# Questions tagged [at.algebraic-topology]

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

**6**

votes

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**4**answers

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

**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 ...

**10**

votes

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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 ...