simplicial complexes in a series of preparatory lectures for the Fall 2013 online course MATH7450 22M305 Topics in Topology Scientific and Engineering Applications of Algebraic Topology Target Audience Anyone interested in ID: 585907
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Slide1
Optional Lecture: A terse introduction to simplicial complexesin a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic TopologyTarget Audience: Anyone interested in topological data analysis including graduate students, faculty, industrial researchers in bioinformatics, biology, computer science, cosmology, engineering, imaging, mathematics, neurology, physics, statistics, etc.
Isabel
K. Darcy
Mathematics Department/Applied Mathematical & Computational Sciences
University
of
Iowa
http://www.math.uiowa.edu/~
idarcy/AppliedTopology.html Slide2
v2
e
2
e
1
e
3
v
1
v
3
2-simplex = oriented face
= (
v
1
, v
2
, v
3
)
1
-simplex = oriented edge
= (
v
1
, v
2
)
e
v
1
v
2
0-simplex = vertex
= v
Building blocks for oriented simplicial complexSlide3
2-simplex = oriented face = f = (v1, v2, v3
)
=
(v
2
,
v
3, v
1) = (v3
, v1, v2
)
– f
= (v2
, v1, v3
) = (v3
, v2, v1
) = (v
1, v3, v
2
)
v
2
e
2
e
1
e
3
v
1
v
3
v
2
e
2
e
1
e
3
v
1
v
3
Building blocks for oriented simplicial complexSlide4
3-simplex = σ = (v1, v2, v3, v4) = (v2, v3, v1,
v
4
)
=
(v
3
, v1, v
2, v4)=
(v2, v1, v4
, v3) = (v
3, v2, v4
, v1) = (v1
, v3, v4, v
2)= (v4
, v2
, v1, v3
) = (v4,
v
3, v2, v
1) = (v
4, v1, v3
, v2)= (v
1, v4, v2, v
3) = (v2, v
4, v3, v1
) = (v3, v
4, v1, v
2) – σ = (v2, v1, v3, v4) = (v3, v
2, v1, v4) = (v1, v3, v
2, v4
)
= (v2
, v4, v1, v
3) = (v3,
v4, v2, v
1) = (v1
, v4,v3
, v2)= (v
1, v2
, v4, v3) =
(v2
, v3, v
4, v1)
= (v3, v1
, v4, v2)
=
(v4, v1
, v2, v3)
= (v4, v
2, v3, v1
)
=
(
v
4
,
v
3
, v
1
,
v
2
)
Building blocks for oriented simplicial complex
v
4
v
3
v
1
v
2Slide5
v2
e
2
e
1
e
3
v
1
v
3
2-simplex = oriented face
= (
v
1
, v
2
, v
3
)
1
-simplex = oriented edge
= (
v
1
, v
2
)
Note that the boundary of this edge is v
2
– v1
e
v
1
v
2
0-simplex = vertex
= v
Note that the boundary
of this face is the cycle
e
1
+ e
2
+ e
3
= (
v
1
,
v
2
)
+ (v
2
,
v
3
)
–
(
v
1
, v
3
)
= (
v
1
, v
2
)
–
(
v
1
, v
3
)
+ (v
2
, v
3
)
Building blocks for oriented simplicial complexSlide6
3-simplex = (v1, v2, v3, v4) = tetrahedronboundary of (v1, v2
, v
3
, v
4
) =
– (
v1, v2, v3
) + (v1, v2, v4)
– (v1, v3, v4) + (v2
, v3, v4) n-simplex = (v
1, v2, …, vn+1)
v
4
v
3
v
1
v
2
Building blocks for oriented simplicial complexSlide7
v2
e
2
e
1
e
3
v
1
v
3
2-simplex = face
=
{v
1
, v
2
, v
3
}
Note that the boundary
of this face is the cycle
e
1 + e2 + e3
= {v1, v2
} + {v2, v3} +
{v1,
v3}1-simplex = edge = {v1, v
2} Note that the boundary of this edge is v2 + v1
e
v
1
v
2
0-simplex = vertex
= v
Building blocks for
an
unoriented
simplicial complex
using
Z
2
coefficientsSlide8
3-simplex = {v1, v2, v3, v4} = tetrahedronboundary of {v1, v2
, v
3
,
v
4
}
={v1, v2,
v3} + {v1, v
2, v4} + {v1, v
3, v4} + {v2, v3
, v4} n-simplex = {
v1, v2, …, vn+1}
v
4
v
3
v
1
v
2
Building blocks for a simplicial complex using
Z
2
coefficients