Let \(X\) be the triangular parachute formed from the standard 2-simplex \(\Delta^2\) by identifying the three vertices with one another.
Compute a presentation for \(\pi_1(X)\).
Show that \(\pi_1(X)\) is isomorphic to a free group \(F_n\) for some \(n\); what is \(n\)?
Solution
Let the three oriented edges of the boundary of \(\Delta^2\) be \(a,b,c\), oriented consecutively around the triangle. After the three vertices are identified, denote their common image by \(v\).
1. The quotient has a CW structure with one \(0\)-cell \(v\), three loop \(1\)-cells \(a,b,c\), and one \(2\)-cell whose attaching word is\[ abc. \]
Identifying the three vertices changes only the boundary of the standard \(2\)-simplex. Each of its three edges becomes a loop based at the common vertex \(v\), while the interior of the simplex remains an open \(2\)-cell. Traversing the boundary once in the chosen orientation traverses the three edge loops in order \[ a,b,c. \] Thus the attaching map of the unique \(2\)-cell represents the word \(abc\) in the free group on the three \(1\)-cells.
2. Therefore\[ \pi_1(X,v) \cong \langle a,b,c\mid abc=1\rangle. \]
1. Hence the displayed presentation follows.
3. The presentation in 2 defines the free group on two generators:\[ \pi_1(X,v)\cong F_2. \]
The single relation \[ abc=1 \] is equivalent to \[ c=b^{-1}a^{-1}. \] Thus \(c\) can be eliminated from the presentation, leaving \[ \langle a,b\mid\ \rangle, \] the free group on \(a\) and \(b\). Equivalently, define \[ \Phi:F(a,b)\longrightarrow\langle a,b,c\mid abc=1\rangle \] by the generators \(a\mapsto a\), \(b\mapsto b\). It is surjective because the relation gives \(c=b^{-1}a^{-1}\). Conversely the assignments \[ a\mapsto a, \qquad b\mapsto b, \qquad c\mapsto b^{-1}a^{-1} \] satisfy the relation \(abc=1\) and therefore define a homomorphism in the reverse direction. The two maps are inverse on generators. Hence the group is \(F_2\), so the requested rank is \[ n=2. \]