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Computational Topology (Jeff Erickson)
Shortest Homotopic Paths
Oh! marvellous, O stupendous Necessity–by thy laws thou dost compel every effect to
be the direct result of its cause, by the shortest path. These [indeed] are miracles.
..
— Leonardo da Vinci,
Codex Atlanticus
(c. 1500)
translated by Jean Paul Richter (1883)
Those who cannot remember the past are condemned to repeat it.
— George Santayana,
Reason in Common Sense
(1905)
A straight line may be the shortest distance between two points,
but it is by no means the most interesting.
— The Doctor [Jon Pertwee],
The Time Warrior
(1973)
2 Shortest Homotopic Paths
2.1 A Few Deﬁnitions
Let
X
be any topological space. A
path
in
X
is a continuous function from the unit interval
[
0,1
]
to
X
,
and a
cycle
in
X
is a continuous function from the standard unit circle
S
1
:
=
{
(
x
,
y
)
∈
R
2

x
2
+
y
2
=
1
}
to
X
. A
loop
is a path whose endpoints coincide; this common endpoint is called the
basepoint
of the
loop. A path or cycle is
simple
if it is injective; a loop is simple if its restriction to
[
0,1
)
is injective. We
refer to paths, cycles, and loops collectively as
curves
.
The
JordanSchönﬂies Theorem
states that for any simple cycle
γ
in the plane, there is a homeo
morphism from the plane to itself whose restriction to
S
1
is
γ
. Thus, the image of any simple cycle
partitions the plane into two components, a bounded
interior
whose closure is homeomorphic to the
disk
B
2
, and an unbounded
exterior
.
For any paths
π
and
π
0
with
π
(
1
) =
π
0
(
0
)
, the
concatenation
π
·
π
0
is the path
(
π
·
π
0
)(
t
)
:
=
(
π
(
2
t
)
if
t
≤
1
/
2,
π
0
(
2
t

1
)
if
t
≥
1
/
2.
The
reversal
π
of a path
π
is the path
π
(
t
)
:
=
π
(
1

t
)
.
A
path homotopy
between paths
π
and
π
0
is a continuous function
h
:
[
0,1
]
×
[
0,1
]
→
X
such
that
h
(
0,
t
) =
π
(
t
)
and
h
(
1,
t
) =
π
0
(
t
)
for all
t
, and
h
(
s
,0
) =
π
(
0
) =
π
0
(
0
)
and
h
(
s
,1
) =
π
(
1
) =
π
0
(
1
)
for all
s
∈
[
0,1
]
. (We will omit the word ‘path’ when it is clear from context.) For all
t
∈
[
0,1
]
, the
function
s
7→
h
(
s
,
t
)
is a path from
π
(
0
)
to
π
(
1
)
. Two paths
π
and
π
0
are
(path) homotopic
if there
is a path homotopy between them; we write
π
’
π
0
to denote that
π
and
π
0
are homotopic. Tedious
deﬁnitionchasing implies that
’
is an equivalence relation. We refer to the equivalence classes as
homotopy classes
, and write
[
π
]
to denote the homotopy class of path
π
.
A homotopy between two paths.
A homotopy from a contractible loop to its basepoint.
We call a loop
‘
is
contractible
if it is pathhomotopic to the constant path mapping the entire
interval
[
0,1
]
to the basepoint
‘
(
0
)
. Two paths
π
and
π
0
with the same endpoints are homotopic if and
only if the loop
π
·
π
0
is contractible. A connected topological space
X
is
simply connected
if every loop
1
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Shortest Homotopic Paths
in
X
is contractible. For example, the plane and the sphere are both simply connected, but the annulus
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 Fall '08
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