Questions tagged [at.algebraic-topology]

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

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29
votes
1answer
821 views

Which of the proofs of the fundamental theorem of algebra can actually produce bounds on where the roots are?

One of the old classic MO questions is a big-list of proofs of the fundamental theorem of algebra. Here is a second big-list question about this big list: Which of the FTA proofs can, even in ...
1
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0answers
92 views

Based loops objects in model categories

If I have a pointed model category then for I can define based loops objects as homotopy pullbacks: $\require{AMScd}$ \begin{CD} \Omega X @>>> *\\ @V V V @VV V\\ * @>>>...
2
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0answers
173 views

Simply put Floer homology

I would like to understand what exactly Floer homology $HF_*(Y)$ for a simply connected compact 3-manifold $Y$ is. I understand there are many variants of Floer homology (and cohomology) but I would ...
5
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1answer
191 views

Kähler manifold with even-only singular cohomology

Given a simply connected smooth projective variety (hence a compact Kähler manifold) with singular cohomology generated in even degrees, do we know that there is a Morse function on it such that all ...
7
votes
1answer
336 views

Descent properties of topological Hochschild homology

Question: What is the finest topology on $\mathrm{CAlg}$ (commutative ring spectra) for which THH (Topological Hochschild Homology) satisfies descent? Adaptations of the arguments appearing in ...
4
votes
1answer
153 views

Homotopy fibre sequence and left Bousfield localization

Let $\mathcal{M}$ be a pointed model category and $\mathcal{C}$ a class of maps in $\mathcal{M}$ for which the left Bousfield localization ${\rm L}_{\mathcal{C}}\mathcal{M}$ exists (see Hirschhorn, ...
3
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0answers
136 views

Is the inclusion $\Delta^{op} \to \Gamma^{op} = Fin_\ast$ homotopy cofinal?

There's a canonical functor $i: \Delta^{op} \to Fin_\ast$. For example, one uses the pullback $i^\ast$ to turn a $\Gamma$-space into a simplicial space, and then takes a geometric realization to ...
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0answers
46 views

manifold bounded by compact manifold with $b_1=0$

Let $X$ be a non-compact manifold without boundary. Suppose that $b_1(X)=0$. Suppose $Y$ is a codimension zero compact submanifold with corner. Q Can we find a compact submanifold $Z$ with smooth ...
3
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1answer
243 views

Lectures on triangulations of manifolds by Robion Kirby

I was looking for the book mentioned in the title. Seemingly it was not published, but copies are available in several mathematical libraries. Google books does not provide preview. I am wondering if ...
13
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2answers
363 views

The $K$-theory homology of the Eilenberg-MacLane spectrum

Let $KU$ be the complex $K$-theory spectrum and $H\mathbb{Z}$ be the Eilenberg-MacLane spectrum. For $n\in \mathbb{Z}$, it is known what the homology groups $KU_{n}(H\mathbb{Z})$ are?
4
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0answers
113 views

Borromean Lines of three $\mathbb{R}^1$ in $\mathbb{R}^3$ and analogous Milnor link invariants

It is know that Borromean rings can be detected by Milnor invariant $$ \bar{\mu}(\gamma_1,\gamma_2,\gamma_3)= \# (\Sigma_1 \cap \Sigma_2 \cap \Sigma_3)-\frac{1}{2}\sum_{I,J,K}\epsilon_{IJK} \sum_{\...
3
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2answers
232 views

Algebraic curve intersecting square-grid

Let us subdivide the unit square into square-grid cells with sidelength $w$. This will give us roughly $w^{-2}$ cells. Formally $$ g_{ij} = \{(wi, wj) + (x,y) : 0\leq x,y\leq w \},$$ for $i,j = 0,\...
3
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1answer
225 views

The spectrum of the Hodge Laplacian on a Riemannian manifold

The Hodge Laplacian operator on differential forms on a (compact?) Riemannian manifold carries useful information about the topology of the manifold. In particular, the multiplicity of the zero ...
1
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0answers
120 views

A topological property of curves on the plane $\mathbb{R}^2$

Let $\gamma\colon [0,1]\to \mathbb{R}^2$ be a continuous injective map. Is it true that for any inner point $t\in (0,1)$ there exist an open neighborhood $U$ of $\gamma(t)$ and a homeomorphism $f\...
3
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0answers
105 views

When are pullbacks of simple homotopy equivalences still simple homotopy equivalences?

Let $f : X \to Y$ and $g : Z \to Y$ be continuous maps between finite CW complexes. If $f$ is a simple homotopy equivalence, are there conditions on $g$ which guarantee that its pullback $f'$ is a ...
5
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1answer
131 views

Is the volume of relative cycles at least the systole of the manifold?

Let $M$ be a manifold with boundary $\partial M$. Suppose that $M$ is equipped with some structure for which a notion of volume for chains can be defined. For example, if $M$ is triangulated, then the ...
8
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1answer
360 views

Koszulness of some DG-algebras and a paper by Kohno and Oda

This is a follow-up of my previous question Formality of the 2nd ordered configuration space of a closed Riemann surface. At page 131 of [B], R. Bezrukavnikov states Proposition 4.1, in which he ...
11
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0answers
365 views

The homotopy theory presented by a Waldhausen category

Waldhausen introduced his categories for the purposes of defining algebraic $K$-theory of suitable categories. From a modern perspective, it looks like he was really doing two things at once: ...
0
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0answers
120 views

Splitting of Atiyah-Hirzebruch Spectral Sequence

Suppse E is a cohomology theory which has Kunneth Formula, i.e $ E(A \wedge B)= E(A) \otimes_{E(pt)} E(B) $. What happens to the Atiyah Hirzebruch Spectral sequence while we compute $ E(A \wedge B) $?...
4
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0answers
57 views

“Robust” Noninjectivity of a Continuous Mapping of a Sphere into the Plane

Let $X=\mathbb{S}^2$ and $Y=\mathbb{R}^2$ and $f:X\to Y$ a continuous mapping. Is it true that there must exist a nonempty set $V\subset f(X)$, open in $f(X)$ (in the subspace topology), such that ...
6
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1answer
164 views

Higher categorical analogue of the equivalence between the category of representations of a monoid and the category of the monoid ring of the monoid

In classical algebra, there is a notion of "monoid rings" such that the functor taking monoids to the monoid rings is the left adjoint to the forgetful functor from the category of commutative rings ...
13
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2answers
585 views

Are manifolds admitting a circle foliation covered by manifolds with a (non-trivial) circle action?

More precisely, is there a criterion that decides the above question? I am particularly interested in the smooth setting: is a smooth manifold with a smooth regular foliation by circles covered by a ...
3
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0answers
127 views

Properties of triangulations of homeomorphic CW complexes

Let $X$ and $Y$ be two triangulable CW complexes which are homeomorphic. Is it true that there exists a triangulation of $X$ and a triangulation $Y$ which have a common subdivision?
7
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0answers
190 views

Which spaces are most naturally presented simplicially?

It's often a great technical convenience to know that you can "do homotopy theory" with simplicial sets. But if you really get down to it geometrically, it can be awkward. In general, if I have a CW ...
11
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0answers
255 views

What would cohomological localization be good for?

An open problem in algebraic topology is whether arbitrary cohomological localizations of simplicial sets (or, equivalently, topological spaces) can be proven to exist in ZFC. It's provable in ZFC ...
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0answers
254 views

Constructions that can be seen as objects representing a functor

Some constructions can be seen as objects representing a functor. For example, Consider a topological group $G$ and a functor $\mathcal{F}:\text{Top}\rightarrow \text{Gpd}$ defined as $M\mapsto \...
6
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0answers
219 views

Funtoriality of twisted K-theory

I posted this question on math.stackexchange, but received no answer there. In order to avoid the XY problem I will first state what I want, then what I think is the solution and how that failed ...
0
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1answer
244 views

Are all manifolds in $\mathbb R^n$ homeomorphic to a smooth manifold?

My question is whether all manifolds that can be embedded in $\mathbb R^n$ are homeomorphic to a smooth manifold? I know that every smooth manifold can be triangulated which I think is a result of ...
4
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0answers
338 views

A particular pushout of homologicaly rational spaces

Let $R^{\delta}$ be the topological group of additive real numbers (with discrete) topology and let $R$ be the topological group of additive real number with the standard topology. Let $X$ be a (...
4
votes
1answer
169 views

$G$-torsor for topological space compared to that for sheaf of groups

I just read about the definitions about torsor of sheaf of groups and get a bit confused. How does the notion of $G$-torsor for a topological space compared to that of a sheaf of groups? Is there a ...
0
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1answer
150 views

Inclusion of closed submanifolds of a manifold

Consider a smooth compact manifold $M$ of dimension $n$, with or without boundary. Choose a submanifold $N$ of $M$ of dimension $k$, where $1 \leq k \leq n - 1$, such that $N$ is either without ...
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0answers
76 views

Local coefficient systems in cohomology

Let $\mathcal F$ be a locally constant sheaf with values in $\mathbb C$ on a nice enough space, say a compact manifold. The etale space of $\mathcal F$ defines a covering $p: \tilde X \to X$. Is ...
12
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1answer
489 views

Open subspaces of CW complexes

I am looking at the paper Covering homotopy properties of maps between CW complexes or ANRs by Mark Steinberger and James West and a claim is made in the proof of their first main theorem ...
1
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0answers
94 views

Quotient by finite subgroups are biholomorphic

Let $X$ be a complex manifold and let $G$ and $H$ be two finite subgroups of its automorphism group $Aut(X)$. Suppose we are given that $X/G$ and $X/H$ are bi-holomorphic complex manifolds. What can ...
3
votes
1answer
94 views

Proof that $Sing^IX$ is $I$-invariant for an interval object in a site by “simplicial decomposition”

I just saw a proof showing that $Sing^{A^1}X$ is $A^1$ invariant where they use an algebraic topological analogue of ``simplcial decomposition'' defined as follow: The argument works by showing that ...
4
votes
1answer
127 views

Homologous quotient of fundamental groupoid

Let $X$ be a connected space and $\Pi_1(X)$ be its fundamental groupoid. We consider the homologous relation $\mathcal R$ on every morphism space: $f,g\in \Pi_1(X)(p,q)$ are related if the singular ...
11
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0answers
294 views

Examples of non-proper model structure

I have recently been thinking about left and right semi-model categories and in which case they can be promoted to Quillen model structure, and I have come to the conclusion that that absolutely all ...
3
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0answers
101 views

Reference request :Conjugation action in the mod 2 cohomology of Integral Eilenberg Maclane spectrum

Due to work of Stanley Kochman in "Integral cohomology operations. Current trends in algebraic topology, Part 1 (London, Ont., 1981), pp. 437–478, CMS Conf. Proc., 2, Amer. Math. Soc., Providence, ...
1
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1answer
132 views

$S^1$ normal bundle on divisor and Serre spectral sequence

Let $D\subset X$ be a smooth divisor in a smooth complex variety. On $D$ we have the normal bundle $N$. Removing the zero section and retracting we get an $S^1$ bundle. Call this bundle $N'$. Now I'd ...
2
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0answers
68 views

Cohomology of colored braid groupoids

Consider braids on $n$ strands and pick $n$ distinct labels $1, \dots, n$. There is a groupoid $\mathcal P_n$ whose objects are tuples $(l_1, \dots, l_n)$ of labels and whose morphisms are braids, ...
10
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1answer
304 views

Examples of topos that are not ordinary spaces

In [SGA6] we find: Mais nous lui conseillons néanmoins, de préférence, de s'assimiler le langage des topos, qui fournit un principe d'unification extrêmement commode. Lets motivate this advice ...
3
votes
2answers
250 views

Homology of a loop-suspension space and action of $\mathcal{D}_1$-operad

If $X$ is a based connected topological space, it is well-known what the homology of $\Omega\Sigma X$ is: according to the Bott-Samelson theorem, it is a tensor algebra over reduced homology of $X$. (...
4
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0answers
105 views

Which ring spectra are homotopy limits of simpler ones?

Most surely I will tag this by reference request: I am sure very much is known about this question, I am just too ignorant to even guess where to look. What makes me feel especially foolish is the ...
5
votes
1answer
135 views

Which homotopy types can be realized as the classifying space of a right-cancellative discrete monoid?

McDuff showed that every connected homotopy type can be realized as the classifying space of a discrete monoid, but the monoid she constructs has lots of idempotents. Question: Which homotopy types ...
2
votes
1answer
227 views

When is the cohomology of a fiber bundle a tensor product?

Let $F\rightarrow E\rightarrow B$ be a fiber bundle. Let $\pi_1$ be the fundamental group of $B$ with base point say $b_0$. In the following we are considering cohomology with coefficients in $\mathbb{...
4
votes
1answer
228 views

On Thurston's triangulations of sphere

I have two questions from Thurston's paper [1]. In the paper [1], Thurston talks about classifying certain classes of triangulations of the sphere. Here a triangulation of a sphere a Topological ...
7
votes
1answer
225 views

Homotopy in $X$ and homology in $X \times I$

Suppose $X^n$ and $M^{n-2}$ are manifolds, and $f_1,f_2 : M \to X$ to two homotopic embeddings of $M$ into $X$. We can then embed $M$ into both boundary components in $X \times I$ using $f_1$ and $...
10
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0answers
362 views

Words and ranks

Let me state two problems that look very much alike. The first one can be solved putting together answers that different people have given to some questions I asked here a few weeks ago. The second ...
18
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0answers
438 views

Are simplicial finite CW complexes and simplicial finite simplicial sets equivalent?

Edit Originally the question was whether an arbitrary diagram of finite CW complexes can be approximated by a diagram of finite simplicial sets. In view of Tyler's comment, this was clearly asking for ...
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0answers
98 views

simplicial nomenclature and homology

Suppose I have a simplicial complex $K$ constructed by taking two simplicial complexes $K_1$ and $K_2,$ and coning off ever vertex of $K_1$ to all of $K_2$ and vice versa (so, a direct generalization ...