Let \(X\) be the topological space obtained by identifying three distinct points on the torus \(S^1\times S^1\). Calculate the fundamental group of \(X\).
Solution
1. Let\[ T=S^1\times S^1, \qquad A=\{p_0,p_1,p_2\}\subseteq T \]be the three points that are identified.Then\[ X=T/A. \]
This is just notation for the quotient in the problem: \(T/A\) denotes the space obtained by collapsing the three-point subspace \(A\) to one point and leaving every other point distinct.
2. The quotient \(T/A\) is homotopy equivalent to the mapping cone\[ T\cup_A CA, \]where \(CA\) is the cone on the three-point space \(A\).
Choose a triangulation of the torus having \(p_0,p_1,p_2\) as vertices. Then \(A\hookrightarrow T\) is the inclusion of a subcomplex, hence a cofibration.
For a cofibration \(A\hookrightarrow T\), the ordinary quotient \(T/A\) is homotopy equivalent to the homotopy cofiber, represented by the mapping cone \[ T\cup_A CA. \] Thus the stated quotient may be replaced, for purposes of computing fundamental groups, by this mapping-cone model.
3. The mapping-cone model is homotopy equivalent to\[ T\vee S^1\vee S^1. \]
The cone \(CA\) is a triod: it has a cone vertex \(c\) and three edges joining \(c\) to \(p_0,p_1,p_2\).
First collapse the edge joining \(c\) to \(p_0\). It is a contractible subcomplex, so collapsing it is a homotopy equivalence. After this collapse, the cone vertex is identified with \(p_0\), and the remaining two cone edges are \(1\)-cells attached to \(T\) with endpoint pairs \[ (p_0,p_1) \qquad\text{and}\qquad (p_0,p_2). \]
Since \(T\) is path-connected, choose paths in \(T\) from \(p_1\) to \(p_0\) and from \(p_2\) to \(p_0\). These paths give homotopies of the two attaching maps \[ S^0\longrightarrow T \] to attaching maps that send both endpoints of each \(1\)-cell to \(p_0\). Adjunction spaces obtained from homotopic CW attaching maps are homotopy equivalent. Therefore the two attached intervals may, up to homotopy, be replaced by two loops based at \(p_0\). The resulting space is \[ T\vee S^1\vee S^1. \]
4. Hence\[ \pi_1(X) \cong \pi_1(T)*\ZZ*\ZZ. \]
By 2 and 3, \[ X\simeq T\vee S^1\vee S^1. \] The Seifert–van Kampen theorem gives the fundamental group of a wedge of path-connected CW complexes as the free product of their fundamental groups. Thus \[ \pi_1(X) \cong \pi_1(T)*\pi_1(S^1)*\pi_1(S^1) \cong \pi_1(T)*\ZZ*\ZZ. \]
5. Therefore\[ \boxed{ \pi_1(X) \cong \ZZ^2*F_2 \cong \left\langle a,b,c,d\ \middle|\ [a,b]=1\right\rangle . } \]
The torus has \[ \pi_1(T)\cong\ZZ^2 \cong \langle a,b\mid [a,b]=1\rangle, \] while the two additional circle factors generate the free group \[ F_2=\langle c,d\rangle. \] Taking the free product produces exactly the displayed presentation.