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X. Fix x " X. The evaluation map evx : (G, e) ’! (X, x) is defined by
evx(g) = g · x .
This induces the evaluation homomorphism
evx : À1(G, e) ’! À1(X, x) .
#
The main reference for the following is [?, §4]. We will see that in general the
image of À1(G, e) in À1(X, x) is a central subgroup of À1(X, x) and independent of
the base point.
1.10.1 Lemma. Let g " P (G, e) and p " P (X, x) (paths emanating from e and x,
respectively). Then the three paths
g(2t) · x, 0 d" t d" 1/2,
(A)
g(1) · p(2t - 1), 1/2 d" t d" 1
(B) g(t) · p(t), 0 d" t d" 1,
p(2t), 0 d" t d" 1/2,
(C)
g(2t - 1) · p(1), 1/2 d" t d" 1
are homotopic by fixed end-point homotopies.
22 1. TRANSFORMATION GROUPS
Proof. Schematically, we will have a diagram
Figure 2: middle:g(t) · p(t)
Introduce a path in P (G, e) by
g(2t), 0 d" t d" 1/2,
(g " cg(1))(t) =
g(1), 1/2 d" t d" 1
and a path in P (X, x) by
x, 0 d" t d" 1/2,
(cx " p)(t) =
p(2t - 1), 1/2 d" t d" 1,
(c denotes the constant path). Now g " cg(1) g implies (g " cg(1)) · p g · p relative
"I. But (g " cg(1)) · p = (g · p) " g(1) · p is the path labeled (A), while g · p is the path
labeled (B). Thus the path(A) is homotopic to (B) via a fixed end-point homotopy.
A similar argument relates (C) to (B).
1.10.2 Theorem. The image of evx : À1(G, e) ’! À1(X, x) is a central subgroup of
#
À1(X, x) which is independent of choices of x.
Proof. Suppose g " P (G, e) and p " P (X, x) are closed loops. Then evx is
#
induced by
g(t) ’! g(t) · x.
Noting that g(1) = e and p(1) = x, we have
g(2t) · x, 0 d" t d" 1/2,
g(t) · x " p(t) =
g(1) · p(2t - 1), 1/2 d" t d" 1
while
p(2t), 0 d" t d" 1/2,
p(t) " (g(t) · x) =
g(2t - 1) · p(1), 1/2 d" t d" 1.
In view of Lemma 1.10.1, these two loops represent the same element of À1(X, x),
[g] · [p] = [p] · evx ([g]),
#
thus the image of evx lies in the center of À1(X, x).
#
Let y be another point in X and C a path from x to y. Then C induces an
isomorphism C# : À1(X, y) -’! À1(X, x) by sending a loop based at y to a loop
C " " C based at x. If = g(t) · x, with = g(t) · y, then C " " C (for the
homotopy just moves along C). Choosing a different path C will send to a loop
homotopic to . Therefore, evx(À1(G, e)) is independent of choice.
"
1.11. LIFTING CONNECTED GROUP ACTIONS 23
1.11. Lifting Connected Group Actions
1.11.1 Theorem ([?, §4], [?, Chapter I, §9]). let G be a path-connected topological
group acting on a space X which admits covering space theory. Let K be a normal
subgroup of À1(X, x) containing the image of evx : À1(G, e) ’! À1(X, x) and put
#
Q = À1(X, x)/K. Then
(1) The G-action on X lifts to a G-action on XK which commutes with the
covering Q-action.
(2) If in addition, X is completely regular, G is a connected Lie group acting
properly on X, then the G-action on XK is proper and the induced action
c&
of Q on W = G\XK is also proper. Consequence of pre-
vious theorems??
Proof. (1) The G-action on XK is described as follows. Given u " G and x " XK,
select a path g " P (G, e) with g(1) = u, and choose a path p which represents x.
We define u · x to be the common equivalence class of the three paths listed in
Lemma 1.10.1. In particular, u · x is represented by a path (g · x) " (u · p) connecting
x and u · y, where y = p(1). Suppose g " P (G, e) also has g (1) = u, and p also
represents x. Then
((g · x) " (u · p)) " (g · x) " (u · p ) (g · x) " ((g · x) " (g · x)) " (u · p) " (u · p ) " (g · x)
((g " g ) · x) " ((g · x) " u · (p " p ) " (g · x)).
Since K contains image{evx : À1(G, e) ’! À1(X, x)}, (g"g )·x represents an element
#
of K.
Now consider the path (g ·x)"u·(p"p )"(g ·x) = (g ·x)"g (1)·(p"p )"(g ·x).
The map
F (t, s) = (g (st) · x) " g (s) · (p(t) " p (t)) " (g (st) · x)
gives a homotopy from F (t, 0) = p(t)"p (t) to F (t, 1) = (g ·x)"g (1)·(p"p )"(g ·x).
Also, since p " p represents an element of K, the definition of u · x does not depend
on the choices of g and p.
Recall that the covering action of Q on XK is induced from the action of
À1(X, x) on X by juxtaposition. Suppose ± " À1(X, x) is represented by a closed
loop at x. By Lemma 1.10.1, the part 0 d" t d" 3/4 of the two paths
ñø
ôø
òøg(2t) · x, 0 d" t d" 1/2,
u · (4t - 2), 1/2 d" t d" 3/4,
ôø
óøu · p(4t - 3), 3/4 d" t d" 1;
and
ñø
ôø 0 d" t d" 1/2,
òø (2t),
g(4t - 2) · x, 1/2 d" t d" 3/4,
ôø
óøu · p(4t - 3), 3/4 d" t d" 1.
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