§4. The topology of the functions Arg(z) and LOg z
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onto 'Y; this is what one calls a lifting of ' Y to r. One should pay attention
to the fact that in general A(t) depends on t itself and not only on the point
'Y(t), as shown by the map t ~ exp(it) of [0,471"] in C* with A(t) = t.
It is clear, Once again thanks to the intermediate value theorem, that the
liftings to r of a given path ' Y (resp. the uniform branches of the argument
along 'Y) can be deduced from one another by a vertical translation by (resp.
by addition of) a multiple of 271" independent of the parameter t. In other
words, the liftings to r of a continuous path traced in C* are entirely determined by their initial points, and a uniform branch of the argument or of
Cog along a path in C* is entirely determined by its value at the origin of the
path. If, for example, ' Y is t ~ exp(it), 0 ~ t ~ 671", the liftings of'Y to rare
arcs of the helix that makes three complete turns around the axis of r, the
trajectories of a person using a staircase to mount three storeys.
(viii) Liftings of a path: existence. Let us now show that every continuous
path ' Y : I ---+ C* possesses a lifting to r, or, what amounts to the same,
that there always exists a uniform branch of the argument along 'Y.
First, such a function exists on every sufficiently small neighbourhood of
any point to of I. Choose an open disc Dc C* with centre 'Y(to) and, on D, a
uniform branch A(z) of the argument in accordance with what we saw above;
since the function ' Y is continuous, 'Y( t) E D for all t sufficiently close to to;
the function t ~ Ab(t)], defined on a neighbourhood of to and continuous
like z ~ A( z), clearly answers the question.
This done, put I = [a, b], choose a number a E Argb(a)], and consider all
pairs (J, AJ) formed by an interval J = [a, u[ of origin a, with u < b, and of a
continuous uniform branch AJ(t) of the argument in J satisfying AJ(a) = a.
The function AJ satisfying these conditions is unique as we saw above. If
now (J', AJ') and (J", AJ") are two such pairs, we have AJ'(t) = AJ"(t) on
J' n J". Writing K = [a, v [ for the union of all these intervals J, we can then
unambiguously define a function AK(t) on K by putting AK(t) = AJ(t) for
all J such that t E J. The function AK(t) trivially satisfies AK(a) = a and
AK(t) E Arg['Y(t)] for all t E K. Since every t E K is interior to one of the
intervals J, the function AK is continuous on K, so that K, which satisfies
the conditions imposed on the intervals J, is the largest of these intervals.
Now there exists on a neighbourhood of v in I a continuous function
A(t) satisfying A(t) E Arg['Y(t)] on this neighbourhood. For t < v sufficiently
close to v, the two functions AK(t) and A(t) are simultaneously defined and
continuous; they are therefore equal, up to a (constant) multiple of 271"; after
subtracting it from A(t) we may assume that A(t) = AK(t) for all t < v
sufficiently close to v. This allows us to extend the function AK(t) up to the
point v, and beyond v if v < b. The second eventuality is excluded since K
is the largest of the intervals J considered above, qed.
In conclusion, for any continuous path t ~ 'Y(t) traced in C* it is possible
to choose the argument of 'Y(t) so that it is a continuous function of t. This
419
onto 'Y; this is what one calls a lifting of ' Y to r. One should pay attention
to the fact that in general A(t) depends on t itself and not only on the point
'Y(t), as shown by the map t ~ exp(it) of [0,471"] in C* with A(t) = t.
It is clear, Once again thanks to the intermediate value theorem, that the
liftings to r of a given path ' Y (resp. the uniform branches of the argument
along 'Y) can be deduced from one another by a vertical translation by (resp.
by addition of) a multiple of 271" independent of the parameter t. In other
words, the liftings to r of a continuous path traced in C* are entirely determined by their initial points, and a uniform branch of the argument or of
Cog along a path in C* is entirely determined by its value at the origin of the
path. If, for example, ' Y is t ~ exp(it), 0 ~ t ~ 671", the liftings of'Y to rare
arcs of the helix that makes three complete turns around the axis of r, the
trajectories of a person using a staircase to mount three storeys.
(viii) Liftings of a path: existence. Let us now show that every continuous
path ' Y : I ---+ C* possesses a lifting to r, or, what amounts to the same,
that there always exists a uniform branch of the argument along 'Y.
First, such a function exists on every sufficiently small neighbourhood of
any point to of I. Choose an open disc Dc C* with centre 'Y(to) and, on D, a
uniform branch A(z) of the argument in accordance with what we saw above;
since the function ' Y is continuous, 'Y( t) E D for all t sufficiently close to to;
the function t ~ Ab(t)], defined on a neighbourhood of to and continuous
like z ~ A( z), clearly answers the question.
This done, put I = [a, b], choose a number a E Argb(a)], and consider all
pairs (J, AJ) formed by an interval J = [a, u[ of origin a, with u < b, and of a
continuous uniform branch AJ(t) of the argument in J satisfying AJ(a) = a.
The function AJ satisfying these conditions is unique as we saw above. If
now (J', AJ') and (J", AJ") are two such pairs, we have AJ'(t) = AJ"(t) on
J' n J". Writing K = [a, v [ for the union of all these intervals J, we can then
unambiguously define a function AK(t) on K by putting AK(t) = AJ(t) for
all J such that t E J. The function AK(t) trivially satisfies AK(a) = a and
AK(t) E Arg['Y(t)] for all t E K. Since every t E K is interior to one of the
intervals J, the function AK is continuous on K, so that K, which satisfies
the conditions imposed on the intervals J, is the largest of these intervals.
Now there exists on a neighbourhood of v in I a continuous function
A(t) satisfying A(t) E Arg['Y(t)] on this neighbourhood. For t < v sufficiently
close to v, the two functions AK(t) and A(t) are simultaneously defined and
continuous; they are therefore equal, up to a (constant) multiple of 271"; after
subtracting it from A(t) we may assume that A(t) = AK(t) for all t < v
sufficiently close to v. This allows us to extend the function AK(t) up to the
point v, and beyond v if v < b. The second eventuality is excluded since K
is the largest of the intervals J considered above, qed.
In conclusion, for any continuous path t ~ 'Y(t) traced in C* it is possible
to choose the argument of 'Y(t) so that it is a continuous function of t. This