# Questions tagged [at.algebraic-topology]

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

**14**

votes

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

141 views

### Moishezon manifold with vanishing $b_2$

Does there exist a closed Moishezon manifold with zero second Betti number?

**4**

votes

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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 ...