# Questions tagged [at.algebraic-topology]

Homotopy theory, homological algebra, algebraic treatments of manifolds.

7,398
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### Spaces homotopy dominated by $S^2 \times S^2\times S^2$

We say that a topological space $A$ is homotopy dominated by a topological space $X$ if there exist continuous maps $f:A\to X$ and $g:X\to A$ such that $g\circ f\simeq 1_A$.
Let $X$ be $S^2 \times S^2 ...

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### Is there a general definition of twisted Real equivariant cohomology theory?

There are some classical examples of Real equivariant cohomology theories and twisted cohomology theories, including equivariant KR-theory in Atiyah and Segal's paper, and the more general ...

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### Question regarding affine fibre bundles

Let $f:X\to Y$ be a morphism of affine varieties such that it is a fibre bundle with fibre $F$. Let $\pi_1(Y)=\Gamma$ be a free group (non abelian) of finite rank and $\pi_1(F)$ is a finite group $G$ ...

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### Diagonal maps, Goodwillie calculus, and $T(n)$ local homotopy theory

Here is a collection of facts that all seem true, but together seem to give a nonsensical solution:
After $T(n)$-localization, all natural transformations $F \sim G$ between homogenous functors $F,G:...

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### Phantom maps in chromatic homotopy theory

It is well-known that there are no non-trivial phantom maps in rational homotopy theory.
Is this also true in Chromatic homotopy theory?
Precisely, let $T(n)$ be the telescope on a $v_n$-self map of a ...

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### Are the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ always nontrivial in the bordism rings for $n>2$?

In my recent research, I need to know if the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ are always nontrivial in the unoriented and oriented bordism rings for $n>2$. (For the ...

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### The Hochschild–Serre spectral sequence and cup products

Let $X$ be a variety over a field $k$ with separable closure $k_s$. Let $A$, $B$ be étale sheaves on $X$. Consider now the Hochschild–Serre spectral sequences.
\begin{align*}
E_2^{pq}: H^p(k, H^q(X_{...

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### TAQ-complete spectra and homotopy completion

I read an article about Homotopy completion and top. Quillen homology from Harper-Hess.
There for any operad $O$ in spectra and $O$-algebra $X$ a completion tower
$$\kappa: X \to holim_{n\geq 1} \...

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### What is $TP(\mathbb{Z}_p)$?

Let $TP$ be periodic topological cyclic homology. What is $\pi_* TP(\mathbb{Z}_p)$?
(i) I know that $\pi_* TP(\mathbb{F}_p) \cong \mathbb{Z}_p[v^{\pm 1}]$ with $v$ in degree $-2$ by IV.4.8 of Nikolaus-...

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### What is definition of branched covering?

What is definition of branched covering in the page 10 of following paper ?
In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. ...

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### Geometric interpretation of shuffle product

Let $A=k\mathbb \Pi$ be the group algebra of an abelian group $\Pi$ and let $B(A)=\bigoplus_{k=0}^\infty\,B^k(A)$ be the unnormalized bar complex of $A$ with generators $[a_0,\dots,a_k] \in B^k(A)=A^{\...

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### Almost acyclicity of the complex of configuration spaces of noncollinear points in projective plane over finite fields

Let $F$ be a finite field with many elements, say more than 7 for example, and $X$ be the corresponding projective plane $\mathbb{P}^2(F)$.
For a set of points in $X$, if any three of them are ...

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### Complex vector bundles on compact complex manifolds

The complex vector bundles on complex projective space $\mathbb{CP}^n$ are explicitly classified for low dimensions. When $n\leq 3$, they are exactly the holomorphic vector bundles; when $n\geq 4$ we ...

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### Motivation of the fundamental theorem of covering spaces

The fundamental theorem of covering spaces states that for a nice topological space $X$, there is an equivalence of categories between covering spaces over $X$ and left $\pi_1(X)$-sets. "...

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### Space of algebraic maps, homotopy type of a CW complex

Considering the algebraic maps between two complex varieties denoted by $C_{alg}(X,Y)$, as a subspace of continuous maps with compact-open topology. Does $C_{alg}(X,Y)$ have homotopy type of a CW ...

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### Homotopy groups of homotopy fixed points of a $\mathbb{Z}\left[\frac{1}{\lvert G\rvert}\right]$-local orthogonal spectrum

Let $G$ be a finite group and $X$ an orthogonal $\mathbb{Z}\left[\frac{1}{\lvert G\rvert}\right]$-local spectrum with an $G$-action that is trivial on $\pi_*X$.
I want to show that then the map $X^{hG}...

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113
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### Fourier transform for constructible sheaves on spheres

Let $S_1 = S_2 = S^d$ be two copies of the $d$-dimensional sphere. Let $p_i : S_1 \times S_2 \to S_i$ be the projection, $j : U \to S_1 \times S_2$ the inclusion of the complement of the diagonal and $...

3
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### What about a Cayley n-complex for n>2?

Let $G$ be a finitely presented group. The Cayley graph of the finite generating set is a $1$-complex where the $0$-cells are the elements of $G$ and the $1$-cells are given by the generators (...

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### distance between two orthogonal projection matrices and its covering number

Let $X, Y \in \mathbb{R}^{n\times p}$ such that $\Vert X- Y \Vert_{HS} \leq \delta$ (Hilbert-Schmidt norm). Also, assume that both $X, Y$ have full column rank. Let the orthogonal prpjection operator ...

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### Presentation complex of a finite perfect group and its features

Let $G$ be a finite perfect group and consider $X_G$, its presentation complex. I have the following questions:
Is there any special property of $X_G$ due to the group's perfectness?
What can we say ...

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### Is there a Dold-Kan theorem for circle actions?

There are several interesting equivalences of "Dold-Kan type" in the setting of stable $\infty$-categories. Namely, let $\mathcal C$ be a stable $\infty$-category. Then the following 3 ...

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### Results on compact slices in a regular foliation

Let $(M,\mathcal{F}$) be a smooth and regular foliation (not necessarily of comdimension 1). I am wondering if there are known (partial) results on the existence of compact, connected submanifolds $F\...

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### Moving subvarieties to avoid a certain point

This seems like a problem that should be easy if true, but I wasn't able to prove it in general or disprove it. Given a complex projective variety $X$ and a sub-variety $Y$ and a closed point $p$ on $...

4
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### Fibrant replacement of an injective model category of enriched diagrams

Take a topologically enriched small category $\mathcal{P}$ and the category of enriched diagrams of spaces $[\mathcal{P},\mathrm{Top}]_0$. We work with the category of $\Delta$-generated spaces ...

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### Product on cellular cochains of the real Grassmannian

The real Grassmannian $Gr(k,n)$ of $k$-planes in $\Bbb R^n$ admits a Schubert cell decomposition, with one cell for each Young diagram $\lambda$ of height $\leq k$ and width $\leq (n-k)$; the ...

9
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### Examples of 6-manifolds without an almost complex structure

Question: I am searching for examples for closed (hence orientable ), smooth $6$-manifolds without an almost complex structure.
Finding such an example is equivelant to finding a manifold where the ...

10
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### Can the Bousfield class of projective space be computed directly?

Recall that the Bousfield class of a spectrum $E$, written $\langle E\rangle$, is the class of spectra $X$ such that $X\wedge E$ is not contractible. For example the Bousfield class of any of the ...

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### Coefficient of the top Pontryagin class in $L$-genus

The $L$ genus can be expressed as combinations of the Pontryagin classes with the first few terms as follows:
$$L_1=\frac{1}{3}p_1,$$
$$L_2=\frac{1}{45}(7p_2-p_1^2),$$
$$L_3=\frac{1}{945}(62p_3-...

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### Intersection number for 4 manifold with boundary

Let $X$ be a closed oriented smooth $4$-manifold. Suppose there is an embedding $\Sigma\to X$, it is known that the self-intersection number satisfies $[\Sigma]\cdot [\Sigma]=\pm\int_\Sigma c_1(N)$, ...

6
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### Čech cohomology is isomorphic to singular cohomology

The singular cohomology with integer coefficients of a projective variety is isomorphic to the Čech cohomology of the constant sheaf of integers on this variety.
If the above statement is correct then ...

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### Is the decomposition of the homotopy type of a complex into a product and into a smash product unique?

Is it true that if $A_1\times A_2\times ... \times A_n = B_1\times B_2\times .. \times B_m$, where $A_i, B_j$ are homotopy types of connected complexes not decomposable into a product, then the ...

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### On the fundamental dimension of a polyhedron

Fundamental dimension of a finite polyhedron $P$ is defined as : $$Fd(P)=\min \{ \dim (X):X\; \text{and} \; P \; \text{have the same homotopy type}\}.$$
My question is that: if $A$ is homotopically ...

3
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1
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### Can one define relative Hurewicz maps using the Dold-Thom theorem

Let $A\to X$ be a (Hurewicz) cofibration of path-connected topological spaces. Then we have a long homotopy sequence $$ \dots\to \pi_i(A)\to \pi_i(X)\to \pi_i(X,A)\to \dots; $$ here we fix a base ...

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1
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### Computation of cohomology of Eilenberg-Maclane spaces

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\Spf{Spf}$Background:
If $E$ is a complex-oriented spectrum, then $E^*(K(\mathbb{Z}/p^k,1))$ sits inside a long exact ...

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133
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### Fiber bundle orientability vs manifold orientability

This question seems like a pretty straightforward generalization of a result from vector bundles but its been on MSE for over a week with no answers so I'm reposting https://math.stackexchange.com/...

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### Can moonshine be explained by the $q$-expansion map $\mathrm{tmf} \to KU[[q]]$?

If we form an equivariant version of both sides of the $q$-expansion map $\mathrm{tmf} \to KU[[q]]$ (with everything being $p$-completed), should the equivariant $q$-expansion map should have ...

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### Are there polyhedra $P_1$ and $P_2$ such that $N\cong \pi_1 (P_1)$ and $G/N\cong \pi_1 (P_2)$?

Let $G\cong \pi_1 (P)$ where $P$ is a polyhedron with finitely generated homology groups $H_i (\tilde{P};\mathbb{Z})$ for $i\geq 2$. Let $N\leqslant G$ be a normal subgroup. Are there polyhedra $P_1$ ...

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### Infinite-dimensional "algebraic varieties"

This question was formerly posted on MSE but did not receive any answer or comment, so I'm re-asking it here.
Let $H$ be the infinite-dimensional seperable complex Hilbert space, and $P(H)$ denote its ...

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### What is the Todd class *really*?

My question is about how to think about the Todd class.
Usually this is presented via Grothendieck Riemann Roch (GRR): if $X$ is a smooth projective scheme over a field $\mathbf{C}$, the chern ...

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### (Lower) homotopy groups from triangulations

Both cohomology and homotopy groups capture global topological information of a manifold $X$. It is interesting to ask if they can be computed from local data. A triangulation $T$ natural presentation ...

4
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### Poincaré dual of the Alexander dual of the fundamental class of a knot is given by a Seifert surface

Let $K\subset S^3$ be an oriented knot and let $F:\overline{B^2}\times K\rightarrow S^3$ be a thickening with self linking number $0$. I will denote $F(B^2\times K)$ by $(B^2\times K)$ for simplicity. ...

3
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### “Combinatorial” moves between cell complexes

EDITED:
A pair of finite simplical complexes are equivalent if and only if they are related by a finite sequence of the Pachner moves.
Is there a similar thing on finite cell complexes? That is, are ...

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### What is the average degree of a d-simplex?

I am a beginner in network topology topics and while I was reading an article about simplicial complexes where the authors had used random simplicial complexes, I came across a formula using "...

3
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### (Homotopy) colimit and manifold

Suppose that I have an arbitrary regular CW complex. By associating a topological space to each vertex of the CW complex, I can have a diagram of topological spaces, denoted by $D$, over the CW ...

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### Question about spin map

I'm confused with the following definition of a spin map.
A spin map is a map $f: N\to M$ between differentiable manifolds such that their second Stiefel-Whitney classes are related $\omega_2(N)=f^*\...

4
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1
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### Are Landweber exact spectra determined by their coefficient ring?

Let $E$ be a Landweber exact ring spectrum. That is, we have a map of homotopy ring spectra $MU\rightarrow E$ and an isomorphism of homology theories $E_*X\simeq MU_*X\otimes_{MU_*}E_*$. Is the ...

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### Vanishing cycles and injectivity of the specialisation map

Consider a proper algebraic map between complex varieties $f : X \to D$ ($D$ is the unit disk), which is a submersion over $D^*$. I would like to know if they are any condition on $f$ such that the ...

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### Definition of union of simplicial complex and a subset

(Cross-posted from MSE: https://math.stackexchange.com/questions/4425225/definition-of-union-of-simplicial-complex-and-a-subset)
Consider a simplicial complex $\Delta$ with vertex set equal to some ...

3
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### Loop spaces of manifolds with boundary

I would like to ask the same question that was already asked here:
loop homology product for oriented compact manifolds with boundary.
I hope it is okay to open a new question. The original question ...

14
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2
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### Is the decomposition of the homotopy type of a complex into a bouquet unique?

Is it true that if $A_1 \vee A_2 \vee .. \vee A_n = B_1 \vee B_2 \vee .. \vee B_m$, where $A_i, B_j$ are homotopy types of complexes not decomposable into a bouquet, then the multisets $A_i$ and $...