Find a presentation for the fundamental group of \(P^2 \# T\).
Solution
Goal: Compute a presentation for the fundamental group of the connected sum \(P^2 \# T\) of the real projective plane \(P^2\) and the 2-torus \(T = T^2\).
1. Cell complex and polygonal scheme construction:Proof:
1.1. The standard polygonal scheme for \(P^2\) is a 2-gon with boundary word \(a^2\), deformation retracting on a punctured disk to the loop \(a\).
1.2. The standard polygonal scheme for \(T^2\) is a 4-gon with boundary word \(b c b^{-1} c^{-1}\), deformation retracting on a punctured disk to the wedge of loops \(b\) and \(c\).
1.3. The connected sum \(P^2 \# T\) is formed by removing an open 2-disk from each surface and gluing along the resulting boundary circles \(S^1\).
1.4. The resulting 2-manifold has a CW structure with:- One 0-cell \(x_0\).- Three 1-cells (generating loops) \(a, b, c\).- One 2-cell attached along the boundary path obtained by concatenating the original boundary schemes:\[w = a^2 b c b^{-1} c^{-1}.\]
2. Presentation via Seifert-van Kampen Theorem:\(\pi_1(P^2 \# T) \cong \langle a, b, c \mid a^2 b c b^{-1} c^{-1} = 1 \rangle\).Proof:
2.1. Decompose \(P^2 \# T = U \cup V\) where \(U \simeq P^2 \setminus \{\text{pt}\}\) and \(V \simeq T^2 \setminus \{\text{pt}\}\).
2.2. \(U\) deformation retracts to \(S^1\) with \(\pi_1(U) = \langle a \mid - \rangle\), and the boundary cycle corresponds to \(a^2\).
2.3. \(V\) deformation retracts to \(S^1 \vee S^1\) with \(\pi_1(V) = \langle b, c \mid - \rangle\), and the boundary cycle corresponds to \(b c b^{-1} c^{-1}\).
2.4. The intersection \(U \cap V \simeq S^1\) is connected, with generator \(\gamma\) mapping to \(a^2 \in \pi_1(U)\) and \((b c b^{-1} c^{-1})^{-1} \in \pi_1(V)\).
2.5. By the Seifert-van Kampen theorem, the amalgamated free product is:\[\pi_1(P^2 \# T) \cong \pi_1(U) *_{\pi_1(U \cap V)} \pi_1(V) \cong \langle a, b, c \mid a^2 = c b c^{-1} b^{-1} \rangle \cong \langle a, b, c \mid a^2 b c b^{-1} c^{-1} = 1 \rangle.\]
3. Equivalent presentation via Dyck’s Theorem:By Dyck’s Theorem, \(P^2 \# T \cong \#^3 \mathbb{R}P^2\) (the non-orientable surface of genus 3), which has the equivalent standard presentation:\[\pi_1(P^2 \# T) \cong \langle x, y, z \mid x^2 y^2 z^2 = 1 \rangle.\]
4. Conclusion:A presentation for \(\pi_1(P^2 \# T)\) is \(\langle a, b, c \mid a^2 b c b^{-1} c^{-1} = 1 \rangle\). Q.E.D.