The Real Projective Plane Triangulated

Triangulation The most efficient way to triangulate the real projective plane RP 2 is the following: A ¨¨ HH ¨ H ¨¨ HH ¨¨ HH ¨ H CDB @ @ @ @ (1) @@ @@ BFEC ¨ HH ¨ H @ ¨ H @ ¨ H ¨¨ HH@ ¨ H@ ¨ A

This is a hexagon with diametrically opposite points identified. All triangles are filled in.

Connected sum Simplicially, the connected sum S#T of two triangulated surfaces S and T is obtained by deleting one 2-simplex from each and identifying the corre- sponding faces in S and T of these two simplices. (There are choices involved, but they do not affect the result, in dimension two.) For example, to construct RP 2#RP 2, we may start by removing the 2-simplex AEF from (1) and redrawing to obtain the triangle

ABCA @ @ @@ FDB @ @ (2) @@ EC @ @ @@ A in which two of the edges are identified. The diagonal edge, AF EA, is the remaining boundary circle. You can check that all the necessary simplices are there and that no extra ones crept in. To form the connected sum, we take a second copy of RP 2, with vertices A0, B0 etc., except that we identify A0 = A, E0 = E, and F 0 = F . We simply reflect (2) in

110.615 Algebraic Topology JMB File: rp2tri, Revision B; 6 Nov 2003; Page 1 2 The Real Projective Plane Triangulated

the diagonal and relabel to obtain the square ABCA @ @@ B0 FDB @ (3) @@ C0 D0 EC @ @@ A B0 C0 A in which the top edge is identified with the right edge and the left edge is identified with the bottom edge. The M¨obius band To recognize RP 2 with a 2-simplex removed as the M¨obius band, we redraw (2) as the following rectangle AEF @ @@ CBDC (4) H HH @ HH@ H@ FA in which the left and right edges are to be identified, with a twist. The Klein bottle To see that RP 2#RP 2 is indeed the Klein bottle, we start from the M¨obiusband (4) and attach another one, corresponding to the second copy of RP 2. However, we do this in a non-obvious way, by distributing the second M¨obius band above and below (4) to obtain the rectangle

C0 B0 D0 C0 @ @ @@ @@ AEF @ @@ CBDC (5) H H @ HH H @ HH@ FA ¨ @ ¨ ¨¨ @ ¨ @ ¨¨ C0 B0 D0 C0 Here, the top and bottom edges are identified, and the left and right edges are iden- tified with a twist.

110.615 Algebraic Topology JMB File: rp2tri, Revision B; 6 Nov 2003; Page 2