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Questions tagged [algebraic-topology]

Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, fundamental groups, covering spaces, and beyond.

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How are topological invariants obtained from TQFTs used in practice?

Topological quantum field theories (TQFTs) are studied for different reasons, as exemplified in the following places: Atiyah, Topological quantum field theory Lurie, Topological Quantum Field Theory ...
Melissa's user avatar
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31 votes
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614 views

Where does one learn how to apply categorical algebra and higher abstractions to algebraic topology?

Tl;Dr: I know higher category theory and algebra is used ubiquitously in advanced algebraic topology. However, every time I ask someone, or try to find out, how one actually learns to apply the higher ...
FShrike's user avatar
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31 votes
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733 views

Definition of Bordism - Gluing Manifolds with Structure

In general, when you have a "cobordism category" (as defined in Stong's Notes on Cobordism Theory) you define two objects $M,N$ to be bordant when there exist $U,V$ such that $M \amalg \partial U \...
Arpon's user avatar
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23 votes
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Mapping cylinder is a CW complex

If you read the question entirely, it is not a duplicate. The first time I asked the question, I already gave the link to the similar question and explained why the answer is not satisfying. If you ...
Nre's user avatar
  • 515
22 votes
0 answers
582 views

Relation between non-vanishing Vector Fields on $\mathbb{T}^2$ and Fundamental Group Maps

Let X be a vector field on $\mathbb {T}^2$, we say that $\varphi: \mathbb {R} \to \mathbb {T}^2$ is a periodic orbit of $X $, if $\varphi $ is a periodic function and $\varphi'(t) = X (\varphi(t)), \...
Matheus Manzatto's user avatar
22 votes
0 answers
804 views

Bousfield–Kan spectral sequence for homotopy colimits

Let $\mathcal{J}$ be a small category and let $X : \mathcal{J} \to \mathbf{sSet}$ be a diagram. We define its homotopy colimit $\newcommand{\hocolim}{\mathop{\mathrm{ho}{\varinjlim}}}\hocolim X$ ...
Zhen Lin's user avatar
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21 votes
0 answers
326 views

Finite-Dimensional Homogeneous Contractible Spaces

Suppose that $X \subset \mathbb{R}^n$ is compact, homogeneous and contractible (and thus connected). Does $X$ have to be a point? I couldn't think of a non-trivial example, and there isn't a ...
John Samples's user avatar
20 votes
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743 views

Why universal G-bundles are contractible?

Let $G$ be a nice topological group and $E\to B$ a universal $G$-bundle. I'm interested in a proof of contractibility of $E$ using only the universal property of it. I also know that if there is a ...
Mostafa's user avatar
  • 1,674
19 votes
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1k views

how to prove that every low-dimensional topological manifold has a unique smooth structure up to diffeomorphism?

It is a deep fact of low-dimensional manifold topology that the notion of isomorphism coincides for the three categories Top, PL, and Diff. In other words, every topological manifold of dimension $n\...
symplectomorphic's user avatar
19 votes
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1k views

Learning roadmap to Topological Quantum Field Theories from a mathematics perspective

I want to learn TQFT's and am looking for review articles or books. My mathematics knowledge is limited to one year of graduate course in Algebra (Groups,Rings,Fields,Categories, Modules and ...
user90041's user avatar
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18 votes
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In Algebraic Topology, why do we want to localize spaces?

I'm reading May's More Concise Algebraic Topology and the first half of the book seems to be written under the assumption that the reader has the motivation that we want to localize the underlying ...
Naiche Cimarron Downey's user avatar
17 votes
0 answers
440 views

Lifting criterion for quotient spaces of non-free group actions

Suppose $X$ and $Y$ are path-connected and locally path-connected, and a group $G$ acts freely on both spaces discretely by homeomorphisms. Let $p_X$ and $p_Y$ be the projections onto the quotient ...
David Sheard's user avatar
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17 votes
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Was Atiyah's proof of the odd order (Feit-Thompson) theorem false?

I read last year that Atiyah thought he had found a proof of the odd order theorem of only 12 pages, using $K$-theory, and that people were trying to figure out if it was correct or not. But I never ...
frafour's user avatar
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16 votes
0 answers
430 views

The homology groups of an infinite product of spaces

Suppose $I$ is some index set and let $\{X_i \}_{i\in I}$ be a collection of topological spaces (as nice as you like them to be). What is known about the (say) singular homology of $X := \prod_{i\in I}...
Timm von Puttkamer's user avatar
15 votes
0 answers
425 views

How to prove that the center of the fundamental group of $T_g$ is trivial for $g \geq 2$?

Where $T_g$ is a closed orientable surface of genus g. I want a proof using covering space theory. I know a proof that uses the notion of hyperbolic groups and Riemanian geometry using uniformization ...
Infinity's user avatar
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15 votes
0 answers
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Cup/cap product: sheaf cohomology vs singular cohomology

Is anyone aware of a good resource which deals with how the cup/cap products of sheaf cohomology classes are a generalization of those in singular cohomology? I would say that I already understand the ...
Eric Auld's user avatar
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15 votes
0 answers
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Constructing $\mathbb{P^n}$ "bundle" with different $n$

Can we find a complex/smooth manifold and map $f\colon X\to\mathbb{C}^3$ such that it is a $\mathbb{P}_\mathbb{C}^m$ bundle over linear subspace $\mathbb{C}^1$ and $\mathbb{P}_\mathbb{C}^n$ bundle ...
user avatar
15 votes
0 answers
332 views

Algebraic methods to compute the cohomology ring of the complex topology of a variety?

Suppose $V$ is an affine (resp. projective) subvariety of the affine (resp. projective) space $\mathbb A_\mathbb C^n$ (resp. $\mathbb P_\mathbb C^n$) with vanishing ideal $I\subseteq\mathbb C[X_1,\...
Yai0Phah's user avatar
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15 votes
0 answers
710 views

Computing the homology class of a curve using Mayer-Vietoris

I am trying to compute the homology classes of various curves on a $6$-punctured torus $X$. I can easily see that the homology group is isomorphic to $\mathbb Z^7$ by letting $X=A\cup B$ where $A$ is ...
Alex Becker's user avatar
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14 votes
0 answers
632 views

Whitehead product and a homotopy group of a wedge sum

Note : this question has been crossposted on the mathematics Overflow. Let $X$ be an $n$-connected ($n\geqslant1$) CW-complex and $Y$ be a $k$-connected ($k\geqslant1$) CW-complex. My goal is to prove ...
Anthony's user avatar
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14 votes
0 answers
911 views

De Rham Cohomology of $M \times \mathbb{S}^1$

Let $M$ be a closed (compact, without boundary) $m$-dimensional manifold. I want to prove that $H^{k+1}(M \times \mathbb{S}^1) = H^k(M) \oplus H^{k+1}(M)$. ($H^k$ is the $k$-th De Rham cohomology ...
Second's user avatar
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14 votes
0 answers
247 views

How to prove that my sum coincides with a sequence on the Online Encyclopedia of Integer Sequences?

I have a topological space $X$ whose reduced $\bmod 2$ Betti numbers (that is to say, the dimension of the $\bmod 2$ reduced homology) I computed to be $$\small \dim \tilde{H}_t(X; {\mathbb{Z}}_2) = \...
user20619's user avatar
  • 323
14 votes
0 answers
1k views

Fixed Points of a Reflection

This question is about problem 2.C.5 from Allen Hatcher's Topology. The statement of the problem is as follows: Let $M$ be a closed orientable surface embedded in $\mathbb{R}^3$ in such a way that ...
NHow's user avatar
  • 141
13 votes
0 answers
591 views

Adjunction of pointed maps is a homeomorphism?

What interests me the most is if the case of exponential law is true under the assumptions claimed for example on nlab: if $X, Y$ are Hausdorff and $Y$ is locally compact, then $F^0(X, F^0(Y, Z))\cong ...
Jakobian's user avatar
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13 votes
0 answers
927 views

Topologist's sine curve is a simply-connected space

I am trying to solve the following problem from Hatcher's Algebraic Topology and have written a solution. Could you help me checking my solution, whether I am right? Thanks in advance. $Y$ is simply-...
Sumanta's user avatar
  • 9,644
13 votes
0 answers
324 views

The étale topos of a scheme is the classifying topos of...?

By a theorem of Joyal and Tierney, every Grothendieck topos is the classifying topos of a localic groupoid. It has been proved (e.g. C. Butz. and I. Moerdijk. Representing topoi by topological ...
W. Rether's user avatar
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13 votes
0 answers
852 views

Visualising $\pi_2(S^2)$ and $\pi_2(\mathbb{R}P^2)$

I would like to know how to visualise the elements of $\pi_2(S^2)$ as unit vector fields on the sphere. For instance, the generator $a$ of $\pi_2(S^2)$ would be visualised as a 'hedgehog' ...
gj255's user avatar
  • 2,830
13 votes
0 answers
594 views

Open questions in Topological K-Theory

I am interested in knowing about current research in the Topological K-Theory, especially its interactions with String Theory. About one and a half decade back, there were some papers by Physicists (e....
user90041's user avatar
  • 5,400
13 votes
0 answers
837 views

An equivalence between group cohomology and sheaf cohomology

I'm recently reading group cohomology from Serre's book local fields, and he uses there the following terminology $H^q(G,A)$ the q-th degree cohomology of G with coefficent in $A$. So i started to ...
Mimmo's user avatar
  • 279
13 votes
0 answers
668 views

Intuitive Approach to Sheaf and Cech Cohomology

Sheaf and Cech cohomology $H^*(X,\mathcal{F})$ (which give the same result when applied to good enough topological spaces) are a useful generalisation of the concepts of de Rham and Dolbeault ...
Jjm's user avatar
  • 3,011
12 votes
0 answers
217 views

Local systems defined by higher homotopy groups

I should mention I have very little background in algebraic topology and don't really know much about homotopy groups besides the definition. I am aware that for a topological space $X$ and a point $x ...
Thomas Manopulo's user avatar
12 votes
0 answers
191 views

Are neighbourhood deformation retracts transitive?

I have the following definition of a neighbourhood deformation retract (or NDR, for short): A pair $(X, A)$ is called an NDR if $A\subseteq X$ is closed and there exists a neighbourhood $A\subseteq V\...
mxian's user avatar
  • 2,019
12 votes
0 answers
329 views

Construction of the Universal Covering Space via Compact-Open Topology

Recently I've been self-studying the theory of covering spaces from "Introduction to Topological Manifolds", by John M. Lee. At the end of Chapter 11, there is an explicit construction of ...
Johnny El Curvas's user avatar
12 votes
0 answers
2k views

What's the intuitive, geometric meaning of torsion in homology groups

If we consider the homology groups $H_n(X)$ of a topological space $X$—say a CW complex for example—one can interpret every free summand of $H_n(X) \cong \mathbb{Z} ^k \oplus T$ as an $n$-dimensional "...
user267839's user avatar
  • 7,211
12 votes
0 answers
298 views

How to prove this is a fibre bundle?

Let $M^{2n}$ be $2n$ dimensional toric manifold over a simple polytope $P^n$. Let $\pi : M^{2n} \longrightarrow P^n$ be the orbit map of the torus action. Let $F^k$ be a $k$ dimensional face of $P^n$. ...
R_D's user avatar
  • 7,352
11 votes
0 answers
191 views

Embedding smooth homology sphere

Let $\Sigma$ be a smooth, $n$-dimensional homology sphere. In a paper that I am reading, the author states that there exists a smooth homotopy $n$-sphere $S$, such that the connected sum $\Sigma \# S$ ...
The_Rookie's user avatar
11 votes
0 answers
222 views

Infinite (co)-homology

Lately, I've been wondering if it was possible to define singular homology also with infinite-dimensional simplices. For example we could define an infinite dimensional simplex as: $$\Delta_{\infty}:=\...
Kandinskij's user avatar
  • 3,676
11 votes
0 answers
184 views

Fundamental group of Homeo($\mathbb{R}^n$)

My question is easy to formulate: What is known about the homotopy groups of Homeo($\mathbb{R}^n$)? Especially, what is its fundamental group? (A guess would be $\mathbb{Z}$ for $n=2$ and $\mathbb{Z}/...
Christian's user avatar
  • 378
11 votes
0 answers
107 views

Topology and Categories for the Visually Impaired

Are there any books (and/or other reference material) for teaching topology and/or category theory for the visually impaired? For example, a teacher may have experience with tactile learning tools ...
ZxJx's user avatar
  • 420
11 votes
0 answers
189 views

Algebraic Geometric Analogue of Brown's Representability

Brown's representability theorem is very usefull to show that the functor $$X \rightarrow H^i(X,A)$$ is representable. I would be interested to see if there exists an analogue of this statement in ...
curious math guy's user avatar
11 votes
0 answers
324 views

Many definitions of Hochschild homology and cyclic homology

It appears that there are more definitions of cyclic homology than there are people working on cyclic homology. As a newcomer, this confuses me to no end. I've written a list of definitions that some ...
nagger's user avatar
  • 131
11 votes
0 answers
337 views

Cohomology of $(S^2\times S^2)/\mathbb{Z}_4$

There was a similar question. Let $X=(S^2\times S^2)/\mathbb{Z}_4$ where $\mathbb{Z}_4$ acts on $S^2\times S^2$ as $(x,y)\mapsto(-y,x)$. My question: What are the cohomology rings of $X$ with ...
Borromean's user avatar
  • 641
11 votes
1 answer
538 views

Space with semi-locally simply connected open subsets

A topological space $X$ is semi-locally simply connected if, for any $x\in X$, there exists an open neighbourhood $U$ of $x$ such that any loop in $U$ is homotopically equivalent to a constant one in $...
mfox's user avatar
  • 621
11 votes
0 answers
2k views

Difference between bordism and cobordism

I have looked around for hours and although I have seen many definitions of bordism and cobordism (for some authors these two coincide and for some other not (without mentioning explicitly what's the ...
user430191's user avatar
11 votes
0 answers
351 views

G-equivariant isomorphism inducing isomorphisms on quotients

Suppose that $X$ and $Y$ are smooth quasi-projective varieties over $\mathbb{C}$ with a holomorphic map $f:X \to Y$ inducing isomorphisms $f_* : H_i(X;\mathbb{Q}) \to H_i(Y;\mathbb{Q})$ for all $i \...
jacob's user avatar
  • 193
11 votes
0 answers
146 views

Relationship between Haefliger structures and principal $\Gamma^q$-bundles

A Haefliger structure on a smooth manifold is a cocycle with coefficient in the Haefliger groupoid $\Gamma^q$. This generalises the notion of foliation of codimension $q$. We know that the classifying ...
W. Rether's user avatar
  • 3,120
11 votes
0 answers
246 views

Codimension $1$ Embedding into $\mathbb{R}^{n+1}$

I am trying to determine which homotopy types can be realized by $n$-manifolds that have codimension one embeddings into $\mathbb{R}^{n+1}$. Suppose I have $X^n \subset \mathbb{R}^{n+1}$ and an ...
user39598's user avatar
  • 1,554
11 votes
0 answers
423 views

Homotopy classes of functions from a finite CW complex

I am given the following problem: taken $X$ finite CW complex and $Y$ a space such that for every basepoint $y \in Y$ the group $\pi_i(Y,y)$ is finite $\forall i \leq \text{dim} X$ then the set $[X, Y]...
N.B.'s user avatar
  • 2,109
11 votes
0 answers
325 views

Galois cohomologies of an elliptic curve

I am studying basic theory of elliptic curves. I encountered Galois cohomology. But two introductory textbooks I read used only $H^0$ and $H^1$. I am curious why higher cohomologies did not appear. I ...
user avatar
11 votes
0 answers
638 views

Fundamental class of the connected sum of two closed orientable manifolds

I need to find a representation of $ [M \mathbin\sharp N] \in H_n(M \mathbin\sharp N) $ in terms of the fundamental classes $[M]$ and $[N]$. My idea is that $$ [M \mathbin\sharp N] = [M]+[N]$$ which ...
Riccardo's user avatar
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