Denote $X=mP^2$ the sphere glued with $m$ Mobius bands. It has a polygon representation $a_1a_1...a_ma_m$, i.e. it's a quotient by a $2m$-sides polygon $P$. Let $o$ be the center point of $P$, $x_0$ a vertex of $P$.

Denote $X_1=P\backslash\{o\}$, then $X_1$ is homotopy to $\partial P$, a bouquet of $m$-circles. So $$ \pi_1(X_1,x_0)=<a_1,...,a_m>, $$

the free group generated by loops $a_1,...,a_m$.

Consider the inclusion map $$ f:\pi_1(P\backslash\{o\},x_0)\to \pi_1(P,x_0), $$ it's a surjective homomorphism. $a_1^2a_2^2...a_m^2\in \pi_1(P\backslash\{o\},x_0)$ is mapped to the trivial elment of $\pi_1(P,x_0)$.

Denote $[a_1^2...a_m^2]$ the normal subgroup generated by $a_1^2...a_m^2$, we know that $[a_1^2...a_m^2]\subset Ker f$. How to prove that $$ Ker f=[a_1^2...a_m^2] $$ without using the conclusion of Van-Kampen theorem? In this simple case, I prefer a direct proof.

  • $\begingroup$ I think it follows from the fact that the presentation complex of the presentation is the space whose fundamental group you are trying to compute. $\endgroup$ – HJRW Mar 15 at 17:52
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    $\begingroup$ Anything that uses a decomposition of the surface will be so close to Seifert-Van Kampen in spirit that I don't see how one could view it as different. Off the top of my head the only way I would imagine computing fundamental groups of surfaces without a decomposition of the surface would be by knowing the universal cover, and the group of covering transformations. This is effectively how the fundamental group of the circle is (typically) computed. $\endgroup$ – Ryan Budney Mar 15 at 23:07
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    $\begingroup$ The first such computation is due to Poincare (Papers on Fuchsian functions), about 30 years before van Kampen's theorem. He was using fundamental polygons in the hyperbolic plane. A modern reference would be Beardon's book on discrete groups or, more generally, Ratcliffe's book on hyperbolic manifolds. $\endgroup$ – Moishe Kohan Mar 16 at 4:45
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    $\begingroup$ The key words you want are "Poincare polygon theorem", or perhaps "Poincare fundamental polygon theorem". This question has some information. Entering those key words in my browser, this was the first hit. And, of course, look at the references in the answer of @MoisheKohan. $\endgroup$ – Lee Mosher Mar 16 at 20:12

As Ryan Budney points out, the only way to not use the ideas behind the Van Kampen theorem is to covering space theory. In the case of surfaces, almost all of them have rather famous contractible universal covers: $\mathbb R^2$ in the case of a torus and Klein bottle, and the hyperbolic plane for surfaces of higher genus. Ironically, dealing with our remaining surface -- showing that the 2-sphere is simply connected -- seems to need a Van Kampen style argument (or, as has been rightly pointed out in the comments, some approximation theorem): not hard, but curiously different.

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    $\begingroup$ Simply connectedness of the twosphere might be proven using sard. A continuous curve is homotopic to a smooth one, its image misses at least a point by Sard. Call such a choice of point $x$. One can contract the curve to $-x$. $\endgroup$ – Thomas Rot Mar 16 at 1:10
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    $\begingroup$ Or just cellular approximation $\endgroup$ – Connor Malin Mar 16 at 1:22
  • $\begingroup$ It's been a long time, but I recall the textbook by Singer and Thorpe takes this perspective. $\endgroup$ – Ryan Budney Mar 16 at 4:19
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    $\begingroup$ For $S^2$, an argument which uses the Lebesgue Number Lemma directly is pretty simple (that lemma, of course, is the main technical tool of a "Van Kampen style argument"). $\endgroup$ – Lee Mosher Mar 16 at 20:14

Any presentation of a given space as a CW-complex immediately gives rise to a presentation of the fundamental groupoid, and hence also the fundamental group.

Specifically, given such a presentation as a CW-complex C, the objects of the fundamental groupoid G are precisely the vertices of C. Each 1-cell of C yields a generating 1-morphism of G. Finally, each 2-cell of C yields a relation for G, induced by the boundary attaching map: the boundary traverses finitely many 1-cells, and each such 1-cell contributes the corresponding generating morphism or its inverse, depending on whether we traverse it in the original direction (given by the structure of a CW-complex) or its inverse.

Another approach is to observe that the given polygon can be considered as a simplicial set, with vertices o and x_0, 1-simplices a_1, …, a_n, and b_1, c_1, b_2, c_2, …, b_n, c_n (radial spokes that connect o and x_0), and 2n different 2-simplices, with the (2i-1)st and 2ith 2-simplices having edges b_i, c_i, a_i respectively c_i, b_{i+1}, a_i. Now one can read off the fundamental groupoid in exactly the same manner as before.

Once a system of generators and relations for the fundamental groupoid is written down, it can be easily converted to a system of generators and relations for the fundamental group.

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    $\begingroup$ Doesn’t knowing this gives a presentation of the fundamental groupoid require van Kampen? $\endgroup$ – Connor Malin Mar 15 at 22:58
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    $\begingroup$ @ConnorMalin: No, it does not. For example, you can use the CW-approximation theorem to represent elements in the fundamental group(oid) as cellular maps from [0,1] to the 1-skeleton and homotopies between them as cellular maps from a bigon to the 2-skeleton. From there you can deduce a presentation of the above type. And for simplicial sets it's even easier, since maps are already simplicial. $\endgroup$ – Dmitri Pavlov Mar 16 at 5:25

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