# Does $\pi_1(S^1)\approx \mathbb{Z}$ mean the same thing as $\pi_1(S^1)=\mathbb{Z}$?

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?

• 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. – Marco All-in Nervo Mar 6 at 5:38
• @MarcoAll-inNervo Yeah I noticed that they used both notations, which is why I was even more confused! Thanks for the clarification. – Thy Art is Math Mar 7 at 2:06