# Questions tagged [at.algebraic-topology]

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

**8**

votes

**1**answer

146 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 ...

**2**

votes

**0**answers

89 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 $b_2$ ...

**5**

votes

**1**answer

121 views

### Moishezon manifold with vanishing $b_2$

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

**4**

votes

**0**answers

50 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 ...

**3**

votes

**0**answers

96 views

### Sections of sheaves on limit spaces

Let $\{U_{\nu}\}_{\nu\in I}$ be an inverse system of topological spaces over a filtered index set $I$ with continuous transition maps.
Let $A_0$ be a sheaf of abelian groups on $U_{\nu_0}$, for some $...

**7**

votes

**0**answers

139 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

225 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 ...

**5**

votes

**0**answers

113 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

364 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 ...

**11**

votes

**3**answers

609 views

### Comparisons of convenient categories for algebraic topology

I've heard that there are many convenient categories for algebraic topology. Such categories often have many nice properties like being cartesian closed, complete, cocomplete, the forgetful functor ...

**5**

votes

**1**answer

196 views

### Topology of connected subsets of the $3$-torus

Consider the $3$-torus $Y=T^3$, a subset $\Sigma\subset Y$, and $\Sigma^*=Y\setminus\overline\Sigma$.
We assume both $\Sigma$ and $\Sigma^*$ to be open, connected, and smoothly bounded.
I am ...

**7**

votes

**2**answers

281 views

### Can one disjoin any submanifold in $\mathbb R^n$ from itself by a $C^{\infty}$-small isotopy?

Let $M$ be a manifold and $V$ be an oriented vector bundle. It's well known that if the Euler class of $V$ is non zero, then $V$ can't have a non-vanishing section. The converse is not true, see ...

**8**

votes

**3**answers

428 views

### Two homeomorphic non-diffeomorphic complex manifolds

Does there exist a closed topological manifold supporting two non-diffeomorphic smooth structures both of which admit a compatible complex structure? Also the same question, but for symplectic ...

**4**

votes

**0**answers

160 views

### Kaehler manifold of dimension 6 not homotopy equivalent to a complex submanifold of $\mathbb{C}P^n$

Does there exist a closed Kaehler manifold of real dimension 6 that is not homotopy equivalent to a complex submanifold of $\mathbb{C}P^n$ for some integer $n$?

**6**

votes

**1**answer

148 views

### Relationship between induced maps at homotopy groups level for maps $f:S^2\to S^2$

It is a known result that based point maps $f:S^2\to S^2$ are classified by their degree. That is, by the induced map at $\pi_2$-level $f_{*2}:\pi_2(S^2)\to\pi_2(S^2)$ (the subindex $*2$ just mean it ...

**4**

votes

**1**answer

155 views

### Characteristic classes of the bundle of trace free, skew adjoint endomorphisms

In "Floer Homology groups in Yang-Mills theory", Donaldson says that if we take an $U(2)$-vector bundle $E$ and we construct the bundle $\mathfrak{g}_E$ of trace-free, skew adjoint automorphisms of $...

**2**

votes

**0**answers

143 views

### Deformation invariance of homotopy type

Let $\mathscr{X}\to \Delta$ be a flat family of projective varieties over the unit disk so that each fiber $X_t$ has canonical singularities and its canonical sheaf $\omega_{X_t}$ is $\mathcal{Q}$-...

**9**

votes

**0**answers

131 views

### An equivalent definition for $\text{Spin}^c$-structures

I'm interested in proving the following proposition ([G], Remark page 48):
Prop: A $\text{Spin}^c$-structure over an oriented vector bundle is equivalent (after stabilizing if the fiber dimension ...

**6**

votes

**1**answer

361 views

### Unstable Greek letter elements

A theorem of Hopkins and Mahowald states that the Thom spectrum of the map $\Omega^2 S^3 \to B\mathrm{GL}_1(\mathbb{S}_{(p)})$ classifying the element $p$ is exactly $\mathrm{H}\mathbf{F}_p$. Let $T(1)...

**5**

votes

**1**answer

265 views

### A question about HTT Lemma 5.5.2.1

I have a question about the statement of Lemma 5.5.2.1 in Lurie's `Higher Topos Theory'.
``Let $S$ be a small simplicial set, let $f: S\rightarrow \mathcal{S}$ be an object of $\mathcal{P}(S^{op})$, ...

**2**

votes

**1**answer

81 views

### First countable geometric realization of a simplicial group

Suppose we have a simplicial group $G$.
What do we need from $G$ to get first countable $BG$, where $BG$ is a geometric realization of $G$?

**2**

votes

**1**answer

108 views

### Relation between Optimal Transport Cost and Difference between Topological Invariants?

I was working on some mathematics of Wasserstein GAN and found out a seemingly interesting research problem but I am not quite sure whether it has already been studied in some recent literature of ...

**8**

votes

**1**answer

384 views

### Formality over $\mathbb{R}$ vs formality over $\mathbb{Q}$

On ncatlab page on formality, it is stated that Deligne--Griffiths--Morgan--Sullivan proved that the real homotopy type of a closed Kaehler manifold is formal. Later, Sullivan "improved" this to $\...

**7**

votes

**1**answer

130 views

### Relation between the Casson-Gordon invariants $\sigma(M, \chi)$ and $\sigma_r(M, \chi)$

Setting: There are two objects in knot theory that are commonly referred to as the Casson-Gordon invariants: the invariant $\sigma$, and the invariant $\tau$ (see for example A. Conway’s notes ...

**5**

votes

**1**answer

157 views

### Approximate Homology of a Large Simplicial Complex

I can use software to calculate the Betti numbers $\beta_0,\beta_1,\beta_2,\dots$ of a finite simplicial complex.
This is prohibitive for large complexes, built on say > 100,000 nodes.
Is there some ...

**0**

votes

**0**answers

35 views

### Inverse limits of relative homology in Euclidean space [migrated]

Let $C$ be a compact set in Euclidean space and $i$ an integer. Is the inverse limit
$$\underset{C\subset U}{\lim_\leftarrow}H_i(U,C)$$
over all open sets $U$ containing $C$ of relative homology ...

**2**

votes

**0**answers

145 views

### Can a birational morphism between two smooth varieties of the same betti numbers exist?

I am considering a birational morphism $f:X\longrightarrow Y$ where $X$ and $Y$ are smooth projective varieties and I want to deform $X$ to another given smooth projective $Z$. It is given that $X$ ...

**5**

votes

**1**answer

110 views

### Obstructions to realisation of dual finite spectra as suspension spectra

Suppose $X$ is a finite dimensional CW-complex with top cell at dimmension $n$ and consider its S-dual denoted by $DX$. I wonder if there are any obstructions to find a space $Y$ and an interger $k\...

**4**

votes

**0**answers

143 views

### Is the Thomason model structure on Cat simplicial? Is it a monoidal model category?

The Thomason model structure on the category of small categories is transferred from the Quillen model structure on simplicial sets along the right adjoint $Ex^2 \circ N$ (where $N$ is the nerve), i.e....

**2**

votes

**0**answers

123 views

### Explicit construction of Steenrod squares vs. “A general algebraic approach…”

I am working on understanding Peter May's "A genaral algebraich approach to Steenrod operations", so for this purpose I am trying to compare his framework with explicit construction of Steenrod ...

**15**

votes

**2**answers

530 views

### Stably equivalent but not homotopy equivalent

What are some examples of (compact, say) manifolds $X$ and $Y$ that are stably equivalent, i.e. $\Sigma^{\infty}X_+\simeq\Sigma^{\infty}Y_+$, but are not homotopy equivalent?

**-4**

votes

**1**answer

133 views

### Topological spaces without retracts [closed]

Is there a way to see whether a topological space $\Omega$ does not allow retractions $r: \Omega \mapsto B$, with $B$ a given subspace of $\Omega$ ?
In other words: when is a space not retractable ...

**9**

votes

**2**answers

421 views

### A question on the fundamental group of a compact orientable surface of genus >1

Let $G=\pi(X,x)$ be the fundamental group of a compact orientable
surface of genus $g\ge 2$. It is well known that a presentation of
$G$ is
$$G=\langle x_1,y_1,\dots,x_g,y_g \ | \ [x_1,y_1]\cdots
[x_g,...

**3**

votes

**1**answer

305 views

### When an isomorphism on cohomology with $\mathbb{Z}$-coefficients being trivial implies it is trivial with $\mathbb{Z}/2$ coefficients?

Suppose $G$ is a group acting freely on a finite CW-complex $X$. Take an element $g$ of $G$. Let $g^*_\mathbb Z$ and $g^*_{\mathbb{Z}/2}$ be the corresponding induced maps on cohomology with $\mathbb ...

**5**

votes

**0**answers

121 views

### Categorification-like statement in the cobordism group?

Suppose we consider a $d$-cobordism group classifying manifolds with $H$-structure and with a classifying space $BG$ of a group $G$, written as
$$
\Omega^{d}_{H}(BG)= \mathbb{Z}_m \oplus \dots,
$$
...

**32**

votes

**2**answers

3k views

### Why is Voevodsky's motivic homotopy theory 'the right' approach?

Morel and Voevoedsky developed what is now called motivic homotopy theory, which aims to apply techniques of algebraic topology to algebraic varieties and, more generally, to schemes. A simple way of ...

**6**

votes

**2**answers

382 views

### Explicit computation of the Burnside ring

I would like to see explicit computations of the Burnside ring $A(G)$ when $G$ is a small Abelian group, such as $G=\mathbb{Z}/2,\mathbb{Z}/2^n,\mathbb{Z}/p^n$ where $p$ is an odd prime and $n\...

**6**

votes

**0**answers

100 views

### Pin cobordism v.s. “KO” theory in low or in any dimensions

Fact: The spin cobordism is equivalent to "KO" theory in low dimension if we only consider the 2-torsion.
This is related to a question and an answer supports the claim.
Here we denote the $p$-...

**6**

votes

**0**answers

138 views

### Arf-Brown-Kervaire invariant and a surjective map $G \to Pin^-$

We know that the Arf-Brown-Kervaire (abk) invariant is a bordism invariant of
$$
\Omega_2^{Pin^-}(pt)=\mathbb{Z}/(8\mathbb{Z}),
$$
where the $\mathbb{Z}/(8\mathbb{Z})$ is generated by a 2-manifold $M^...

**8**

votes

**1**answer

320 views

### Does $\mathbb{Z}_p$ acts freely on $ \mathbb{C}P^{n_1} \times \mathbb{C}P^{n_2} \times \cdots \times \mathbb{C}P^{n_k}$?

Is it true that for prime $p\neq 2 $, $k > 1$ and $n_1,n_2,\dots,n_k\geq 1$, the cyclic group $\mathbb{Z}_p$ has no continuous free action on $ \mathbb{C}P^{n_1} \times \mathbb{C}P^{n_2} \times ...

**9**

votes

**1**answer

133 views

### Discrete Pin structures

It is clear that an oriented manifold $M^n$ (with dimension $n$) admits spin structures if and only if its second Stiefel-Whitney class $[w^2]\in H^2(M,\mathbb Z_2)$ vanishes. In the construction of ...

**11**

votes

**1**answer

300 views

### Do solenoids embed into Möbius strips?

I found a strange attractor which looks a lot like a solenoid. The attractor continuum is the closure of a continuous line which limits onto itself, and it is locally homeomorphic to Cantor set times ...

**5**

votes

**0**answers

56 views

### Triple data for Pontrjagin dual of the Spin bordism group

It is said that the Pontrjagin dual of the 3-dimensional Spin bordism group of $BG$ for $G$ a finite group,
$$
\text{Hom}(Ω^{spin}_3(BG),\mathbb{R/Z}),
$$
can be expressed by triples of cochains $$(w, ...

**1**

vote

**0**answers

49 views

### Monoidal structure on left dg-modules over a brace algebra

Relating to my other question: Modules over Hopf Algebras and $E_2$-algebras
Preliminary: Let $A$ be an associative dg-algebra that is also an algebra over the brace operad. Let $M$ and $N$ be left ...

**5**

votes

**0**answers

83 views

### Induced new structures on Poincare dual manifolds

"R. C. Kirby and L. R. Taylor, Pin structures on low-dimensional manifolds (1990)" shows
Given a spin structure on $M^3$, the submanifold $\text{PD}(a)$ can be given a natural induced $\text{Pin}^-$...

**5**

votes

**0**answers

270 views

### Elementary questions about Morse-Bott functions

Let $M$ be a manifold, $F$ be a Morse-Bott function, $c$ be a critical level, and $M_c$ be the corresponding critical submanifold. Let us assume that $M_c$ is connected, and the index of $M_c$ is ...

**5**

votes

**1**answer

358 views

### Using spectral sequence show that if $p\neq 2$, then $\mathbb{Z}_p$ cannot act freely on $\mathbb{C}P^n$

If $p\neq 2$, then the cyclic group $\mathbb{Z}_p$ has no free continuous action on $\mathbb{C}P^n$. My question is how to prove the above fact using Leray-Serre spectral sequence associated to the ...

**9**

votes

**0**answers

131 views

### Hochschild-Serre spectral sequence via explicit filtration

Let
$$1 \longrightarrow K \longrightarrow G \longrightarrow Q \longrightarrow 1$$
be a short exact sequence of groups and let $M$ be a $\mathbb{Z}[G]$-module. The Hochschild--Serre spectral ...

**6**

votes

**1**answer

169 views

### Serre spectral sequence degeneration in homology vs cohomology

Let $\pi\colon E \rightarrow B$ be a fiber bundle with fiber $F$. I am not assuming that $B$ is simply-connected. We then have Serre spectral sequences in both rational homology and rational ...

**5**

votes

**0**answers

96 views

### The automorphism group of the fibered cylinder

My collegue (Oleg Gutik) is interested in finding a proper reference to a description of the group $G$ of homeomorphisms $h:\mathbb T\times\mathbb R\to\mathbb T\times\mathbb R$ of the cylinder that ...