3
votes
0answers
141 views

Stable homotopy of spheres non-locally

Are there any results/conjectures about the stable homotopy groups of spheres that relate the picture at different primes? Something like Gauss's reciprocity law in number theory? I know about the ...
2
votes
2answers
332 views

Beautiful constructions in algebraic topology that facilitate one's understanding of homotopy theory [closed]

There is an army of interesting constructions in AT, and Understanding them are usually very helpful for appreciate the theory underneath. So I would like to invite you to share those examples that ...
14
votes
1answer
1k views

What have simplicial complexes ever done for graph theory?

(I am asking in a somewhat tongue-in-cheek fashion, of course, but nevertheless...) Are there examples of results in "classical" [*] graph theory that have been achieved by using simplicial ...
7
votes
4answers
807 views

Coboundaries and Gluing in Cech Cohomology - Intuition?

I'm trying to develop an intuition for Cech cohomology geometrically, but am currently failing. A lot of people seem to say that the groups $H^n$ measure obstructions to gluing local sections to get ...
6
votes
3answers
1k views

Homology versus cohomology of Lie groups

A central advantage of cohomology theory over homology -- at least in terms of richness of structure and strength as an invariant -- is the additional ring structure from the cup product. Recall that ...
3
votes
1answer
339 views

What do we mean by contractible for simplicial objects in a category?

EDIT: removed cruft from this question. Recall that extra degeneracies for an augmented simplicial set $X$ are maps $s_0\colon X_n \to X_{n+1}$ for $n=-1,0,1,2,\ldots$ which satisfy the usual ...
14
votes
0answers
845 views

Origins of the Nerve Theorem

Recently, I've read two papers which have cited the Nerve Theorem, one crediting Borsuk with the result and another Leray. Here is the question: Who was the first to prove the Nerve Theorem?
22
votes
5answers
2k views

Understanding/Mastering Analysis in Topology, necessary?

I have spoken to one professor so far about this, which of course was helpful, and so I am looking for additional opinions: To work with topological tools that were built via analysis, should I be a ...
5
votes
4answers
381 views

(Co)homological characterization of homotopy pullbacks

For a commutative square of spaces (of manifolds, or of simplicial sets): $$S=\left(\begin{array}{ccc} A & \to & B \newline \downarrow & & \downarrow \newline C & \to & ...
1
vote
1answer
362 views

Program for drawing cobordisms [closed]

Perhaps this is not the right place to ask the following question but I did not find any suitable on the web. So I would be very grateful for sharing your experience. What is a good program to draw ...
17
votes
1answer
1k views

The whole plethora of topology

In his answer to a recent MO question, Johannes Ebert sketches the proof of a very nice result (implying that homotopy spheres are parallelizable) which, as he says, involves the whole plethora of ...
0
votes
1answer
286 views

On $\pi_{1}(f(\Omega))$ with $\Omega$ convex

Suppose $\Omega\subset R^{n}$ is an open,convex and bounded set,$f:\Omega\to\mathbb{C}$ is a smooth map. My question: 1)when $\pi_{1}(f(\Omega))=\lbrace 1 \rbrace$? Or in order to make ...
54
votes
16answers
7k views

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 ...
11
votes
5answers
1k views

The definition of homotopy in algebraic topology

In this post, let $I=[0,1]$. Something about the definition of homotopy in algebraic topology (and in particular in the study of the fundamental group) always puzzled me. Most books on the ...
5
votes
3answers
1k views

Advantages of working with CW complexes/spaces over Kan complexes/simplicial sets?

Many topologists express a clear preference for working with CW complexes instead of simplicial sets. One of the reasons is that the cellular chain complex of a CW complex is often easier to work ...
19
votes
2answers
2k views

Does this approach for the Poincare conjecture work?

Several months ago a paper was posted at http://arxiv.org/abs/1001.4164 called "Another way of answering Henri Poincare's fundamental question." The author gave a talk on it today at my institution. ...
19
votes
4answers
3k views

Mathematically mature way to think about Mayer–Vietoris

This question is short but to the point: what is the "right" abstract framework where Mayer-Vietoris is just a trivial consequence?
13
votes
7answers
2k views

Why should I prefer bundles to (surjective) submersions?

I hope this question isn't too open-ended for MO --- it's not my favorite type of question, but I do think there could be a good answer. I will happily CW the question if commenters want, but I also ...
31
votes
5answers
2k views

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 ...
11
votes
3answers
1k views

Applications of homotopy groups of spheres

The study of the homotopy groups of spheres $\pi_i(S^n)$ is a major subject in algebraic topology. One knows for example that nearly all of them are finite groups. Some are explicitly known. There is ...
29
votes
7answers
3k views

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 ...
14
votes
3answers
794 views

How does $\pi_1(SO(3))$ relate exactly to the waiters trick?

I hope this is serious enough. It is a well-known fact that $\pi_1(SO(3)) = \mathbb{Z}/(2)$, so $SO(3)$ admits precisely one non trivial covering, which is 2-sheeted. Another well known fact is that ...
6
votes
4answers
379 views

Examples of the varying strengths of topological invariants

In my first algebraic topology class, I remember being told that the simplest reason for homology was to distinguish spaces. For example, if is X=circle and a Y= wedge of a circle and a 2-sphere then ...
41
votes
11answers
4k views

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 ...
27
votes
11answers
8k views

Homological Algebra texts

I would like to hear the communities' ideas on good Homological Algebra textbooks / references. The standard example is of course Weibel (which I'll leave for someone else to describe). As usual, ...