Abstract
We have worked quite hard to find a space whose fundamental group is non-trivial. We should capitalize on this result and see if we can find other, related spaces whose fundamental groups can now be computed easily as a result of our hard work. An example where this approach is successful is for product spaces.
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Notes
- 1.
This homomorphism is sometimes also written \(\pi _1(f)\).
- 2.
Reminder: Suppose that \(f : A \rightarrow B\) and \(g : B \rightarrow A\) are two maps such that \(f \circ g = {{\,\mathrm{id}\,}}\). Now, if \(g(x) = g(y)\), then applying f to both sides yields \(x=y\), hence g is injective. Also, in order to solve the equation \(f(x) = y \) given y, we simply use \(x := g(y)\). Hence f is surjective.
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Bray, C., Butscher, A., Rubinstein-Salzedo, S. (2021). Tools for Fundamental Groups. In: Algebraic Topology. Springer, Cham. https://doi.org/10.1007/978-3-030-70608-1_10
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DOI: https://doi.org/10.1007/978-3-030-70608-1_10
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