makes them interesting?

Topological automorphic forms (TAF) were introduced by Mark Behrens and Tyler Lawson in 2007, being to Shimura varieties what topological modular forms (TMF) is to the moduli stack of elliptic curves. ...
2
votes
1answer
192 views

Does the Hopf construction work for $S^0$?

Given a $0$-connected $H$-space $X$, we can prove that left and right multiplication are (weak) homotopy equivalences. This allows us to construct the Hopf fibrations for $S^1$, $S^3$ and $S^7$. I ...
5
votes
0answers
79 views

homotopy classes of maps from $S^3/\Gamma$ to $X$

Given a finite subgroup $\Gamma <SO(4)$ and a topological space $X$ whose $i$th homotopy group $\pi_i(X)$ are known for any $i$, is there any way to compute $[S^3/\Gamma,X]$? Here $[S^3/\Gamma,X]$...
11
votes
2answers
2k views

Why not a Stacks project for Homotopy Theory?

The lack of resources bridging the gap between what one finds in Hatcher's algebraic topology text and modern research on homotopy theory has been brought several times before on MathOverflow [1, 2, 3]...
5
votes
1answer
196 views

Classifying space of semidirect product of groups

Assume that $G$ and $H$ are two groups and $G\rtimes _\phi H$ is their semidirect product. My question is, how does the classifying space $B(G\rtimes_\phi H)$ of $G\rtimes _\phi H$ relate to $BG$ and $...
12
votes
1answer
497 views

Does the Grothendieck construction satisfy Fubini's thorem

Suppose we are given a functor $F:(A\times B)^{\operatorname{op}}\to \operatorname{Set}$. It's well-known that the Grothendieck construction in this case evaluates as $\int_{A\times B}F = (A\...
6
votes
1answer
265 views

Model structure on the category of topological groups

Consider the category $TopGr$ of topological groups. I want to know that this is a model category (can one understand its model structure by understanding a model structure on the category of enriched ...
3
votes
1answer
134 views

Approximation of homotopy avoiding a point in $\mathbb{R}^3$

For a proof that $\mathbb{R}^3\setminus \mathbb{Q}^3$ is simply connected using Baire category theorem I need to approximate an homotopy $H : [0,1]\times \mathbb{S}^1 \to \mathbb{R}^3$ from a loop $\...
3
votes
0answers
70 views

Torus bundle over spheres

I was wondering what is the classification of all torus bundles over spheres? That is, to classify the fibration $$ T^m \hookrightarrow M \to S^n. $$ It is well known that if $n=1$, all fibrations ...
4
votes
0answers
87 views

Second homology of finitely presented group with free abelianisation

It is known that for a presented group $G=F/N$ we have $$H_2(G;\mathbb{Z}) \cong \frac{[F,F]\cap N}{[F,N]}.$$ In general, the right side seems to be difficult to calculate. I am in the special ...
7
votes
0answers
242 views

About Kan-Thurston theorem

The Kan-Thurston theorem obsesses me recently, which is proved by Daniel Kan and William Thurston, Every connected space has the homology of a K(π,1), Topology Vol. 15. pp. 253–258, 1976. Later, there ...
6
votes
0answers
117 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
126 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
189 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
114 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
92 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 ...
7
votes
1answer
401 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
141 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
61 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
1answer
237 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 ...
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votes
0answers
82 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
86 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
82 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
113 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
152 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 ...
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vote
0answers
97 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
46 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
262 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
179 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
142 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
549 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 ...
11
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 ...
9
votes
0answers
159 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
149 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
119 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
568 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
270 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
837 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 ...