Don't the path components form a group, in general, with the operation being "component containing the composition of representative homeomorphisms"? So it's incurious to ask anything less than "what group?"

For the torus, I'm pretty sure that the answer is SL^{\pm}(2, \mathbb{Z}): https://en.wikipedia.org/wiki/Special_linear_group#SL%C2%B1(n,F) In particular, wlog let the torus be R^2 / Z^2, or equivalently a square with its edges appropriately identified. Now consider the path which traverses the four edges of this square (say, anticlockwise) and returns to the starting vertex. Apply the homeomorphism to this path, and then lift it to the universal cover R^2; this now bounds a tile of unit area whose vertices form a parallelogram. We can apply a homotopy to 'straighten' this tile into a parallelogram and henceforth only consider homeomorphisms given by linear maps. The result follows. I don't have the same intuition about the Klein bottle, unfortunately.

On Thu, Dec 20, 2018, 8:32 PM Dan Asimov <dasimov@earthlink.net wrote:

...

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? -----------------------------------------------------------------------

—Dan

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