# Easy papers on fundamental groups (for beginners)

I'd like to read some papers concerning fundamental groups, for example, papers written to explain some basic facts about homotopy explicitly for undergraduate students.

All the papers I have requires many background knowledge (homology, for example) but I'd a paper for young students.

I know that there are many good books but usually in books we find the theory explained in a row, or in the order just to read and follow. I'd like to start some research on a low level .

Suggestions are welcome. Best wishes.

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The fundamental group has been a basic tool in the topologist's kit for over a century. All modern research presupposes familiarity with it. If you want to read research papers where the fundamental group is explained, you might consider looking up Poincare's original series of papers introducing it (discussed here: map.mpim-bonn.mpg.de/Poincar%C3%A9%E2%80%99s_homology_sphere). – Neal Dec 11 '12 at 15:41

There are many books in Algebraic Topology that discuss the fundamental group without talking about homology (singular/cellular or simplicial). Resources I have used:

1. Hatcher - Algebraic Topology (used in last semester's MATH 4204 at ANU)
2. Bredon - Geometry and Topology
3. Rotman - Algebraic Topology

There is no doing research without going through the basics and slogging it out in understandinh the full theory first.

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Thanks. I know those books, but as I said, I'd like some paper around 10 pages in English with some non standard computations. In a book we can find almost done. I'd like to motivate the research. For examples, papers from 'The American Mathematical Monthly' (I didn't find any). I'll start with those books but I'd like to finish with some paper. – Sigur Dec 11 '12 at 11:03
@Sigur In my experience those books are perfectly fine in explaining the general theory of the fundamental group. – user38268 Dec 11 '12 at 11:05
I agree with you. I like all those books. But I have to insert some paper reading to the project. The writing style is different. It is important to read some paper with some applications, even if basic ones. – Sigur Dec 11 '12 at 11:10
@Sigur The applications are plenty in those books. To name just two: The Brouwer fixed point theorem for $D^2$ and The fundamental theorem of algebra. And you don't need to know of universal coefficients, the Ext functor or Mayer - Vietoris to understand the proofs of these facts. – user38268 Dec 11 '12 at 11:11
Almost any paper in three-manifold theory uses the fundamental group in an essential way. For example, this paper on homology spheres I found through Wikipedia mentions $\pi_1$ in the very second sentence: jstor.org/… – Neal Dec 11 '12 at 15:44

I find that most books on algebraic topology make things more difficult for students than seems necessary by not using paths of "arbitrary length" so that the paths under composition form a category, that is composition is associative and each path has a left identity and a right identity. That is one can define a path (of length $r$) for some $r \geqslant 0$ in $X$ to be a map $f: [0,r] \to X$; or to be a pair $(f,r)$ where $r \geqslant 0$ and $f: [0, \infty) \to X$ is constant on $[r, \infty)$.

Second the notion of the fundamental groupoid $\pi_1(X,A)$ of $X$ on a set $A$ of base points was introduced by me in 1967, and has many advantages over the usual fundamental group. See my answer to this question: http://mathoverflow.net/questions/40945/compelling-evidence-that-two-basepoints-are-better-than-one

This tool allows more powerful theorems with in many cases simpler or clearer proofs, and is developed and applied in my book Topology and Groupoids, the 2006 edition of a book published in 1968; this groupoid $\pi_1(X,A)$ is used in no other topology text in English, to my knowledge. See also this downloadable book Categories and Groupoids.

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