In Hatcher's Algebraic Topology, one of the first results encountered is that the fundamental group of the circle, $\pi_1(S^1)$, is isomorphic to the integers $\mathbb{Z}$. notated in the book as $\pi_1(S^1)\approx\mathbb{Z}$.

However, in other documents I have found online (take the solution to problem 16 (a) in this document, for instance), I keep seeing the equality ($=$) symbol in the place of the isomorphic ($\approx$) symbol, i.e. instead of $\pi_1(S^1)\approx\mathbb{Z}$ it is written as $\pi_1(S^1)=\mathbb{Z}$.

So my question is: which is it? That is, what notation is technically correct? Or are they both correct?

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    $\begingroup$ Yes, they mean the same thing. Really, the equal sign is not correct, because $\pi_1$, as a set, cointains element of type $[\gamma]$, i.e class of homotopic equivalent paths, not numbers. When you see an equal sign followed by a number group, really it's a isomorphism. Even if the paper you linked, sometime the author uses the equal sign, sometime the isomorphism sign. $\endgroup$ – Marco All-in Nervo Mar 6 at 5:38
  • $\begingroup$ @MarcoAll-inNervo Yeah I noticed that they used both notations, which is why I was even more confused! Thanks for the clarification. $\endgroup$ – Thy Art is Math Mar 7 at 2:06

There is a good article about 'equality': Mazur - When is one thing equal to some other thing.

  • $\begingroup$ That was a fantastic read, and was exactly what I needed to see. $\endgroup$ – Thy Art is Math Mar 7 at 2:07

Definitely not equality, just isomorphisim. Equality is a much stronger property (of sets) while isomorphism just means equivalence in that current category you're in (so groups for your case).

  • $\begingroup$ That's what I figured, but I wanted to make sure! Thanks for the answer. $\endgroup$ – Thy Art is Math Mar 7 at 2:08

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