# Questions tagged [at.algebraic-topology]

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

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### 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**

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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

**0**answers

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**

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**1**answer

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

**1**answer

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

**1**answer

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**

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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

**1**answer

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**

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**1**answer

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**

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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**

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**1**answer

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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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**

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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

**1**answer

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**

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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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

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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

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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**

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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**

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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**

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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

**2**answers

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

**1**answer

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

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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

**1**answer

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

**1**answer

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.
...

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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

**1**answer

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 ...

**2**

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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

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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?...

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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**

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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

**1**answer

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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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. ...

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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 ...

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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 ...

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votes

**1**answer

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**

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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

**4**answers

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

**2**answers

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**

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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**

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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**

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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**

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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

**1**answer

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

**1**answer

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 ...