Computing the Fundamental Group of the Circle

Loops on the Circle and Their Lifts

As a loop travels around the circle, it may go around once, several times, or travel forward and then back. The endpoint of the loop alone cannot distinguish these possibilities: every based loop starts and ends at the same point. A lift of the loop to the real line records the travel before it is wrapped onto the circle. Its endpoint will provide an integer that counts the loop’s net winding.

Let \(S^1=\{z\in\mathbb{C}:|z|=1\}\) have the subspace topology inherited from the usual topology on \(\mathbb{C}\), and use \(1\in S^1\) as basepoint. We use the covering map \(p:\mathbb{R}\to S^1\) given by \(p(t)=e^{2\pi i t}\), where \(\mathbb{R}\) has its usual topology. In particular, \(p(t)=1\) exactly when \(t\in\mathbb{Z}\).

The standard path-lifting result for the covering map \(p\) says that every path \(\alpha:I\to S^1\) with \(\alpha(0)=1\) has a unique continuous lift \(\widetilde{\alpha}:I\to\mathbb{R}\) satisfying \(\widetilde{\alpha}(0)=0\) and \(p\circ\widetilde{\alpha}=\alpha\). We call this the normalized lift. If \(\alpha\) is a based loop, then \(p(\widetilde{\alpha}(1))=\alpha(1)=1\); hence \(\widetilde{\alpha}(1)\in\mathbb{Z}\).

Winding Number and the Loop Class

To assign a winding number to a class of loops, rather than to a chosen representative, we need the number to be unchanged under based homotopy. The homotopy invariance of winding number is established in the earlier material on homotopy invariance of winding number: if two based loops are homotopic through based loops, their normalized lifts have the same endpoint. Thus \(w([\alpha])=w(\alpha)\) is well defined on \(\pi_1(S^1,1)\).

Concatenation Adds Winding Numbers

For based loops \(\alpha,\beta\), their concatenation \(\alpha*\beta\) is defined by \((\alpha*\beta)(t)=\alpha(2t)\) for \(0\le t\le\frac12\) and \((\alpha*\beta)(t)=\beta(2t-1)\) for \(\frac12\le t\le1\). The pieces agree at the joining time because \(\alpha(1)=1=\beta(0)\).

Proof. Let \(A\) and \(B\) be the normalized lifts of \(\alpha\) and \(\beta\). Write \(m=A(1)\) and \(n=B(1)\), both integers. Define \(L:I\to\mathbb{R}\) by \(L(t)=A(2t)\) for \(0\le t\le\frac12\) and \(L(t)=m+B(2t-1)\) for \(\frac12\le t\le1\). At \(t=\frac12\), the first formula gives \(A(1)=m\), and the second gives \(m+B(0)=m\), so \(L\) is continuous. It starts at \(A(0)=0\). On the first half, \(p(L(t))=\alpha(2t)\). On the second half, \(p(L(t))=p(m+B(2t-1))=p(B(2t-1))=\beta(2t-1)\), since \(m\in\mathbb{Z}\) and \(p(u+m)=p(u)\). Thus \(L\) is the normalized lift of \(\alpha*\beta\), and \(L(1)=m+B(1)=m+n\). This proves the winding-number formula. Since \(W\) is well defined on classes and concatenation represents the group operation, the formula says exactly that \(W\) is a homomorphism. \(\square\)

The Fundamental Group of the Circle

Proof. The map is a homomorphism by the concatenation result. We prove that it is surjective and injective.

For surjectivity, let \(n\in\mathbb{Z}\) be arbitrary, including \(n=0\) or a negative integer. Define \(\gamma_n(t)=p(nt)\). Then \(\gamma_n(0)=p(0)=1\) and \(\gamma_n(1)=p(n)=1\), so it is a based loop. Its normalized lift is \(t\mapsto nt\), whose endpoint is \(n\). Therefore \(W([\gamma_n])=n\), and every integer occurs.

For injectivity, suppose \(W([\alpha])=W([\beta])=n\). Let \(A,B:I\to\mathbb{R}\) be the normalized lifts. They have the same endpoints: \(A(0)=B(0)=0\) and \(A(1)=B(1)=n\). Define \(H(s,t)=p((1-s)A(t)+sB(t))\). This is continuous. For every \(s\), its value at \(t=0\) is \(p(0)=1\), and its value at \(t=1\) is \(p((1-s)n+sn)=p(n)=1\). Also, \(H(0,t)=\alpha(t)\) and \(H(1,t)=\beta(t)\). Thus \(H\) is a homotopy through based loops, so \([\alpha]=[\beta]\). The map \(W\) is injective and therefore is an isomorphism. \(\square\)

This result computes the group, not just the winding number of individual examples: two based loops represent the same element of \(\pi_1(S^1,1)\) exactly when they have the same winding number. In particular, the constant loop corresponds to \(0\), and concatenation of classes corresponds to addition in \(\mathbb{Z}\).

A Common Pitfall

Common mistake: Since a based loop ends at \(1\), one might conclude that the endpoint of its lift must be \(0\). That confuses the endpoint in \(S^1\) with the endpoint in \(\mathbb{R}\). For example, the loop \(\alpha(t)=p(2t)\) ends at \(p(2)=e^{4\pi i}=1\), but its normalized lift ends at \(2\), not \(0\). The integer is essential: reducing the lifted endpoint modulo integers erases the information that distinguishes different loop classes.