3
votes
1answer
30 views

Intuition behind prism operators to prove homotopy invariance of homology

I'm trying to understand the proof of homotopy invariance of induced maps on homology. However, I do not really understand the intuition behind this proof and especially what the prism operators (as ...
2
votes
1answer
54 views

The term “Homotopy” was given by whom?

I want to know the names who define the term 'Homotopy' in algebraic topology in 1907. Are they Dehn and Paul Heegaard? What is the full name of Dehn? Thank you in advance.
6
votes
0answers
70 views

Origins of the name “Q” and “R” for cofibrant and fibrant replacement functors.

In a model category $\mathscr M$ (in the modern sense, i.e. closed and with functorial factorizations), there is a notion of fibrant and cofibrant replacement functors. Specifically, for any object ...
4
votes
1answer
66 views

How do topologists count infinite dimensional holes?

For example, it seems like there "should" be an infinite dimensional hole (or perhaps many) in $S^1 \times S^1 \times \ldots$. (Or perhaps none...) Is there an invariant that would count it? What ...
3
votes
3answers
172 views

How to explain why free loop space, or based loop space, is infinite dimensional to non-math people.

I am giving a math talk to non-mathematicians. I was wondering how to explain how the free loop space, or based loop space, of a topological space is infinite dimensional so that a non-mathematician ...
5
votes
1answer
107 views

Suggestion about Algebraic Topology talk

following the content of the title I am writing here to ask some suggestions concerning a talk I will be presenting at my university in a week or two. The main topic I chose is the fundamental ...
2
votes
0answers
39 views

Cross product of cohomology classes: intuition

Let $X$ and $Y$ be topological spaces and consider cohomology over a ring $R.$ Hatcher (in his standard Algebraic Topology text) defines the cross product of cohomology classes $$H^k(X) \times ...
6
votes
4answers
715 views

Topological groups, why need them?

I'm reading through Munkres and Armstrong's books on topology. However, I find topological groups to be really complicated objects! I feel they are twice as hard to deal with then just groups and ...
5
votes
1answer
32 views

Surprising constructions in algebraic topology that facilitate one's understanding of underlying theory

I am recently come into the world of algebraic topology and find it a fascinating place with lots of beautiful constructions that challenge one's intuition. Also, understanding these constructions are ...
2
votes
2answers
86 views

Where do non-associative rings appear?

When reading papers on algebraic topology, I often find the term "associative ring". The multiplication structure of a ring is normally assumed to be associative, therefore I guess that ...
1
vote
1answer
99 views

What is the contribution of group theory to topology?

An answer for a question on MathOverflow.net which asked for some recommendations on textbooks for books in topology received the following comment: "It's a great book to introduce applied ...
1
vote
4answers
239 views

The fundamental group of Cayley graph

Today I read a math book and find interested in Theorem. Every group has its graph representation. And we call it cayley graph. Now we sort out the question Firstly, if we have a group, ...
0
votes
0answers
41 views

Efficient ways of finding homotopies

This semester, I'm taking an introductory course on algebraic topology and I'm facing difficulties in finding a homotopy between two continuous paths. I understand the homotopy (homotopy relative to ...
0
votes
1answer
67 views

Path or One-Dimensional Manifold

The terminology of curve, path and one dimensional manifold is always in the textbooks about topology, differential manifold and riemannian geometry. The definition of path and one dimensional ...
3
votes
2answers
108 views

Properties of $\pi_n$ from a category theoretical point of view

This will be a more open question. I would like to have a better understanding about how the homotopy group functors behave with different constructions such as limits or colimitis. Some examples: ...
13
votes
2answers
273 views

Motivation for introducing algebraic topology?

What kind of topological questions does algebraic topology answer where point set topology is not enough? Phrased differently: Where is the line (or maybe intersection) between point set topology ...
4
votes
0answers
108 views

Relationship between paradoxes in logic and geometry/topology

Though I've been reading for years, this is my first question here. Believe it or not, I've tried the search feature- apologies if this is a duplicate. The main point of this post can be summarized ...
32
votes
13answers
2k views

How To Present Algebraic Topology To Non-Mathematicians?

I am writing my master thesis in algebraic topology (fundamental groups) and as a system in my school students must write about one page about their theses explaining for non mathematicians the ...
4
votes
3answers
207 views

Geometry and Physics

I have to do a presentation on Geometry and Physics. I am asking it here (rather than physics.se) because I have to focus on Geometry More than Physics. The intended audience is Undergraduate Seniors ...
8
votes
2answers
197 views

Ideas for a present to my topology teacher

Tomorrow is the my final lecture in my favorite course, algebraic topology. I want to give a present to my prof. as a keepsake, something along the lines of this only something I can make due ...
3
votes
0answers
174 views

Imagining four or higher dimensions and the difference to imagining three dimensions

I’m very interested in how people envision four or higher dimensions. And I’m especially interested in how geometers and topologists who actually work in four dimensions do. Now I know of the video ...
8
votes
1answer
236 views

what's the role of fiber bundles play in understanding the base space?

Suppose $\pi: E\to B$ be a fiber bundle, and assume $B$ is connected, I want to know: 1, What do the fiber bundles (e.g., the covering spaces) on $B$ help understand the topology, or the structure on ...
4
votes
3answers
221 views

Interesting theorems/facts about identification spaces

I am now studying algebraic topology (still at the beginning). I am now studying identification spaces, adjunction spaces,... As I still don't know how these concepts are going to be used, I think I ...
6
votes
3answers
1k views

Topology Prerequisites for Algebraic Topology

Note: There is another question of the same title, but it is different and asks for group theory prerequisites in algebraic topology, while i want the topology prerequisites. I am a physics ...
24
votes
3answers
1k views

Roadmap to study Atiyah-Singer index theorem

I am a physics undergrad and want to pursue a PhD in Math (geometry or topology). I study it almost completely by myself, as the program in my country offers very less flexibility to take non ...
1
vote
1answer
118 views

Intuition Regarding Homotopic Spaces

I am just starting to do some algebraic topology (very basic stuff) so have obviously just been introduced to the notion of homotopys, contractible spaces and homotopic spaces. It's this last one that ...
77
votes
3answers
8k views

Topology: The Board Game

Edit: I've drawn up some different rules, a map and some cards for playing an actual version of the game. They're available at my personal website with a Creative Commons Attribution 4.0 license. ...
6
votes
1answer
197 views

Is there any deep connection between algebraic topology and homological algebra on rings?

There is a deep connection between algebraic topology and homological algebra on groups. A group $G$ can be interpreted as the fundamental group of a covering space $Y \rightarrow X$. (Co)Homology ...
8
votes
2answers
371 views

Fast paced book in point-set topology to move on to algebraic topology

I am sorry, if this is a repetition of previous questions. But my case is sightly different. I am a physics undergrad who wants to shift to pure maths, and I want to study topology. The supreme ...
4
votes
1answer
113 views

Seek results in group theory obtained by applping algebraic topology tools

After studying the first section, especially the section 1.3 (covering spaces) in Hatcher's book, it is obvious that many classical topics in group theory, especially in infinite group theory, have ...
13
votes
1answer
1k views

Why did a famous mathematician say that algebraic topology was dead?

I found Novikov said that algebraic topology was dead in the early 1970's in this article. Segal had been one of Atiyah's first students, working on equivariant K-theory, and then other ...
11
votes
3answers
356 views

Module Theory for the Working Student

Question: What level of familiarity and comfort with modules should someone looking to work through Hatcher's Algebraic Topology possess? Motivation: I am taking my first graduate course in ...
11
votes
3answers
734 views

Category Theory usage in Algebraic Topology

First my question: How much category theory should someone studying algebraic topology generally know? Motivation: I am taking my first graduate course in algebraic topology next semester, and, ...
22
votes
3answers
2k views

Research in algebraic topology

I have started studying algebraic topology with the help of Armstrong(Basic), Massey, and Hatcher. If I plan to do research in algebraic topology in future: What else should I study after ...
3
votes
0answers
86 views

Importance of Poincare's Conjecture [duplicate]

Possible Duplicate: What is the importance of the Poincaré conjecture? It is rather well known that Poincare conjecture was proved by Perelman in 2003 given the amount of coverage it ...
2
votes
1answer
226 views

How much connection is there between Commutative Algebra and Algebraic Topology?

How much connection is there between Commutative Algebra and Algebraic Topology? I am looking for general highlights, not complex details.
1
vote
1answer
263 views

Radical Applications of Algebraic Topology

Are there any radical applications of algebraic topology? For example, in probability theory we look at sample spaces. Suppose the sample space is a torus (for example). Would computing homology ...
8
votes
0answers
243 views

The status of $\mathbb{R}$ in homotopy theory.

The definition of a path as a continuous map $I \rightarrow X$ is a completely natural one. But this raises two questions in my mind. First, what properties of the interval give rise to useful ...
13
votes
4answers
460 views

Useful fibrations

What are the most useful fibrations that one be familiar with in order to use spectral sequences effectively in algebraic topology? There's at least the four different Hopf fibrations and $S^1\to ...
8
votes
3answers
420 views

What is combinatorial homotopy theory?

Edit: After a discussion with t.b. we agreed that this question aims to a different answer from this one, for more information you can read the comment below. Many times I've heard people ...
8
votes
5answers
665 views

Algebraic topology, etc. for Mac Lane's “Categories for the Working Mathematician”

[NOTE: For reasons that I hope the question below will make clear, I am interested only in answers from those who have read Mac Lane's Categories for the working mathematician [CWM], or at least have ...
1
vote
1answer
189 views

Algebraic Topology in Simple Terms

I just wanted to clarify a few basic concepts in algebraic topology. Suppose one space is my room ($\text{Room} \ A$). Suppose the other space is another room in my house ($\text{Room} \ B$). So ...
34
votes
2answers
1k views

How much rigour is necessary?

I am taking a course in Algebraic Topology. We are using Hatcher as a textbook. One of the main problems I am facing with the textbook is its level of rigour. Example: On Pg 10, Hatcher mentions in ...
21
votes
3answers
1k views

Why algebraic topology is also called combinatorial topology?

I remember reading somewhere(at least more than once) that algebraic topology is also known by the name "Combinatorial Topology" which essentially tags the subject fundamentally with some counting ...
0
votes
1answer
123 views

Differences between finite and infinite CW complex categories

I am currently reading Dyer's Cohomology Theories and he in the very beginning makes the assumption to work in the category of finite CW complexes. This kills of many interesting objects ($K(G,n)$, ...
3
votes
1answer
364 views

Need for computation in pure Mathematics at the highest level?

I'm fourth year undergrad student and I've noticed the skills that I've built up to do computation isn't actually being used. A good example is algebraic topology, I've never really used calculus in ...
5
votes
0answers
284 views

homology groups, physical interpretation

One can roughly interpret Betti numbers as the number of n-dimensional holes of our space. Is there a reasonable interpretation for torsion coefficients? Moreover, for 'physical' applications, are ...
46
votes
2answers
2k views

Sheaf cohomology: what is it and where can I learn it?

As I understand it, sheaf cohomology is now an indispensable tool in algebraic geometry, but was originally developed to solve problems in algebraic topology. I have two questions about the matter. ...
26
votes
4answers
974 views

Geometry or topology behind the “impossible staircase”

This question on the topology of Escher games reminded me of a question I've had in my head for a little while now. Is there anything interesting geometric or topological that can be said about the ...
11
votes
3answers
820 views

Homology - why is a cycle a boundary?

I have a question about the basic idea of singular homology. My question is best expressed in context, so consider the 1-dimensional homology group of the real line $H_1(\mathbb{R})$. This group is ...