Questions tagged [at.algebraic-topology]

Homotopy theory, homological algebra, algebraic treatments of manifolds.

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4 votes
1 answer
87 views

Topology of a smoothing of an isolated singularity

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 ...
5 votes
1 answer
125 views

If the number of ends of Freudenthal space is infinite, then its space of ends is homeomorphic to the Cantor set?

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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22 votes
1 answer
947 views

A property of even continuous functions on the sphere

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 $k\neq 0$) ...
9 votes
1 answer
158 views

Links and non-orientable surfaces

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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4 votes
1 answer
250 views

What is the circle-equivariant cohomology of the real projective plane

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 ...
1 vote
0 answers
144 views

The key step in Serre's method on higher homotopy groups

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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3 votes
1 answer
149 views

Is $\Sigma^\infty_+ O(n)^\vee$, the Spanier-Whitehead dual of the orthogonal group, an $A_\infty$-ring spectrum?

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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5 votes
0 answers
93 views

The fundamental group of the complement of badly embedded open $n$-ball in $\Bbb R^n$

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 $\...
  • 607
4 votes
1 answer
76 views

Alexander polynomials for a certain family of closed braids

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}\...
4 votes
1 answer
176 views

If $\pi_\ast A$ is graded-commutative, then is $A_\ast$ a lax monoidal functor?

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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6 votes
0 answers
118 views

Relative version of Browder's theorem on H-spaces

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 ...
2 votes
1 answer
252 views

Explicit generators from Serre spectral sequence

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 $H_p(B, H_q(F, \mathbb{Q}))$...
3 votes
0 answers
154 views

Is every tree a deformation retract of the disk?

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 ...
7 votes
1 answer
356 views
+400

The center of $\mathbf{hTop}$

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. ...
1 vote
0 answers
58 views

about codimension two foliation

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 ...
6 votes
0 answers
115 views

Resolutions of unbounded complexes: Condition ($\ast$) in Spaltenstein's paper

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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5 votes
0 answers
101 views

How to see $\delta_2(\hat{\chi}(V))=\chi(V)$ in differential cohomology?

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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8 votes
0 answers
215 views

Colimits of symmetric groups

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 $\...
27 votes
2 answers
557 views

Is there a flat manifold with trivial first homology?

Is there a closed flat manifold whose fundamental group has trivial abelianization? The famous Hantzsche–Wendt flat manifold has fundamental group with finite abelianization.
3 votes
0 answers
76 views

"Standard computations" with stable Hopf invariants

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/...
17 votes
1 answer
374 views

Topology of the space of embedded genus $g$ surfaces in $S^3$

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 $\...
0 votes
0 answers
139 views

Cup-product in cohomology and Hopf algebra

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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2 votes
0 answers
125 views

Pushout homotopy squares in motivic homotopy theory

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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5 votes
1 answer
192 views

Stable torus that is not a torus [duplicate]

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?
7 votes
2 answers
275 views

Injectivity of the cohomology map induced by some projection map

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 votes
1 answer
162 views

How many pairwise non-homeomorphic non-empty closed subsets of the Cantor set are there? [duplicate]

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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3 votes
1 answer
135 views

Equivariant K-theory for products of groups?

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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3 votes
1 answer
68 views

Explicit examples of Classical, Flat $U(2)$-connections on a torus link complement with non-trivial holonomy

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_{\...
4 votes
1 answer
135 views

Closed good cover of a triangulable space

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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13 votes
1 answer
389 views

Applications of equivariant homotopy theory to representation theory

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 ...
2 votes
0 answers
61 views

Model structure for dga of (endormorphism) vector bundle valued differential forms

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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2 votes
0 answers
99 views

Posets whose homotopy type can be efficiently studied without fibrant replacement?

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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4 votes
1 answer
194 views

When are filtered colimits of (trivial) cofibrations still (trivial) cofibrations?

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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3 votes
1 answer
214 views

Homology of braid groups and loop spaces

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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4 votes
1 answer
297 views

"Singular homology = simplicial homology" relative to a fibration

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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3 votes
0 answers
75 views

Long exact sequence in Borel-Moore homology

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^{...
3 votes
1 answer
98 views

Definition of S-reducibility and reducibility of a space

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 ...
10 votes
1 answer
432 views

Is every retraction homotopic to a smooth retraction?

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 $n$-...
19 votes
2 answers
2k views

Why the sphere spectrum is more correct than $\mathbb{Z}$?

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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11 votes
1 answer
458 views

Computing KO^-1 of RP^3 without AHSS

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 ...
7 votes
1 answer
180 views

Relation between cohomology operations and the Adams spectral sequence

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

On an Artin (?) subgroup of braid groups

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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4 votes
0 answers
123 views

Let $p : \tilde{M} \to M$ be the universal cover. Can we ever deduce curvature properties of $M$ from the curvature of $\tilde{M}$?

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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3 votes
1 answer
280 views

Are there strictly connective smooth proper algebras over $\mathbb{F}_p$?

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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1 vote
0 answers
198 views

Is the equivalence $\mathrm{CRing}^{\mathrm{op}}\simeq \mathrm{AffSch}$ related to the homotopy hypothesis?

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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25 votes
3 answers
4k views

Higher Topos Theory- what's the moral?

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 ...
1 vote
0 answers
83 views

Moore space over a group with infinite generator

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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7 votes
1 answer
326 views

Is cohomology with local coefficients a representable functor?

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'' ...
0 votes
1 answer
138 views

Proving the induced map on the cohomology is an isomorphism

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 ...
5 votes
1 answer
156 views

If $X = X_1 \cup \cdots \cup X_n$ is shellable, then is $(X_1 \cup \cdots \cup X_k)\cap X_{k+1}$ shellable?

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}...
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