Calculate \(H_1(P^2 \# T)\). Assuming that the list of compact surfaces given in Theorem 75.5 is a complete list, to which of these surfaces is \(P^2 \# T\) homeomorphic?
Solution
1. Cellular presentation and computation of \(H_1(P^2 \# T)\):
1.1. The connected sum \(X = P^2 \# T\) admits a standard CW decomposition with one 0-cell, three 1-cells \(c, a, b\), and one 2-cell attached along the boundary loop \(c^2 [a, b] = c^2 a b a^{-1} b^{-1}\).
connected sum of polygonal schemas for \(P^2\) (\(c^2\)) and \(T\) (\(aba^{-1}b^{-1}\)).
1.2. The cellular chain complex for \(X\) is:\[ 0 \longrightarrow \mathbb{Z} \xrightarrow{\partial_2} \mathbb{Z}c \oplus \mathbb{Z}a \oplus \mathbb{Z}b \xrightarrow{\partial_1} \mathbb{Z} \longrightarrow 0. \]
cellular homology definition.
1.3. The boundary map \(\partial_2: C_2(X) \to C_1(X)\) maps the generator of \(C_2(X)\) to:\[ \partial_2(e^2) = 2c + a + b - a - b = 2c. \]
abelianizing the boundary loop word \(c^2 a b a^{-1} b^{-1}\).
1.4. The first homology group is:\[ H_1(P^2 \# T) = \frac{\mathbb{Z}c \oplus \mathbb{Z}a \oplus \mathbb{Z}b}{\operatorname{im}(\partial_2)} = \frac{\mathbb{Z}c}{2\mathbb{Z}c} \oplus \mathbb{Z}a \oplus \mathbb{Z}b \cong \mathbb{Z}_2 \oplus \mathbb{Z} \oplus \mathbb{Z} \cong \mathbb{Z}^2 \oplus \mathbb{Z}_2. \]
quotient of free abelian groups.
2. Identification with standard compact surface:
2.1. \(P^2 \# T\) is non-orientable because it contains an embedded Möbius band from the \(P^2\) summand.
non-orientability of \(P^2\).
2.2. Compute the Euler characteristic using \(\chi(A \# B) = \chi(A) + \chi(B) - 2\):\[ \chi(P^2 \# T) = \chi(P^2) + \chi(T) - 2 = 1 + 0 - 2 = -1. \]
\(\chi(P^2) = 1\) and \(\chi(T) = 0\).
2.3. By the Classification Theorem for compact surfaces, every closed connected non-orientable surface is homeomorphic to the connected sum of \(k\) projective planes \(k P^2 = P^2 \# \cdots \# P^2\), which has Euler characteristic \(\chi(k P^2) = 2 - k\).
Theorem 75.5.
2.4. Setting \(2 - k = -1\) gives \(k = 3\).Thus \(P^2 \# T \cong P^2 \# P^2 \# P^2 = 3P^2\) (the connected sum of three projective planes, Dyck’s surface).
Dyck’s Theorem / classification invariants.
3. Conclusion:\(H_1(P^2 \# T) \cong \mathbb{Z}^2 \oplus \mathbb{Z}_2\), and \(P^2 \# T\) is homeomorphic to \(P^2 \# P^2 \# P^2\) (\(3 P^2\)). Q.E.D.
1 and 2.