# Questions tagged [at.algebraic-topology]

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

6,147
questions

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votes

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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**

vote

**0**answers

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\\
* @>>>...

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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**

votes

**1**answer

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

**1**answer

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

**1**answer

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**

votes

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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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vote

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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**

votes

**1**answer

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**

votes

**2**answers

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**

votes

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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**

votes

**2**answers

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**

votes

**1**answer

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**

vote

**0**answers

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**

votes

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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**

votes

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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**

votes

**1**answer

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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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:
...

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votes

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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) $?...

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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**

votes

**1**answer

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**

votes

**2**answers

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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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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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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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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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 \...

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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**

votes

**1**answer

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 ...

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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**

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**1**answer

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**

votes

**1**answer

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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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**

votes

**1**answer

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**

vote

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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 ...

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votes

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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

**1**answer

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 ...

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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 ...

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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, ...

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vote

**1**answer

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 ...

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votes

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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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**1**answer

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

**2**answers

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 $X$. (...

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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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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 $...

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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 ...

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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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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 ...