§ 2. Differential Forms of Degree 1
179
previous formulas in order to obtain for 1/3 ≤ u ≤ 2/3, the arc 1 ≤ t ≤ 2 of
ν s (t) arising from the deformation of μ 1 :
u −→
σ(3su, 2)
for 0 ≤ u ≤ 1/3 ,
σ [s, 3(1 − u)] for 1/3 ≤ u ≤ 2/3 ,
σ [3s(1 − u), 1] for 2/3 ≤ u ≤ 1 ,
(7.5”)
define a fixed endpoint-homotopic path to μ 1 . When s = 1, by assumption,
the path ν s reduces to a point c. Up to parametrization, the paths (5’) and
(5”) are then clearly identical. As they are (fixed-endpoint) homotopic to
respectively μ 0 and μ 1 , the same also holds for these.
Homotopy theory includes many small calculations of this type. After
having done them once or twice, diagrams are substituted for them, to which
the reader is in turn free to substitute his small calculations.
Any star domain G with respect to a point a is clearly simply connected:
homotheties with centre a and ratio s ∈ [0, 1] deform closed paths into a
unique point. More generally, the same holds for any contractible domain G
in the sense of the previous chapter. The main difference with the case of
domains in C is that in R
n , a simply connected domain is not necessarily
contractible (counterexample: the open subset bounded by two concentric
spheres).
(iii) The Banach space C
1/2 (I; E). As in n
◦ 3, (ii) of Chapter VIII, the
computation (3) of the derivative of F (μ + sν) can be interpreted in terms of
differential calculus in a Banach space.
28 The vector space C
1/2 (I, E) of C
1/2
maps μ : I −→ E, where E is the ambient Cartesian space, can be equipped
with the norm
|
|
|μ| |
| = μ I + μ
I
inspired from distribution theory (Chap. V, § 10), or maybe it is the other way
round ; C
1/2 (I, E) then becomes a complete space, i.e. a Banach space: the
proof is the same as in Chapter VIII. If C
1/2 (I; G) is the set of all admissible
paths in the given open subset G of E, then C
1/2 (I; G) is clearly an open
subset of C
1/2 (I, E), i.e. for any μ ∈ C(I; G), any path ν ∈ C(I, E) with
sufficiently small |
|
|μ − n| |
|, or merely sufficiently small μ − ν I , is also in
C
1/2 (I; G).
The function F (μ) =
ω[μ(t); μ
(t)]dt, defined on the open set C
1/2 (I, G),
is everywhere continuous on it. For this, let μ, ν be two elements in this open
set. To find an upper bound for |F (ν) − F (μ)| for given μ and ν near μ, it is
necessary to search for a uniform upper bound of
ω [ν(t); ν
(t)] − ω [μ(t); μ
(t)] = p i [ν(t)] Dν
i (t) − p i [μ(t)] Dμ
i (t)
if ω = p i (x)dx
i . The problem is similar to that of the proof of the continuity of
a product: the uniform continuity of the p i (x) in a compact neighbourhood
28 To understand the rest of the book, reading this n
◦ is not essential.
179
previous formulas in order to obtain for 1/3 ≤ u ≤ 2/3, the arc 1 ≤ t ≤ 2 of
ν s (t) arising from the deformation of μ 1 :
u −→
σ(3su, 2)
for 0 ≤ u ≤ 1/3 ,
σ [s, 3(1 − u)] for 1/3 ≤ u ≤ 2/3 ,
σ [3s(1 − u), 1] for 2/3 ≤ u ≤ 1 ,
(7.5”)
define a fixed endpoint-homotopic path to μ 1 . When s = 1, by assumption,
the path ν s reduces to a point c. Up to parametrization, the paths (5’) and
(5”) are then clearly identical. As they are (fixed-endpoint) homotopic to
respectively μ 0 and μ 1 , the same also holds for these.
Homotopy theory includes many small calculations of this type. After
having done them once or twice, diagrams are substituted for them, to which
the reader is in turn free to substitute his small calculations.
Any star domain G with respect to a point a is clearly simply connected:
homotheties with centre a and ratio s ∈ [0, 1] deform closed paths into a
unique point. More generally, the same holds for any contractible domain G
in the sense of the previous chapter. The main difference with the case of
domains in C is that in R
n , a simply connected domain is not necessarily
contractible (counterexample: the open subset bounded by two concentric
spheres).
(iii) The Banach space C
1/2 (I; E). As in n
◦ 3, (ii) of Chapter VIII, the
computation (3) of the derivative of F (μ + sν) can be interpreted in terms of
differential calculus in a Banach space.
28 The vector space C
1/2 (I, E) of C
1/2
maps μ : I −→ E, where E is the ambient Cartesian space, can be equipped
with the norm
|
|
|μ| |
| = μ I + μ
I
inspired from distribution theory (Chap. V, § 10), or maybe it is the other way
round ; C
1/2 (I, E) then becomes a complete space, i.e. a Banach space: the
proof is the same as in Chapter VIII. If C
1/2 (I; G) is the set of all admissible
paths in the given open subset G of E, then C
1/2 (I; G) is clearly an open
subset of C
1/2 (I, E), i.e. for any μ ∈ C(I; G), any path ν ∈ C(I, E) with
sufficiently small |
|
|μ − n| |
|, or merely sufficiently small μ − ν I , is also in
C
1/2 (I; G).
The function F (μ) =
ω[μ(t); μ
(t)]dt, defined on the open set C
1/2 (I, G),
is everywhere continuous on it. For this, let μ, ν be two elements in this open
set. To find an upper bound for |F (ν) − F (μ)| for given μ and ν near μ, it is
necessary to search for a uniform upper bound of
ω [ν(t); ν
(t)] − ω [μ(t); μ
(t)] = p i [ν(t)] Dν
i (t) − p i [μ(t)] Dμ
i (t)
if ω = p i (x)dx
i . The problem is similar to that of the proof of the continuity of
a product: the uniform continuity of the p i (x) in a compact neighbourhood
28 To understand the rest of the book, reading this n
◦ is not essential.