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Computational Topology (Jeff Erickson)
Homotopy of Curves on Surfaces
The inﬁnite possibilities that each day holds should stagger the mind. The sheer number of
experiences I could have is uncountable, breathtaking, and I’m sitting here refreshing my
inbox. We live trapped in loops, reliving a few days over and over, and we envision only a few
paths laid out ahead of us. We see the same things each day, respond the same way, we think
the same thoughts, every day a slight variation on the last, every moment smoothly following
the gentle curves of societal norms. We act like if we just get through today, tomorrow our
dreams will come back to us.
— Randall Munroe, “Dreams”,
http://xkcd.com/137/
7 Homotopy of Curves on Surfaces
In this lecture, we’ll see efﬁcient algorithms to determine whether a given loop
‘
on a given 2manifold
Σ
is contractible. (We previously considered this problem for polygons with holes in the plane.) To make
the problem concrete, we assume that
Σ
is represented by a polygonal schema
Π
with complexity
n
,
and
‘
is a closed walk of length
k
in the induced embedded graph
G
or its dual
G
*
. After developing
some necessary mathematical tools, I will describe a classical algorithm of Dehn
[
3
]
for the special case
when
G
is a system of loops, which runs in
O
(
k
)
time. Dehn’s algorithm and its later generalizations are
the foundation of
geometric group theory
. Next I will describe a simple extension of Dehn’s algorithm to
more general schemata that runs in
O
(
gn
+
g
2
k
)
time for surface of genus
g
. Finally, following a result
of Dey and Guha
[
4
]
, I will show how to improve the running time to
O
(
n
+
k
)
.
Let
Σ
be a ﬁxed, compact, connected 2manifold. For most of the lecture, for reasons that will
become clear, I will assume that
χ
(Σ)
<
0; thus,
Σ
is
not
the sphere, the projective plane, the torus, or
the Klein bottle. I will brieﬂy reconsider these surfaces at the end of the lecture.
7.1 The Fundamental Group
Recall the following deﬁnitions from Lecture 2. A
path
in
Σ
is a continuous function
π
:
[
0,1
]
→
Σ
. The
concatenation
π
·
σ
of two paths
π
and
σ
with
π
(
1
) =
σ
(
0
)
is the path
(
π
·
π
0
)(
t
)
:
=
(
π
(
2
t
)
if
t
≤
1
/
2,
σ
(
2
t

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

t
)
. We easily observe that
π
·
σ
=
σ
·
π
.
A
loop
is a path
π
whose endpoints coincide; this common endpoint is the loop’s
basepoint
. The
concatenation of two loops is a loop, and the reversal of a loop is a loop.
A
(path) homotopy
between paths
π
and
π
0
is a continuous function
h
:
[
0,1
]
×
[
0,1
]
→
Σ
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
]
. Two paths
π
and
π
0
are
homotopic
, written
π
’
π
0
, if there is a homotopy between
them. A loop is
contractible
if it is homotopic to a constant path at its basepoint.
Homotopy is an equivalence relation; we write
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This note was uploaded on 01/24/2012 for the course CS 598 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.
 Fall '08
 Staff

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