On Mon, Dec 24, 2018 at 10:35 PM Allan Wechsler <acwacw@gmail.com> wrote:
Wikipedia gives a different answer. See the article "Mapping Class Group", in the section "Examples", subsection "Nonorientable surfaces", where they say it's Z2 x Z2.
On Mon, Dec 24, 2018, 9:52 PM Dan Asimov <dasimov@earthlink.net wrote:
Think of the Klein bottle K as a cylinder S^1 x [0,1] after its two boundary circle S^1 x 0 and S^1 x 1 have been identified by a reflection of S^1 via (x, y) |—> (x, -y).
Then you cannot rotate the cylinder after identification (which has become K) around its axis, because its end circles must rotate in opposite directions. The two things you *can* do are combinations of
a) slide the cylinder along its length, and
b) reverse the cylinder's direction.
You can also flip the orientation of the circle. That gives the other Z/2Z Andy
All of the slides are continuously deformable to each other, so there are two equivalence classes, so the group is Z/2. (Not a proof, of course.)
—Dan
----- Here's a puzzle I think I know the answer to, but I don't have a proof:
Let K denote the Klein bottle, the ideal surface you get when you identify the top and bottom edges of the unit square [0,1] x [0,1] normally, by
(x,0) ~ (x,1),
but identify the left and right edges by a flip:
(0,y) ~ (1,1-y).
Puzzle: ----------------------------------------------------------------------- Consider the space Homeo(K) of self-homeomorphisms of the Klein bottle. I.e., Homeo(K) consists of all continuous bijections
h : K —> K
having a continuous inverse.
Two self-homeomorphisms h_0, h_1 of K are *in the same path component* of Homeo(K) if there is a continuous *family*
{h(t) | 0 <= t <= 1}
of homeomorphisms h(t) in Homeo(K) such that h(e) = h_e for e = 0, 1.
The continuity of this family just amounts to there being a continuous map
H : K x [0,1] —> K
such that the restriction of H to any time-slice K x {t}:
H | K x {t} —> K
is the homeomorphism h(t) : K —> K.
QUESTION: ————————— How many path components does Homeo(K) have? ----------------------------------------------------------------------- -----
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com