John Stillwell - Naive Lie Theory.pdf - Index of

196 9 Simply connected Lie groups
The subdivision of the square into small subsquares is done with the
following idea in mind:
• By making the subsquares sufficiently small we can ensure that their
images lie in ε-balls of R n for any prescribed ε.
• The bottom edge of the unit square can be deformed to the top
edge by a finite sequence of deformations d ij , each of which is the
identity map of the unit square outside a neighborhood of the (i, j)-
subsquare.
• It follows that if p can be deformed to q then the deformation can
be divided into a finite sequence of steps. Each step changes the
image only in a neighborhood of a “deformed square,” and hence in
an ε-ball.
To make this argument more precise, though without defining the d ij in
tedious detail, we suppose the effect of a typical d ij on the (i, j)-subsquare
to be shown by the snapshots shown in Figure 9.3. In this case, the bottom
and right edges are pulled to the position of the left and top edges, respectively,
by “stretching” in a neighborhood of the bottom and right edges and
“compressing” in a neighborhood of the left and top. This deformation
will necessarily move some points in the neighboring subsquares (where
such subsquares exist), but we can make the affected region outside the
(i, j)-subsquare as small as we please. Thus d ij is the identity outside a
neighborhood of, and arbitrarily close to, the (i, j)-subsquare.
Figure 9.3: Deformation d ij of the (i, j)-subsquare.
Now, if the (1,1)-subsquare is the one on the bottom left and there are
n subsquares in each row, we can move the bottom edge to the top through
the sequence of deformations d 11 ,d 12 ,...,d 1n ,d 2n ,...,d 21 ,d 31 , .... Figure
9.4 shows the first few steps in this process when n = 4.
Since each d ij is a map of the unit square into itself, equal to the identity
outside a neighborhood of an (i, j)-subsquare, the composite map d ◦ d ij
(“d ij then d”) agrees with d everywhere except on a neighborhood of the