Show that the Klein bottle is homeomorphic to \(P^2 \# P^2\). [Hint: Split the square in Figure 74.11 along a diagonal, flip one of the resulting triangular pieces over, and paste the two pieces together along the edge labelled \(b\).]
Show how to picture the 4-fold projective plane as an immersed surface in \(\mathbb{R}^3\).
Solution
Use the standard square model of the Klein bottle. Its edge scheme may be written \[ aba^{-1}b. \] Cut the square along a diagonal. Flip one of the two resulting triangles and reglue along the edge labelled \(b\), as indicated in the hint. After this cut-and-paste, the boundary word becomes \[ aabb, \] which is the standard polygon scheme for the connected sum \(P^2\#P^2\). Cutting, flipping, and regluing along whole edges does not change the quotient surface, so \[ \boxed{K\cong P^2\#P^2.} \]
Consequently \[ P^2\#P^2\#P^2\#P^2\cong K\#K. \] Take two standard immersions of the Klein bottle in \(\mathbb R^3\), place them in disjoint balls, and choose on each a small disk lying in a regular embedded patch and disjoint from its self-intersection set. Remove the interiors of these two disks. Join the two resulting boundary circles by a thin embedded tube whose interior misses both immersed surfaces. On each original piece use the given Klein-bottle immersion, and on the tube use the embedding just chosen. Along collars of the boundary circles these maps agree after the usual smoothing/reparametrization, so together they give an immersion \[ K\#K\looparrowright\mathbb R^3. \] Via part (a), this is an immersion of the four-fold projective plane. Thus the desired picture is obtained by taking two immersed Klein bottles and joining them by a tube at regular patches.