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Felix Hilsky
masterthesis
Commits
842cb814
Commit
842cb814
authored
Aug 07, 2022
by
Felix Hilsky
Browse files
add version of alternative proof
parent
c6c4ed4e
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1
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tex/proof_continuous_lift_simplified.tex
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842cb814
%! TEX program = lualatex
\input
{
.maindir/tex/header/preamblesection
}
% inputs the preamble only if necessary
\docStart
\providecommand*
{
\ver
}
[2]
{
#1 [#2]
}
% define only if not defined already
% makes sure that this bit is compilable on its own but does not overwrite
% other definitions
\begin{proof}
% of theorem Orientability of continuous line fields
As
\cref
{
thm:projection
_
as
_
a
_
covering
_
map
}
shows,
\ver
{$
P ⫶ 𝕊M → 𝒬
^{
𝕊
}
M
$}{$
P
_
N ⫶ 𝕊
^{
N

1
}
→ 𝒬
^{
𝕊'
}
ℝ
^
N
$}
is a covering map.
Use
$
Q
_{
p
_
0
}$
and a
$
n
_
0
∈ 𝕊M
$
with
$
P
(
n
_
0
)
=
Q
_{
p
_
0
}$
as the distinguished points.
Connected manifolds are pathconnected and locally pathconnected.
\ask
{
it's correct, do I need to proof it? Not hard but needs some lines.
}
% $Q_* ⸨ π_1(M, p_0) ⸩ ⊆ P_* ⸨ π_1(𝕊M, n_0) ⸩$ (for $n_0 ∈ 𝕊_{p_0}M$ such that $Q_{p_0} = P(n_0)$)
% is equivalent to all generators of $π_1(M, p_0)$ being in $P_* ⸨ π_1(𝕊M, n_0) ⸩$.
For the
\enquote
{
if
}
part
let
$
γ ∈ G
$
,
\ie
$
γ ⫶
[
0
,
1
]
→ M
$
continuous with
$
γ
(
0
)
=
γ
(
1
)
=
p
_
0
$
.
$
Q
$
is orientable along
$
γ
$
. That means that there exists
$
n ∈
$
\ver
{$
Γ
_
C
(
\rest
{
𝕊M
}{
γ
([
0
,
1
])
}
)
$}{$
C
(
γ
([
0
,
1
])
, 𝕊
^{
N

1
}
)
$}
with
$
P
(
n
)
=
Q
$
.
If the orientation of
$
Q
$
along this path has
$
n
(
γ
(
0
))
=

n
_
0
$
, use
$

n
$
instead.
Then
$
n ∘ γ
$
is a continuous loop at
$
n
_
0
$
in
$
𝕊M
$
with
$
P
_
*
[
n ∘ γ
]
=
[
P ∘ n ∘ γ
]
=
[
Q ∘ γ
]
=
Q
_
*
[
γ
]
$
Since
$
γ
$
is an arbitrary generator of
$
π
_
1
(
M, p
_
0
)
$
and
$
Q
_
*
$
and
$
P
_
*
$
are group homomorphisms,
we get that
$
Q
_
*(
π
_
1
(
M, p
_
0
))
⊆ P
_
*(
π
_
1
(
𝕊M, n
_
0
))
$
.
Hence there exists
$
n ⫶ C
(
M, 𝕊M
)
$
with
$
P
(
n
)
=
Q
$
by
\cref
{
thm:lifting
_
of
_
continuous
_
maps
}
,
\ie
$
Q
$
is orientable
\ver
{
and
$
n ∈ Γ
_
C
(
𝕊M
)
$}{}
.
For the
\enquote
{
and only if
}
part restrict an orientation for
$
Q
$
on the paths in
$
G
$
.
\end{proof}
% of theorem Orientability of continuous line fields
\docEnd
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