Let \(C \subseteq \mathbb{R}^n\) be convex and \(x_0 \in C\text{.}\) Any loop \(f\) at \(x_0\) is null-homotopic via the straight-line homotopy \(H(s,t) = (1-t)f(s) + t x_0\text{,}\) which stays in \(C\) by convexity and fixes the basepoint. So \(\pi_1(C, x_0) = 0\text{;}\) in particular \(\pi_1(\mathbb{R}^n) = 0\) and \(\pi_1(D^n) = 0\text{.}\)
Section 1.4 First Examples and Computations
We have so far developed a fair amount of theory without computing a single fundamental group. Here are the computations available to us with the tools at hand, together with a statement of the one genuine calculation that will occupy the next lecture.
Example 1.20.
Proposition 1.21.
Let \(X\) and \(Y\) be spaces with basepoints \(x_0, y_0\text{.}\) The projections induce an isomorphism
\begin{equation*}
\pi_1\bigl(X \times Y, (x_0,y_0)\bigr) \xrightarrow{\ \cong\ } \pi_1(X, x_0) \times \pi_1(Y, y_0).
\end{equation*}
Proof.
A map \(Z \to X \times Y\) is the same thing as a pair of maps \(Z \to X\) and \(Z \to Y\text{,}\) and this correspondence is compatible with homotopies, i.e., a homotopy into \(X \times Y\) is precisely a pair of homotopies. Applying this with \(Z = I\) and \(Z = I \times I\) shows that \([f] \mapsto \bigl((p_X)_*[f], (p_Y)_*[f]\bigr)\) is a bijection, and it is a homomorphism because the projections are.
Proposition 1.22.
Let \(A \subseteq X\) be a retract, so there is \(r \colon X \to A\) with \(r \circ i = \mathrm{Id}_A\) for the inclusion \(i\text{.}\) Then \(i_* \colon \pi_1(A, a_0) \to \pi_1(X, a_0)\) is injective.
Proof.
By functoriality \(r_* \circ i_* = (r \circ i)_* = \mathrm{Id}\text{,}\) so \(i_*\) has a left inverse.
Example 1.23.
The punctured plane \(\mathbb{R}^2 \setminus \{0\}\) deformation retracts onto the unit circle via \(H(x,t) = (1-t)x + t\,x/\lvert x \rvert\text{,}\) and the Möbius band deformation retracts onto its core circle. In both cases the fundamental group agrees with that of \(S^1\text{.}\) Likewise \(\mathbb{R}^n \setminus \{0\}\) deformation retracts onto \(S^{n-1}\text{.}\)
We will review the proof of the following in the next lecture.
Theorem 1.24.
\(\pi_1(S^1, 1) \cong \mathbb{Z}\text{,}\) generated by the class of the loop \(\omega(s) = e^{2\pi i s}\text{.}\)
Granting the theorem, Proposition 1.21 computes the fundamental group of the torus,
\begin{equation*}
\pi_1(S^1 \times S^1) \cong \mathbb{Z} \times \mathbb{Z},
\end{equation*}
and more generally \(\pi_1(T^n) \cong \mathbb{Z}^n\text{,}\) while Example 1.23 gives \(\pi_1(\mathbb{R}^2 \setminus \{0\}) \cong \mathbb{Z}\text{.}\)
Remark 1.25.
It is worth stressing that \(\pi_1\) really does take values in \(\mathbf{Grp}\) and not in \(\mathbf{Ab}\text{.}\) The Seifert–van Kampen theorem of Chapter 4 will show that the figure eight \(S^1 \vee S^1\) has fundamental group the free group \(F_2\) on two generators. Contrast this with homology, which is abelian by construction; the fundamental group retains strictly more information.
Remark 1.26.
On the other hand \(\pi_1\) is far from a complete invariant. The spheres \(S^2\) and \(S^3\) are both simply connected but are not homotopy equivalent, as homology already detects. To see the difference homotopically we need the higher homotopy groups \(\pi_n\text{,}\) introduced in Chapter 5. Computing those for spheres is the problem that motivates most of this course.
