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Questions tagged [at.algebraic-topology]

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

3
votes
1answer
136 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 ...
6
votes
0answers
109 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 ...
10
votes
1answer
251 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 ...
13
votes
1answer
211 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
110 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|...
9
votes
1answer
313 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
191 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
59 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 \...
6
votes
0answers
108 views

Moduli space of flat connections of Lie group over a 2-torus

We know that the moduli space of SU($N$) flat connections over a torus, is equivalent to a complex projective space $\mathbb{P}^{N-1}$ Namely, $$ M_{\rm flat}=\mathbb{E} / {S}_N = \mathbb{P}^{N-1} $$ ...
9
votes
1answer
305 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
110 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
96 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
141 views

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

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
149 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
98 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 ...
3
votes
1answer
229 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
127 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 ...
15
votes
2answers
504 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
383 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
210 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
26 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
122 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
129 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
106 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 ...
14
votes
1answer
367 views

A parametric version of the Borsuk Ulam theorem

Is there a topological space $X$, which is not a singleton, and satisfies the following property? For every continuous function $f: X\times S^2\to\mathbb{R}^2$ there exist a point $x\in S^2$ such ...
7
votes
0answers
185 views

Different definitions of Stiefel-Whitney classes

It is quite easy to show that different definitions of the Stiefel-Whitney classes agree by showing that they satisfy the well-known axioms. Nevertheless I have been asking myself wether one can prove ...
15
votes
1answer
427 views

Is a spectrum with trivial homology groups trivial?

If $X$ is a spectrum with trivial (integer-valued) homology groups, does it have to be weakly-equivalent to a point? This is easy to prove for connective spectrum, as a Hurewitz-type argument is then ...
5
votes
1answer
204 views

An operad-like structure, is there a name for it?

Here is an example which I'd like to have a name for. Let $P$ be a compact smooth manifold of dimension $p$, possibly with non-empty boundary. Define $E(k,P)$ to be the space of smooth (codimension ...
4
votes
1answer
191 views

Link between homotopy equivalence of simplicial sets and categorical equivalences

In Higher Topos Theory, a map $f: S \rightarrow T$ of simplicial sets is a categorical equivalence if after applying the functor $\mathfrak{C}[-]$ we have an equivalence of simplicial categories. In ...
4
votes
0answers
172 views

homotopy type of box topology.

Suppose that $X$ is weakly equivalent to a point. Let $I$ be a set. Does $\prod_{i\in I}X$ weakly equivalent to a point, where $\prod_{i\in I}X$ is equipped with box topology ?
7
votes
1answer
159 views

Differentials in Weil model for equivariant cohomology

Why should we define the differential in Weil model as follows? I could understand $\sum_{j,k} c_{jk}^i \theta_j \wedge \theta_k$ plays a role in the formula because it is the dual of the structure ...
5
votes
0answers
74 views

Group cohomology of “twisted” projective SU(N) with various coefficients

Given a group $$ G= PSU(N) \rtimes \mathbb{Z}_2, $$ where $PSU(N)$ is a projective special unitary group. Say $a \in PSU(N)$, $c \in \mathbb{Z}_2$, then $$ c a c= a^*, $$ which $c$ flips $a$ to its ...
5
votes
1answer
191 views

Inequality number of facets simplicial complex

In a recent preprint, Adiprasito proves that if $\Delta$ is a simplicial complex of dimension $d$ that can be embdedded in a $2d$-dimensional homology sphere (say $\Sigma$) that satisfies a version of ...
15
votes
1answer
558 views

Which spaces have trivial K-theory?

What is known about spaces $X$ with the property that $K^*(\text{point})\to K^*(X)$ is an isomorphism? The same question for $K$-homology $K_*(X)\to K_*(\text{point})$; I don't even know whether ...
5
votes
2answers
616 views

Algebra for algebraic topology

My research is in analysis, but it moved to the area that requires algebraic topology. I have some working knowledge in that area, but I always feel that I am on a shaky ground and I need to go back ...
4
votes
1answer
175 views

Homotopy type of smooth manifolds with boundary

It seems very likely to me that every smooth connected $n$-dimensional manifold with non-empty boundary has the homotopy type of a $(n-1)$-dimensional CW complex. Is that true and how to prove it? (...
6
votes
0answers
130 views

A property of the Anderson dual of the sphere spectrum

Let $X$ be a spectrum, and let $I_{\mathbb{Z}}$ be the Anderson dual of the sphere. Using the definition of $I_{\mathbb{Z}}$, it is easy to get short exact sequences $$0\rightarrow Ext(\pi_{n-1}X, \...
11
votes
1answer
689 views

The homology of the orbit space

Suppose we have an acyclic group $G$ and let $X$ be a contractible CW-complex such that $G$ acts freely on $X$ (we do not suppose that the action is proper). Is there a way to understand the homology ...
23
votes
1answer
346 views

Modern survey of unstable homotopy groups?

Toda no doubt made some big strides when computing unstable homotopy groups $\pi_{n+k}(S^n)$ for $k < 20$ which his collaborators later improved upon. The methods he used are documented in his ...
1
vote
0answers
48 views

Filtrations of spectra related to cellular ones and singular homology

I would like to study filtrations of spectra (i.e., objects of the "topological" stable homotopy category $SH$; a filtration of a spectrum $E$ is a sequence of compatible maps $E_{\le i}\to E$) whose ...
0
votes
0answers
144 views

deformations of Lie algebroids

In the paper "Deformations of Lie brackets"- by I. Moerdijk and M. Crainic, they define deformations of a Lie algebroid as follows: Let $A$ be a fixed vector bundle, and $I\subset \mathbb{R}$ and ...
2
votes
0answers
138 views

Characteristic classes in term of cocycles

Giving a vector (principal) bundle is equivalent to give a family of cocycles ${g_{\beta \alpha}: U_\alpha\cap U_\beta \to G}$ where $G$ is the structure group of the bundle. Chern classes are ...
6
votes
1answer
158 views

Precise reference for the equivalence of $E_n$ algebras and locally constant factorization algebra?

I've seen the following theorem attributed to Lurie: Theorem. There is an equivalence of $(\infty,1)$-categories between $E_n$ algebras and locally constant factorization algebra on $\mathbf{R}^n$. ...
1
vote
0answers
89 views

What are the “ouverts convenables” used to prove Brieskorns lemma?

In the proof of Brieskorns lemma, see 3.3 here, Brieskorn mentions that we take "ouverts convenables" satisfying some properties, but, as far as I can tell, never specifies what these opens actually ...
5
votes
0answers
119 views

DG-Modules over CDG-algebras in the sense of rational homotopy theory

I don't know if this question is elementary or not. Suppose we have a rationalization $X_\mathbb{Q}$ of a simply connected topological space $X$. Then we can construct a CDG-algebra - a minimal ...
7
votes
1answer
335 views

Inverting homotopy groups of spectra

Let $X$ be a spectrum. Is there a canonical construction/functor that would associate to this spectrum, an inverse spectrum $X'$, in the sense that $$\pi_*(X)\cong \pi_{-*}(X')?$$ To be more precise, ...
33
votes
1answer
904 views

Does there exist a continuous 2-to-1 function from the sphere to itself?

I am interested in the following question: Does there exist a continuous function $f:S^2\to S^2$ such that, for any $p\in S^2$, $|f^{-1}(\{p\})|=2$? I suspect the answer is no, but I don't know ...
4
votes
1answer
148 views

Is the category of rational Lie algebras monoidal?

I hate to ask such a naive question, but here goes. Suppose $A$ and $B$ are rational Lie algebras, i.e. rational vector spaces together with a bracket. Then, $A\otimes_{\mathbb{Q}} B$ is a rational ...
1
vote
0answers
101 views

Whitehead Theorem for maps

Let us consider two simply-connected CW complexes. Combining the theorems of Whitehead and Hurewicz we have that a map between them is an equivalence if and only if its induced map on integral chains ...
2
votes
0answers
60 views

resolution of differential graded algebras.

Suppose that we have tree maps of differential graded algebras $A\rightarrow B$, $A\rightarrow C$ and $A\rightarrow D$ such taht $A\rightarrow C$ is a trivial cofibration of differential graded ...