# The Fundamental Group of the Circle, Part 2

/**EDIT 11/22/16: ***I encourage the reader to skip this post and proceed directly to Part 3. Part 2 merely contains the justification of a shortcut that we never actually use in the remainder of this series. In particular, we are certainly *not* proving 'well-definedness,' even though that's what I claim. (Apologies, dear readers!) For more, see my explanation in the comments below. *

*Algebraic Topology*that the fundamental group of the circle is isomorphic to $\mathbb{Z}$. Recall our outline:

__Part 1__: Set-up/observations

__Part 2:__ Show $\Phi$ is well defined

__Part 3:__ Show $\Phi$ is a group homomorphism

__Part 4:__ Show $\Phi$ is surjective

__Part 5:__ Show $\Phi$ is injective (Note: parts 4 and 5 require two lemmas whose proofs we will defer until part 5)

__Part 6:__ Prove the two lemmas used in parts 4 and 5

## Claim: Φ is Well-Defined

To this end, let $\widetilde{\gamma}_n$ be any *other* path from $0$ to $n$ in $\mathbb{R}$. Then $\widetilde{\omega}_n\simeq \widetilde{\gamma}_n$ are homotopic by the straight line homotopy $F:I\times I\to\mathbb{R}$ where $F(s,t)=(1-t)\widetilde{\omega}_n+t\widetilde{\gamma}_n$.

- $G(s,0)=p(F(s,0))=p(\widetilde{\omega}_n(s))=\omega_n(s)$
- $G(s,1)=p(F(s,1))=p(\widetilde{\gamma}_n(s))=\omega_n'(s)$.

Next time, we'll show that $\Phi$ is in fact a group homomorphism.