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

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106
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Proofs of Bott periodicity

K-theory sits in an intersection of a whole bunch of different fields, which has resulted in a huge variety of proof techniques for its basic results. For instance, here's a scattering of proofs of ...
98
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3answers
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Is R^3 the square of some topological space?

The other day, I was idly considering when a topological space has a square root. That is, what spaces are homeomorphic to $X \times X$ for some space $X$. $\mathbb{R}$ is not such a space: If $X ...
81
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12answers
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Why is the fundamental group of a compact Riemann surface not free ?

Consider a compact Riemann surface $X$ of genus $g$. It is well-known that its fundamental group $\pi_1(X)$ is the free group on the generators $a_1,b_1,...,a_g,b_g$ divided out by the normal ...
79
votes
11answers
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Spectral sequences: opening the black box slowly with an example

My friend and I are attempting to learn about spectral sequences at the moment, and we've noticed a common theme in books about spectral sequences: no one seems to like talking about differentials. ...
73
votes
6answers
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Counterexamples in algebraic topology?

In this thread Books you would like to read (if somebody would just write them...), I expressed my desire for a book with the title "(Counter)examples in Algebraic Topology". My reason for doing so ...
69
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17answers
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Occurrences of (co)homology in other disciplines and/or nature

I am curious if the setup for (co)homology theory appears outside the realm of pure mathematics. The idea of a family of groups linked by a series of arrows such that the composition of consecutive ...
66
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6answers
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What properties make $[0,1]$ a good candidate for defining fundamental groups?

The title essentially says it all. Consider the category $\mathfrak{Top}_2$ of triples $(J,e_0,e_1)$ where $J$ is a topological space, and $e_i \in J$. There is an obvious generalization of the ...
63
votes
4answers
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Which manifolds are homeomorphic to simplicial complexes?

This question is only motivated by curiousity; I don't know a lot about manifold topology. Suppose $M$ is a compact topological manifold of dimension $n$. I'll assume $n$ is large, say $n\geq 4$. ...
59
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10answers
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Motivation for algebraic K-theory?

I'm looking for a big-picture treatment of algebraic K-theory and why it's important. I've seen various abstract definitions (Quillen's plus and Q constructions, some spectral constructions like ...
56
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16answers
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Are there examples of non-orientable manifolds in nature?

Whilst browsing through Marcel Berger's book "A Panoramic View of Riemannian Geometry" and thinking about the Klein bottle, I came across the sentence: "The unorientable surfaces are never discussed ...
56
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4answers
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Algorithm or theory of diagram chasing

One of the standard parts of homological algebra is "diagram chasing", or equivalent arguments with universal properties in abelian categories. Is there a rigorous theory of diagram chasing, and ...
52
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7answers
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Homotopy groups of Lie groups

Several times I've heard the claim that any Lie group $G$ has trivial second fundamental group $\pi_2(G)$, but I have never actually come across a proof of this fact. Is there a nice argument, ...
52
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8answers
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equivalence of Grothendieck-style versus Cech-style sheaf cohomology

Given a topological space $X$, we can define the sheaf cohomology of $X$ in I. the Grothendieck style (as the right derived functor of the global sections functor $\Gamma(X,-)$) or II. the Čech ...
51
votes
2answers
2k views

The topological analog of flatness?

Recall that a map $f:X\to Y$ of schemes is called flat iff for any $x\in X$ the ring $O_{X,x}$ is a flat $O_{Y,f(x)}$-module. Briefly the question is: what is the topological analog of this? Many ...
49
votes
4answers
6k views

Etale cohomology — Why study it?

I know (at least I think I know) that some of the main motivating problems in the development of etale cohomology were the Weil conjectures. I'd like to know what other problems one can solve using ...
49
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0answers
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The topology of Arithmetic Progressions of primes

The primary motivation for this question is the following: I would like to extract some topological statistics which capture how arithmetic progressions of prime numbers "fit together" in a manner ...
47
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15answers
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Teaching homology via everyday examples

What stories, puzzles, games, paradoxes, toys, etc from everyday life are better understood after learning homology theory? To be more precise, I am teaching a short course on homology, from ...
47
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0answers
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Grothendieck-Teichmuller conjecture

(1) In "Esquisse d'un programme", Grothendieck conjectures Grothendieck-Teichmuller conjecture: the morphism $$ G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T}) $$ is an isomorphism. Here ...
46
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11answers
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What is Quantization ?

I would like to know what quantization is, I mean I would like to have some elementary examples, some soft nontechnical definition, some explanation about what do mathematicians quantize?, can we ...
45
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28answers
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Examples where it's useful to know that a mathematical object belongs to some family of objects

For an expository piece I'm writing, it would be useful to have good examples of the following phenomenon: (1) ${\cal X}$ is a parameterized family of somethings. (Varieties, schemes, manifolds, ...
45
votes
6answers
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third stable homotopy group of spheres via geometry?

It is ''well-known'' that the third stable homotopy group of spheres is cyclic of order $24$. It is also ''well-known'' that the quaternionic Hopf map $\nu:S^7 \to S^4$, an $S^3$-bundle, suspends to a ...
44
votes
8answers
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What are the uses of the homotopy groups of spheres?

Pete Clark threw down the challenge in his comment to my answer on Why the heck are the homotopy groups of the sphere so damn complicated?: Have the homotopy groups of spheres ever been applied to ...
43
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8answers
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What is a continuous path?

I would like some help, because I am getting mad trying to answer the following Question: Let $X$ be a topological space, what is a continuous path in $X$? Well, maybe you're already getting ...
42
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6answers
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Which of Quillen's Papers Should I read?

I just heard that Dan Quillen passed on. I am not familiar with his work and want to celebrate his life by reading some of his papers. Which one(s?) should I read? I am an algebraic geometry who is ...
41
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8answers
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Natural transformations as categorical homotopies

Every text book I've ever read about Category Theory gives the definition of natural transformation as a collection of morphisms which make the well known diagrams commute. There is another possible ...
39
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2answers
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Hirzebruch's motivation of the Todd class

In Prospects in Mathematics (AM-70), Hirzebruch gives a nice discussion of why the formal power series $f(x) = 1 + b_1 x + b_2 x^2 + \dots$ defining the Todd class must be what it is. In particular, ...
39
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5answers
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Computational complexity of computing homotopy groups of spheres

At various times I've heard the statement that computing the group structure of $\pi_k S^n$ is algorithmic. But I've never come across a reference claiming this. Is there a precise algorithm ...
39
votes
4answers
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Torsion in homology or fundamental group of subsets of Euclidean 3-space

Here's a problem I've found entertaining. Is it possible to find a subset of 3-dimensional Euclidean space such that its homology groups (integer coefficients) or one of its fundamental groups ...
38
votes
7answers
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Is Mac Lane still the best place to learn category theory?

For a student embarking on a study of algebraic topology, requiring a knowledge of basic category theory, with a long-term view toward higher/stable/derived category theory, ... Is Mac Lane still ...
38
votes
2answers
1k views

$H^4(BG,\mathbb Z)$ torsion free for $G$ a connected Lie group

Recently, prompted by considerations in conformal field theory, I was let to guess that for every compact connected Lie group $G$, the fourth cohomology group of it classifying space is torsion free. ...
37
votes
4answers
2k views

Total spaces of $TS^2$ and $S^2 \times R^2$ not homeomorphic

Hello, I'm looking for an invariant to distinguish the homeomorphism types of homotopy equivalent spaces. Specifically, how does one show that the total spaces of the tangent bundle to $S^2$ and the ...
37
votes
5answers
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Why higher category theory?

This is a soft question. I am an undergrad and is currently seriously considering the field of math I am going into in grad school. (perhaps a little bit late, but it's better late then never.) I ...
36
votes
2answers
770 views

Can a subset of the plane have nontrivial $H_2$ or $\pi_2$?

This is a question that occurred to me years ago when I was first learning algebraic topology. I've since learned that it's a somewhat aesthetically displeasing question, but I'm still curious about ...
36
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11answers
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Why torsion is important in (co)homology ?

I've once been told that "torsion in homology and cohomology is regarded by topologists as a very deep and important phenomenon". I presume an analogous statement could be said in the context of ...
36
votes
2answers
933 views

Is there an analog of Sperner's lemma for the Hopf invariant?

Recall that Sperner's lemma is essentially a combinatorial version of the topological statement "A map from $S^n$ to $S^n$ with degree one cannot be nullhomotopic." My question is, does there exist ...
35
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2answers
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What arithmetic information is contained in the algebraic K-theory of the integers

I'm always looking for applications of homotopy theory to other fields, mostly as a way to make my talks more interesting or to motivate the field to non-specialists. It seems like most talks about ...
34
votes
4answers
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To which extent can one recover a manifold from its group of homeomorphisms

Question. Suppose that $M$ is a closed connected topological manifold and $G$ is its group of homeomorphisms (with compact-open topology). Does $G$ (as a topological group) uniquely determine $M$? ...
34
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6answers
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Can we actually find any fixed points with Brouwer's theorem?

Background At the risk of greatly oversimplifying matters, let me state a heuristic from Granas and Dugundji's beautiful book: fixed point theorems fall into two broad categories. The first class is ...
34
votes
3answers
885 views

Lambda-operations on stable homotopy groups of spheres

The Barratt-Quillen-Priddy theorem says in one interpretation that there is a weak equivalence of spectra $K(FinSet) \simeq \mathbb{S}^0$. In other words K-theory groups of finite sets are the stable ...
33
votes
10answers
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Definition of “simplicial complex”

When I think of a "simplicial complex", I think of the geometric realization of a simplicial set (a simplicial object in the category of sets). I'll refer to this as "the first definition". However, ...
33
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9answers
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List of Classifying Spaces and Covers

I am looking for a list of classifying spaces $BG$ of groups $G$ (discrete and/or topological) along with associated covers $EG$; there does not seem to be such cataloging on the web. Or if not a ...
33
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1answer
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Cochains on Eilenberg-MacLane Spaces

Let $p$ be a prime number, let $k$ be a commutative ring in which $p=0$, and let $X = K( {\mathbb Z}/p {\mathbb Z}, n)$ be an Eilenberg-MacLane space. Let $F$ be the free $E_{\infty}$-algebra over $k$ ...
32
votes
7answers
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What is DAG and what has it to do with the ideas of Voevodsky?

In Toen's and Vezzosi's article From HAG to DAG: derived moduli stacks a kind of definition of DAG is given. I am not an expert and can't see what's the relation between DAG and the motivic cohomology ...
32
votes
11answers
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Compelling evidence that two basepoints are better than one

This question is inspired by an answer of Tim Porter. Ronnie Brown pioneered a framework for homotopy theory in which one may consider multiple basepoints. These ideas are accessibly presented in his ...
32
votes
8answers
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How should one think about pushforward in cohomology?

Suppose f:X→Y. If I decorate that first sentence with appropriate adjectives, then I get a pushforward map in cohomology H*(X)→H*(Y). For example, suppose that X and Y are oriented ...
32
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0answers
814 views

Homotopy type of TOP(4)/PL(4)

It is known (e.g. the Kirby-Siebenmann book) that $\mathrm{TOP}(n)/\mathrm{PL}(n)\simeq K({\mathbb Z}/2,3)$ for $n>4$. I believe it is also known (Freedman-Quinn) that ...
31
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6answers
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What does actually being a CW-complex provide in algebraic topology?

From time to time, I pretend to be an algebraic topologist. But I'm not really hard-core and some of the deeper mysteries of the subject are still ... mysterious. One that came up recently is the ...
31
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5answers
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Why do wedges of spheres often appear in combinatorics?

Robin Forman writes in "A User's Guide to Discrete Morse Theory": The reader should not get the impression that the homotopy type of a CW complex is determined by the number of cells of each ...
30
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7answers
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What part of the fundamental group is captured by the second homology group?

Let $X$ be a connected CW complex. One can ask to what extent $H_\ast(X)$ determines $\pi_1(X)$. For example, it determines its abelianization, because the Hurewicz Theorem implies that $H_1(X)$ is ...
30
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6answers
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Non-examples of model structures, that fail for subtle/surprising reasons?

An often-cited principle of good mathematical exposition is that a definition should always come with a few examples and a few non-examples to help the learner get an intuition for where the concept's ...