# Tagged Questions

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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Imagining four or higher dimensions and the difference to imagining three dimensions

Iâ€™m very interested in how people envision four or higher dimensions. And Iâ€™m especially interested in how geometers and topologists who actually work in four dimensions do. Now I know of the video ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
### 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 ...