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Topic: Image Homotopy Semi-Group
Replies: 0

 Davide P. Cervone Posts: 4 Registered: 12/6/04
Image Homotopy Semi-Group
Posted: Oct 29, 1993 11:20 AM

Prof. Conway:

I saw your request for the relations for the image homopty semi-group
mentioned in the geometry-forum.

If we let S be the standard torus, T be the "twisted" torus, and B+
and B- be the left- and right-handed 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 right-handed "twisted" form
(generated in the same way as the twisted torus: make a figure-8 tube
as before, but when the ends are joined, make only a 180 degree twist
so that the top of one figure-8 is joined to the bottom of the other
figure-8), and K- = B- # B- is the left-handed 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) 421--434.

(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) 475--481.

[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 three-space, Bull. Soc.
Math. Belg. 12 (1960) 5--22.

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