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Image Homotopy
Posted:
Oct 29, 1993 7:20 AM


Prof. Conway:
I saw your request for the relations for the image homopty semigroup mentioned in the geometryforum.
If we let S be the standard torus, T be the "twisted" torus, and B+ and B be the left and righthanded versions of the Boy surface, then the defining relations are
S # S = T # T B+ # B+ # B+ # B+ = B # B # B # B S # B+ = B+ # B+ # B S # B = B+ # B # B T # B+ = B # B # B T # B = B+ # B+ # B+
where # represents connected sum.
As a consequence of this, every surface can be written as the connected sum of one of ten basic surfaces together with zero or more handles:
/ S T \ _ B+ B _ # S # ... # S  K+ K K  \____ ____/ \ K+ # B+ K # T K # B / V zero or more
where K = B+ # B is the standard immersion of the Klein bottle with reflective symmetry, K+ = B+ # B+ is the righthanded "twisted" form (generated in the same way as the twisted torus: make a figure8 tube as before, but when the ends are joined, make only a 180 degree twist so that the top of one figure8 is joined to the bottom of the other figure8), and K = B # B is the lefthanded version (twist the opposite direction before joining the ends of the tube).
The relevant paper is:
[1] U. Pinkall, Regular homotopy classes of immersed surfaces, Topology 24 (1985) 421434.
(unfortunately, he uses "regular homotopy" to mean the same thing as "image homopty", though he does note that some call it image homopty).
Also of interest are:
[2] U. Pinkall, Tight surfaces and regular homotopy, Topology 25 (1986) 475481.
[3] W. Kuehnel and U. Pinkall, Tight smoothing of some polyhedral surfaces, Proc. Conf. Global Diff. Geom. and Global Analysis, Berlin 1984, Lecture Notes in Mathematics 1156, Springer, Berlin, 1985.
[4] N. H. Kuiper, On surfaces in Euclidean threespace, Bull. Soc. Math. Belg. 12 (1960) 522.
In [3], Kuehnel and Pinkall give examples of tightly immersed surfaces in each immersion class for all but finitely many surfaces for which tight immersions are possible. They left (essentially) twelve classes without examples. Part of my recent doctoral thesis develops new examples for all but two of these cases. One of these remaining two is likely not to admit a tight immersion; the fate of the other is less clear.
I hope this helps.
Davide P. Cervone The Geometry Center dpvc@geom.umn.edu



