(June ’08) Let \(X\) be the triangle parachute formed from the standard \(2\)-simplex \(\Delta^2\) by identifying the three vertices with one another. Compute a presentation for \(\pi_1(X)\) and show that \(\pi_1(X)\) is isomorphic to a free group \(F_n\) (and identify which \(n\)!).
Solution
After the three vertices of \(\Delta^2\) are identified, the three edges become three loops \(a,b,c\) based at the single vertex. The interior of the triangle is one \(2\)-cell. Orient the boundary cyclically; its attaching word is \[ abc. \] Thus van Kampen gives \[ \pi_1(X)\cong\langle a,b,c\mid abc=1\rangle. \] The relation eliminates \(c\): \(c=(ab)^{-1}\). Hence \[ \pi_1(X)\cong\langle a,b\mid\ \rangle\cong F_2. \] Therefore the requested rank is \[ \boxed{n=2}. \]