(May 2016) Construct a \(\Delta\)-complex structure, and use it to compute the simplicial homology groups, for the connected sum of two projective planes.
Solution
Let \(K=\mathbb{R}P^2\#\mathbb{R}P^2\). Take the unit square with corners \(P=(0,0)\), \(Q=(1,0)\), \(R=(0,1)\), \(S=(1,1)\), and identify its sides by the edges \[ a\colon P\to Q \text{ and } R\to S, \qquad b\colon P\to R \text{ and } S\to Q, \] so that the boundary word read from \(P\) counterclockwise is \(a\,b^{-1}a^{-1}b^{-1}\). Add the diagonal \(c\colon P\to S\).
1. The quotient is homeomorphic to \(K\).
The edge \(a\) occurs once with each exponent and \(b\) occurs twice with exponent \(-1\), so the quotient is a closed nonorientable surface. The identifications give \(P\sim R\) and \(Q\sim S\) (from \(a\)) and \(P\sim S\) and \(R\sim Q\) (from \(b\)), so all four corners become one vertex \(v\). With one vertex, two edges, and one face, \(\chi=1-2+1=0\). By the classification of closed surfaces, the nonorientable surface with \(\chi=0\) is \(\#^2\mathbb{R}P^2=K\).
2. The diagonal cuts the square into two \(2\)-simplices \(U=[P,R,S]\) and \(L=[P,S,Q]\), and together with \(v\) and the edges \(a,b,c\) this is a \(\Delta\)-complex structure on \(K\) with\[ \partial_2 U=a+b-c, \qquad \partial_2 L=-a+b+c, \qquad \partial_1=0. \]
In \(U=[P,R,S]\) the faces are \([R,S]=a\), \([P,S]=c\), \([P,R]=b\), each oriented from the lower to the higher vertex, so \(\partial_2 U=[R,S]-[P,S]+[P,R]=a-c+b\). In \(L=[P,S,Q]\) the faces are \([S,Q]=b\), \([P,Q]=a\), \([P,S]=c\), so \(\partial_2 L=[S,Q]-[P,Q]+[P,S]=b-a+c\). Every edge begins and ends at \(v\), so \(\partial_1=0\).
3. \(H_2(K)=0\).
If \(\partial_2(xU+yL)=0\), the coefficients of \(a\) and \(b\) give \(x-y=0\) and \(x+y=0\), so \(x=y=0\). There are no \(3\)-simplices, so \(H_2(K)=\ker\partial_2=0\).
4. \(H_1(K)\cong\mathbb Z\oplus\mathbb Z/2\).
\(H_1(K)=\ker\partial_1/\operatorname{im}\partial_2=\mathbb Z\langle a,b,c\rangle/\langle a+b-c,\,-a+b+c\rangle\). In the basis \(a,b,c'\) with \(c'=a+b-c\), the relations are \(c'\) and \(-a+b+(a+b-c')=2b-c'\). Hence the quotient is \(\mathbb Z\langle a,b\rangle/\langle 2b\rangle\cong\mathbb Z\oplus\mathbb Z/2\).
5. Q.E.D.
With one vertex and \(\partial_1=0\), \(H_0(K)=C_0\cong\mathbb Z\). Together with steps 3 and 4, \[ \boxed{H_0(K)\cong\mathbb Z,\qquad H_1(K)\cong\mathbb Z\oplus\mathbb Z/2,\qquad H_n(K)=0\ (n\ge2).} \]