# Questions tagged [at.algebraic-topology]

Homotopy theory, homological algebra, algebraic treatments of manifolds.

Homotopy theory, homological algebra, algebraic treatments of manifolds.

7,674
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Let $(X,x)$ be an affine variety with a normal isolated singularity and $Y$ be a smoothing of $X$ (for this we mean $Y$ can be realized as a smooth fibre of a deformation of $X$).
Question. Can we ...

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I don't know whether this is the right place to discuss a part of someone's thesis or not. If it is wrong, let me know; I will delete my post.
I am reading this thesis.
Corollary 4.1.15. on page 63 ...

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This question is inspired by On moments of inertia of planar and 3D convex bodies.
Let $f:{\mathbb R}^3\setminus\{0\}\to{\mathbb R}$ be an even homogeneous ($f(kx)=f(x)$ for all real $k\neq 0$) ...

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Let $\Sigma \subset \mathbb{R}^3$ be a compact embedded surface with boundary $\partial \Sigma$ and $i:\Sigma\setminus \partial\Sigma \to \mathbb{R}^3 \setminus \partial\Sigma$ the inclusion.
Is the ...

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Let P denote the real projective plane. It has an action of the circle group S1. (e.g. Let S1 act on the 2-sphere by rotations about an axis, then this action descends to the quotient P). I have a ...

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Let $n \geq 2$ and $X$ be a $(n-1)$-connected simplicial complex. This means that all of the lower homotopy groups $\pi_{k}(X) = 0$ for $k \leq n-1$. My goal is to compute the higher homotopy groups ...

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Recently, Ching and Salvatore have proven that the $E_n$ operad is Koszul self dual. While thinking about the analogous question for the framed $E_n$ operad, I realized there is an obvious first ...

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Let $\mathcal D^n$ be an open subset of $\Bbb R^n$ such that $\mathcal
D^n$ is homeomorphic to $\{x\in \Bbb R^n:|x|<1\}$. Suppose $\Bbb R^n\setminus \mathcal D^n$ is path-connected. How bad can
$\...

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Let $n\geq 3$ be a positive integer and $\kappa=(k_1, \dots, k_n)\in \mathbb{Z}^n$. Denote by $B_n$ the braid group on $n$ strings. Consider the braid on $n+1$ strings $\sigma_\kappa:=\sigma_1^{k_1}\...

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Let $A$ be a homotopy ring spectrum. Then the homology theory $A_\ast : Spectra \to GrAb$ lifts to a homology theory valued in $GrMod(\pi_\ast A)$. If $A$ is homotopy commutative, then this functor $...

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A theorem of W. Browder [1] from 1961 says that an H-space $X$ with finitely generated integral homology (meaning that $H_*(X)$ is finitely generated) has $\pi_2(X) = 0$. This generalizes Cartan's ...

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Let $p: E \to B$ be a locally trivial fibration with fiber $F$. If necessary, suppose that $B$ is simply connected. Suppose that the Serre spectral sequence leaves the term $H_p(B, H_q(F, \mathbb{Q}))$...

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I apologise if this question is not suitable for MathOverflow.
We define a graph here to mean a disjoint union of points and copies of $[0,1]$ quotiented so that the endpoints of any interval lie on a ...

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What is the center of the homotopy category $\mathbf{hTop}$? I strongly believe that it is trivial, but it is hard to prove since $\mathbf{hTop}$ is not concretizable and hence has no small separator. ...

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Are there examples of codimension 2 foliations on closed compact 4-manifolds or 5-manifold
I am curious about examples of codimension
Are there any previous studies or lecture notes of foliation ...

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In the paper "Resolutions of unbounded complexes" (Compositio Math., vol. 65, no. 2, pp. 212-154) N. Spaltenstein generalizes the 6 functor formalism to unbounded complexes of sheaves over ...

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I'm reading the paper "Differential Characters and Geometric Invariants" by Cheeger and Simons. In Page 62 the authors defined the differential Euler character $\hat{\chi}(V)\in \hat{H}^{2n-...

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The infinite symmetric group $S_{\infty}$ of finitely supported permutations of $\mathbb{N}$ can be written as a colimit over the $S_n$'s with respect to the embedding $S_{n} \to S_{n+1}$ that maps $\...

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Is there a closed flat manifold whose fundamental group has trivial abelianization?
The famous Hantzsche–Wendt flat manifold has fundamental group with finite abelianization.

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I am struggling in understanding the proof of Lemma 10.6 of the paper "Mapping class groups and function spaces" by Bodigheimer, Cohen and Peim http://www.math.uni-bonn.de/people/cfb/...

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Consider the space of smoothly embedded genus $g$ surfaces in the 3-sphere under the $C^\infty$ topology:
$$\mathcal E_g:=\operatorname{Emb}(\Sigma_g,S^3)/{\operatorname{Diff}(\Sigma_g)}$$
where $\...

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Let $X$ be a manifold and let its cohomology
$H^*(X;\mathbb{Z})=\bigoplus_{q=0}^\infty H^q(X;\mathbb{Z})$
be a module of finite type without $p^2$-torsion
for any prime integer $p$.
Assume that on
...

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I am using Vladimir Voevodsky's notes on motivic homotopy theory and I am having trouble understanding the proofs of Corollary 2.20 and Lemma 2.21. Both of these deal with homotopy pushout squares and ...

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Let $M$ be a closed manifold such that $M\times \mathbb{S}^1$ is a torus.
Is it true that $M$ is homeomorphic to a torus?

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Given a (compact) Lie group $G$, persumably disconnected, there exists a short exact sequence
$$1\rightarrow G_c\rightarrow G\rightarrow G/G_c\rightarrow 1$$
where $G_c$ is the normal subgroup which ...

5
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My question is more or less related to basic set theory. But I don't know even that. Apologies if I added the wrong tags.
Motivation: How many non-compact (planar) surfaces are there upto ...

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Let $X$ be a $(G \times H)$-space. What is known about the connection between the groups $K_G(X)$, $K_H(X)$ and $K_{G \times H}(X)$? The $G$ and $H$ action on $X$ come from the canonical inclusions $G ...

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I am looking for non-trivial examples of flat $U(2)$ connections over the complement of a torus link $\mathcal{S}^3-L$ i.e.
$\mathcal{A}:\mathcal{S}^3-L \longrightarrow \mathfrak{U}(2)$ such that $F_{\...

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By a good closed cover of a topological space $X$, I mean a collection of closed subspaces of $X$, such that the interior of them cover $X$, and any finite intersection of these closed subspaces is ...

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Equivariant homotopy theory focuses on spaces together with some group action on them. Jeroen van der Meer and Richard Wong have a paper where they use equivariant methods to compute the Picard group ...

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I was browsing and came across this discussion on the model structure for a dga. They mostly explain the commutative case but then say some things about the non-commutative case.
Context Consider a ...

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Let $P$ be a poset and $NP$ its nerve. In order to study the homotopy type of $NP$ via the tools of simplicial homotopy theory, we generally need to take a Kan-fibrant replacement of $NP$, e.g. by ...

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Let $\mathcal{M}$ be a
locally finitely presentable model category, cofibrantly generated by
two sets $\mathcal{I}$ and $\mathcal{J}$ of cofibrations and trivial
cofibrations with presentable domain ...

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How do Segal's theorems from (Configuration-spaces and iterated loop-spaces. Invent. Math.21:213--221)
imply that there is an isomorphism $H_*(B_\infty,\mathbb{Z})\cong H_*(\Omega^2S^3,\mathbb{Z})$,
...

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Let $p:E\to B$ be a fibration. Suppose $B$ has a simplicial decomposition. For each $n\in\mathbb{Z}_{\ge0}$, let $C_n$ be the free abelian group generated by the set of pairs $(\sigma,\tau)$ where $\...

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The Wikipedia page for Borel-Moore homology states that for a locally compact set $X$ and a closed subset $Z$, if we write $U = X \setminus Z$ we have the following long exact sequence
$$\cdots \to H^{...

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I was going through this paper by Tanaka but I am stuck at Proposition 4.1 given below . I just cannot make sense of the first two lines of the proof. What does it mean when he says S-reducible and ...

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I am not an expert in Differential Topology, so let me apologize if this question admits a straightforward answer. I checked some standard references, but I could not find one.
Let $M$ be a smooth $n$-...

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One may argue that $\mathbb{S}$ is more correct than $\mathbb{Z}$. Can anyone make it more explicitly? For example, what information will be lost if we work in $\mathbb{Z}$ instead of $\mathbb{S}$？
...

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I wanted to compute $\mathit{KO}^{-1}(\mathbb{R}P^3)$ and regrettably I could only think of using the Atiyah Hirzebruch spectral sequence, which seemed like a big overkill but looking at similar ...

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$\newcommand{\Z}{\mathbb Z} \DeclareMathOperator{\Ext}{Ext}
\DeclareMathOperator{\Cone}{Cone}$
I'm trying to understand how higher order cohomology operations are related to the Adams spectral ...

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While working on something apparently unrelated I encountered a "braid-like" group, which is a relatively geometric subgroup of a braid group and seems to be itself an Artin group. It seems ...

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Let $M$ be a $C^{\infty}$-smooth, connected, paracompact manifold with universal cover $\tilde{M}$. Assume $M$ is not simply connected, so that the covering map $p : \tilde{M} \to M$ is not the ...

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Are there any connective $E_1$ rings $R$ over $\mathbb{F}_p$ satisfying the following?
$\pi_*(R)$ is a finite dimensional $\mathbb{F}_p$ vector space
$R$ is compact as a module over $R \otimes R^{op}...

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At the heart of algebraic geometry lies the op-equivalence between commutative rings and affine schemes, i.e.,
$$\mathrm{CRing}^{\mathrm{op}}\simeq \mathrm{Aff\,Sch}.$$
At the heart of homotopy theory ...

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I've often seen Lurie's Higher Topos Theory praised as the next "great" mathematical book. As someone who isn't particularly up-to-date on the state of modern homotopy theory, the book seems ...

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I am not an expert on this topic. I am trying to learn about Moore spaces of type $(G,n)$. where $G$ is abelian and $n\geq 2$.
Let $M$ be a simply connected non-compact $4$-manifold with $H_2(M;\...

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It is well known that the functor of cohomology is representable.
More precisely, given $n\ge1$ and abelian group $G$,
we have $H^n(X;G)\simeq[X,K(G,n)]$.
(Here we probably need some ``nice'' ...

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I was going through a paper by Tanaka where I am stuck at the following map "f" which is given by the composition of these maps. Next, he mentions that the induced map is clearly an ...

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Let $X = X_1 \cup \cdots X_n$ be a shellable complex, where the $X_i$ are the maximal faces, in the shelling order.
Question 1:
Let $0 \leq k \leq n-1$. Then is $(X_1 \cup \cdots \cup X_k) \cap X_{k+1}...