Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [at.algebraic-topology]

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

14
votes
1answer
388 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
198 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
449 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
221 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
207 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
174 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
203 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
79 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 ...
6
votes
1answer
235 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
588 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
657 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
180 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
137 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
799 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 ...
24
votes
1answer
459 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
52 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
155 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
140 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
174 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
132 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
342 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
946 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
155 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
106 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
62 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 ...
13
votes
0answers
338 views

How well-defined is $\bar\kappa$ in the stable $20$-stem?

The $2$-completed stable $20$-stem $\pi_{20}(S)_2$ is cyclic of order $8$. Mimura and Toda (1963, Lemma 15.4) mr=157384 show the existence of a class $\bar\kappa_7 \in \pi_{27}(S^7)$ whose stable ...
6
votes
1answer
111 views

Number of connected components of degree 2 affine algebraic varieties

Suppose an algebraic variety $V$ is given as the solutions to $q$ polynomial equations of degree $\le k$ with real coefficients $$p_1(x_1,\dots,x_m)=0,\dots,p_q(x_1,\dots,x_m)=0$$ for $x\in\mathbb R^m$...
7
votes
0answers
132 views

Cohomology of little disks and dg algebras over $\mathbb{F}_p$

This a alternative form of the question I posted some time ago. We, the people who don't know topology, are told that in characteristic p the formalizm of DG algebras is not quite adequate for ...
4
votes
0answers
56 views

Formality of fixed points in the equivariant localisation

Let $X$ be a complex algebraic variety equipped with an algebraic $\mathbf{C}^{\times}$-action. The Borel construction gives a map $f: \mathrm{E}\mathbf{C}^{\times}\times^{\mathbf{C}^{\times}}X\to \...
11
votes
0answers
214 views

3-fold of general type homeomorphic to rational 3-fold

Is there a smooth (complex projective) 3-fold of general type which is homeomorphic (in the complex topology) to a rational $3$-fold? I am aware of such examples in complex dimension $2$, for ...
5
votes
1answer
81 views

On the existence of a domination map of a finite polyhedron

A continuous map $d:X\to A$ is called domination if there exists a map $u:A\to X$ so that $d\circ u\simeq 1_A$. Is there a domination map $d:P\to P$ of a finite polyhedron $P$ so that $d$ is not a ...
11
votes
0answers
273 views

If an additive group of $\Bbb R^2$ contains a smoothly deformed circle, is it necessarily all of $\Bbb R^2$?

It can be shown that if an additive subgroup of $\Bbb R^2$ contains the unit circle, then it is necessarily all of $\Bbb R^2$. Does this also hold for a suitably smoothly deformed unit circle? ...
2
votes
0answers
121 views

Do we have estimate like $\int_\gamma \alpha \le |\alpha| \cdot |\gamma|$?

Let $(X,g)$ be a compact smooth Riemannian manifold. It is known that $H^1(X, \mathbb R)\cong \mathrm{Hom} (\pi_1(X), \mathbb R)$, namely there is a natural pairing $$ H^1(X) \times \pi_1(X) \to \...
7
votes
1answer
171 views

Invariants in relative cohomology and compact support cohomology of the quotient

Let $\cal H$ be the Poincare upper half-plane and $\overline {\cal H}$ the union of $\cal H$ with the set of cusps $\bf P^1 (\bf Q)$, provided with its usual topology. Let $\Gamma$ a congruence ...
10
votes
1answer
229 views

What does the classifying space of a topological monoid classify?

The classifying space $BG$ of a topological group $G$ classifies principal $G$ bundles. I have come to appreciate this. I hope the following question is appropriate for MathOverflow: What does the ...
10
votes
1answer
411 views

An almost complex structure on $S^2\times …\times S^2 / \mathbb{Z_2}$

Consider the product of $2n$ two-spheres $X_n=(S^2)^{2n}$. This manifold admits an orientation preserving involution that preserves the product structure and acts as the (orientation reversing) ...
14
votes
1answer
326 views

Homotopy fixed points of complex conjugation on $BU(n)$

Stably it is known that $(\mathbb Z\times BU)^{hC_2}\simeq \mathbb Z\times BO$ holds. The homotopy fixed point spectral sequence for KU with complex conjugation action can be completely calculated and ...
6
votes
0answers
131 views

Degeneracy of the Serre Spectral Sequence

I am learning the Serre spectral sequence and I am intrigued about the degeneracy of such at the $E_2$-page. Assuming field coefficients in cohomology for simplicity. In fact, for a Serre fibration $...
8
votes
1answer
177 views

The extension class of a finite Heisenberg group

Let $\mathbb{K}$ be a field of characteristic $\neq 2$ and let $(V, \omega)$ be a symplectic vector space. Then the Heisenberg group $\mathsf{Heis}(V, \, \omega)$ is the central extension of the ...
3
votes
1answer
213 views

Homotopy type of $SO(4)/SO(2)$

A classical result states that the quotient $SO(4)/SO(3)$ is homotopy equivalent to $S^3$. In fact, this can be stated in more general terms since $SO(n+1)/SO(n)$ has the homotopy type of $S^n$. What ...
4
votes
1answer
141 views

The existence of the extension of a non-trivial line bundle

In Three Dimensional Gravity Revisted, Witten studied the Abelian Chern-Simons theory in three dimensions. Let $W$ be a three dimensional manifold. Let $\mathcal{L}$ be a non-trivial line-bundle over ...
8
votes
1answer
173 views

Moishezon manifold vs proper complex variety

Does there exist a closed Moishezon manifold that does not have the homotopy type of the analytification of a smooth proper complex variety (I think we know that every closed Moishezon manifold is ...
3
votes
0answers
108 views

Topology of abstract varieties over $\mathbb{C}$

What are the known restrictions on the topology of complex manifolds corresponding to analytifications of smooth proper algebraic varieties over $\mathbb{C}$? I think they have to have non-zero $b_2$ ...
6
votes
1answer
141 views

Moishezon manifold with vanishing $b_2$

Does there exist a closed Moishezon manifold with zero second Betti number?
4
votes
0answers
69 views

Decomposition of bordism groups for $BG$ where $G$ is a product of two groups

Let a group $G=G_1 \times G_2$, where $G_1$ is a discrete group (can be finite or infinite), $G_2$ be any compact Lie group or finite group. Question: Is there some simple result that we can ...
9
votes
1answer
236 views

Is $\mathbb{C}^n\setminus V(f)$ homotopy equivalent with a “large ball complement”?

Let $f\in\mathbb{C}[x_1,\dots,x_n]$, and let $V(f)$ denote the vanishing locus. Is it true that for large enough $N$, there is a homotopy equivalence $$\mathbb{C}^n\setminus V(f)\simeq B(0,N)\setminus ...
3
votes
0answers
262 views

Topological approach to create a space between clouds

I have a dataset associated with labels. According to https://arxiv.org/pdf/1802.03426.pdf --> UMAP (Uniform Manifold Approximation and Projection) which is a novel manifold learning technique for ...
8
votes
1answer
237 views

Is there a fibration sequence of spectra $K\mathbb{F}_q\to KU\to KU$?

Quillen famously constructed a fibration sequence $BGL(\mathbb{F}_q)^+ \to BU \to BU$ to compute the algebraic K-groups of finite fields, where the second map is $\psi^q-1$. Does this lift to the ...
11
votes
1answer
412 views

homotopy and (co)filtered limits

Suppose we have a (co)filtered digaram $\dots \rightarrow X_{2}\rightarrow X_{1}$ of topological space. Is is true that the natural map $\pi_{0}[\lim X_{i}]\rightarrow \lim \pi_{0}(X_{i})$ is an ...