Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
8,255
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word problem for the fundamental group of complements
It is well known that the finite type (pure) Artin groups have solvable word problem. This was proved by Deligne in 1972. His aim was to show that the complement of a simplicial hyperplane arrangement ...
- 1,017
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1
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Where does the notation $\pi_1(X,x)$ for the fundamental group first appear?
I've spent the last half hour browsing Stillwell's translation of Poincaré's Analysis Situs and Dieudonné's History of Algebraic and Differential Topology, and I haven't found the source of this ...
- 2,362
3
votes
1
answer
456
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Spin structures and divisibility of cohomology classes
Let me begin with some motivation. In calculating the Chern-Simons invariant of a $U(1)$ connection $A$ on a 3-manifold $M$, we can proceed by picking a bounding 4-manifold $X$ with $\partial X = M$ ...
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1
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Combinatorial spin structures
I would like to know how to define spin structures combinatorially, for an oriented smooth manifold equipped with a triangulation. In the case of a 2d manifold, spin structures correspond to ...
- 1,459
8
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1
answer
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Does a bifunctor that's monoidal in each argument take pairs of monoids to a commutative monoid?
Let $\mathcal{C}, \mathcal{D}, \mathcal{E}$ be (symmetric?) monoidal categories, and $H : \mathcal{C} \times \mathcal{D} \to \mathcal{E}$ be a functor that is monoidal in both arguments, ie. $H(C,-)$ ...
user11863
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1
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Homotopy groups of an infinite wedge of 2-spheres
I know Hilton's result about a finite wedge of spheres, and I know that certain homotopy groups (such as the third homotopy group) can be directly calculated for an infinite wedge too.
My question is ...
- 459
10
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1
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Are finite (levelwise) homotopy limits of spectra homotopy invariant?
I found an easy proof that the (levelwise) homotopy limit of a pointwise equivalence of finite diagrams of orthogonal spectra is an equivalence, without assuming that the spectra in the diagrams are ...
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0
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is this a simplicial model category?
A basic result in the theory of model categories is that simplicial sets form a simplicial model category. The same is true for simplicial $k$-algebras. I have two questions related to this. One ...
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Topological Hochschild cohomology?
Let $A$ be a $E_\infty$-ring spectrum. By EKMM, it may be treated as a commutative algebra in the appropriate category. In particular, one may define topological Hochschild homology as $A\wedge_{A\...
- 733
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The normalised cochain complex, totalisation and cosimplicial simplicial $R$-modules
Short Version
Given a cosimplicial space $X_\bullet$, what is the relationship between (co)chains on the totalisation of $X_\bullet$ and the totalisation of the cosimplicial chain complex obtained by ...
- 461
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A problem on infinite dimensional metric space
Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that:
$X_{n}$ is a regular$^1$ CW-complex of constant local dimension$^3$ $n$, it is of finite type$^4$...
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answer
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Finding automorphism groups of simplicial complexes
Question:
Given a finite simplicial complex $K$, what general techniques allow one to efficiently compute (a presentation of) the group $\text{Aut}(K)$ of $K$'s automorphisms?
Since this is ...
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2
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On triangulated categories of pro-objects
Which term is used for model categories whose homotopy categories are triangulated? Stable proper model categories?
I want $Ho(Pro-M)$ to be triangulated ($Pro-M$ is the category of pro-objects of M) ...
- 16.5k
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Lefschetz duality for manifolds with boundary / stratified spaces
Let $M$ be a manifold with corners. Let $F_p$ denote the union of all the codimension $i \geq p$ faces of $M$.
Then I have read that there is a form of Lefschetz duality that says that there is an ...
1
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1
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Standard way to prove that groupoids are homotopy 1-types
It is very well know that groupoids, considered as spaces via the nerve construction, are homotopy 1-types, i.e. aspherical. Here is a sketch of proof: Consider the canonical functor $f:C\rightarrow \...
- 1,540
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3
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Second Stiefel Whitney class of quotients of odd spheres
I don't know much of algebraic topology so the following question could be very silly. Let $G$ a finite subgroup of $U(n)$ that acts linearly (the action induced by the action of $U(n)$ on $\mathbb{C}^...
- 1,707
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votes
1
answer
362
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isotopy classes of embeddings of the torus
Let's consider $S^1$-bundle $E$ over a 2-manifold $M$. How many isotopy classes of embeddings of the torus $\mathbb{T}^2$ in $E$?
For each free homotopy classes $\gamma$ of mappings of the circle ...
- 251
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1
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fibre bundle as a boundary of a fibre bundle
Let $M_{n+1}$ be a fibre bundle with $S_1$ as the base and $n$-dimensional CW complex $F_n$ as the fibre.
Assume $M_{n+1}$ is oriented.
(1) Can one show that $M_{n+1}$ is always a boundary of a CW ...
- 4,716
3
votes
1
answer
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Embedding e_n -> e_m
Let $e_n$ be the operad of (say, rational) chains of the operad of little n-disks. Consider the natural embedding $e_n\to e_m$, where $n<m$ and $n,m>1$ induced by the embedding $\mathbb{R}^n\to \...
- 733
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Signature of compact oriented 4-manifold
I was told that the signature of $S_1\times F_3$ is zero, where $F_3$ is a compact oriented 3-manifold. Let $M_4$ be a fibre bundle with $S_1$ as a the base manifold and $F_3$ as the fibre. Assume $...
- 4,716
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0
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The spheres operad
I have a rather naive question. Consider the space of all maps
$$ S^{j_1} \times S^{j_2} \times \cdots \times S^{j_k} \to S^n $$
for all possible natural numbers $n, k, j_1, \cdots , j_k$.
This ...
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1
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210
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Are characteristic maps of CW complexes Lipschitz up to homotopy?
Let us consider a finite CW complex $X=\cup X_j$ with a given metric, compatible with the topology (maybe a reasonable one coming from some embedding into some $\mathbb{R}^n$).
The characteristic ...
- 1,185
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votes
1
answer
637
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fiber bundle and free action
In Spanier's book " Algebraic topology" a fiber bundle is defined as follows:
A fiber bundle $\xi=(E,B,F,p)$ consists of a total space $E$, a base space $B$ and a fiber $F$ and a bundle projection $p:...
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A classification of smooth $S^1$-actions on $\mathbb CP^3$?
Question 1. Is there a classification of smooth $S^1$-actions with isolated fixed points on the standard $\mathbb CP^3$?
Question 2. What if one additionally imposes the condition that the action ...
- 14k
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votes
3
answers
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Is there a "by hand" proof on the symmetry of the Atiyah class of $TX$?
Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence ...
- 8,707
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votes
2
answers
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Action of involutions on homology
Suppose we have a sufficiently nice (example below) pathwise connected topological space $X$ with a double cover $\pi : \widetilde X \to X$. There is an associated free $\mathbb Z_2$-action on $\...
- 672
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votes
4
answers
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fixed point property for maps of compacts
Definition. A topological space $X$ has the Fixed Point Property (FPP) if every continuous self-map $X\to X$ has a fixed point.
Question. If $X$ and $Y$ are homotopy-equivalent compact metrizable ...
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Toda brackets and factorisation of a sequence of spectra
I've found a paper of Spanier's (Higher Order Operations) where he uses the theory of "carriers" to study $n$-th order operations. The set-up is rather general; for example a particular case defines ...
- 3,655
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1
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Universal covering map from $\mathcal{H}$ to $\mathbb{C}\setminus \mathbb{Z}\oplus i\mathbb{Z}$ (the countably punctured complex plane)
It's a consequence of the uniformization theorem for simply connected Riemann surfaces that the universal cover of $\mathbb{C}\setminus(\mathbb{Z}\oplus i\mathbb{Z})$ ($\mathbb{C}$ punctured at all ...
- 10k
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votes
3
answers
542
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Cobordism and finite sheeted covers of manifolds
Let $M$ be an oriented manifold, not necessarily compact. Let $M'$ be a (finite) $k$-sheeted cover and let $\pi:M'\longrightarrow M$ be the covering map.
Question 1 : Is it true that $M'$ is (...
- 3,403
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2
answers
630
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Reference for Stasheff Operad
I want a reference for Stasheff operad, where operad maps are defined explicitly at the point-set level. I would also like to ask the question that what exactly do one mean by Stasheff operad? Is ...
- 2,013
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Is $\mathcal{D} \bigl( \mathrm{QCoh}(\mathfrak{X}) \bigr)$ compactly generated?
An object $E$ in a triangulated category $\mathcal{T}$ with (small) coproducts is called compact if the functor $\mathrm{Hom}_{\mathcal{T}}(E,-)$ commutes with arbitrary coproducts or, equivalently, ...
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$(BU,f)$ structures on manifolds via stable normal bundles and stable tangent bundles
Background
Consider $BU=colim \, BU_k$ where we take $BU_k$ to be the specific model of classifying space for the group $U(k)\subseteq O(2k)$ given by the quotient space of the infinite real Stiefel ...
- 1,152
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votes
2
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P.J. Hilton notes requested
Does anybody here have the mimeographed notes Homotopy theory and duality, by P.J. Hilton, Cornell University, 1959 ?
I guess that those notes were never published online.
I believe that some ...
9
votes
1
answer
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Construction of Thom-Spectrum for G_2-Structures
The motivation to this question is the paper of Crowley and Nordstrøm "A New Invariant of $G_2$-Structures". I am trying to find a homotopy theoretic interpretation of the following geometric ...
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5
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Inference using Topological Data Analysis: Is it worth it for a regular statistician to learn TDA?
After having read Gunnar Carlsson's Topology and Data I feel enthusiastic to use some topological data analysis (TDA) methods in my current research, mostly in social sciences. We often handle huge ...
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votes
2
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649
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Infinite loop spaces
Let $X, Y$ be infinite loop spaces: $X = QA$ and $Y = QB$, where $A,B$ are connected topological spaces, and $Q$ stands for $\Omega^\infty S^\infty.$ Let $f:X \to Y$ be a continuous map such that $\...
- 2,319
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votes
1
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Topological degree of homogeneous function of degree k [closed]
Let $F:\mathbb{C}\to \mathbb{C}$ be a homogeneous map of degree $k$ (i.e., $F(tx)=t^kF(x)$, $t>0$). It is true that $F$ has topological degree less than or equal to k?
This is true if F is ...
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votes
1
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Stiefel classes and generic sections
I asked this question in math.stackexchange few days ago.
Unfortunately, I haven't seen any simple answer.
One can say that the Stiefel-Whitney classes is dual classes to the locus of linearly ...
- 251
3
votes
2
answers
566
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Dimension of the homology group with coefficients in $\mathbb{Z}/2\mathbb{Z}$
I asked this on math.stackexchange.com, but didn't get a single answer.
Charles Weibel writes in his survey of homological algebra
Riemann defined a surface $S$ to be $(n + 1)$-fold
connected if ...
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votes
1
answer
430
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A_n operad as configuration spaces
$A_\infty$ operad can be described both in terms of Stasheff polytopes and configuration spaces.$A_n$ operad can be described as subspace of Stasheff operad described using Stasheff polytope. Is there ...
- 2,013
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votes
3
answers
1k
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smooth manifolds as real algebraic set (continued)
There are several ways of producing manifolds,say:
1.orbits space of group action
2.connected sum of manifolds
3.underlying topological space of nonsingular algebraic set
....
here,i am ...
- 878
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votes
2
answers
287
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Pseudofree $T^2$ actions on spheres
Is it possible to construct a smooth action of $S^1\times S^1$ on $S^{2n+1}$ ($n\ge 2$) such that no point on $S^{2n+1}$ has an infinite stabilizer?
Note that if such an action exists, it can not be ...
- 14k
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votes
2
answers
2k
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Open Torelli problems
I just finished studying the proof of the Torelli Theorem for K3 surfaces made by Daniel Huybrechts (following the approach of Misha Verbitsky).
This theorem states that two K3 surfaces $X$ and $Y$ ...
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votes
2
answers
245
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Fixed component of an $S^1$ action on $S^n$
Suppose $S^1$ is acting smoothly on $S^n$ and $M$ is a connected component of the set of fixed points of the action. What can be said about $M$?
Is it true that $\pi_1(M)=0$? (sorry this first bit ...
- 14k
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votes
1
answer
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Is there a Wall finiteness obstruction in other settings?
Let $\mathcal{S}$ be the $(\infty, 1)$-category of spaces. Then the compact objects of $\mathcal{S}$ are precisely the retracts of finite CW complexes. These are not the same as the finite CW ...
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2
answers
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Regarding understanding differential geometry [closed]
I am essentially looking for a book that would hold my hand through basic concepts to more complicated ones. I am coming from physics. I am looking to make some connections with Classical mechanics ...
user39066
8
votes
3
answers
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Configuration spaces of the torus
I would like a reference that calculates the rational homology of the unordered configuration spaces of the torus.
- 985
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votes
1
answer
2k
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hyperelliptic involution on a surface
What is the Dehn twist factorization of the hyperelliptic involution on an oriented surface of genus g (with one boundary component)?
- 1,335
7
votes
1
answer
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Iterated Milnor fibrations and Thom's a_f condition
Ok so there's a lot of litterature about nearby cycles functor since it was introduced by Grothendieck and Deligne but I couldn't find any clear answer to the following natural question:
Problem: Let ...
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