lecture 5: triangulations & simplicial complexes (and cell complexes). in a series of...
TRANSCRIPT
![Page 1: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/1.jpg)
Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology
Target Audience: Anyone interested in topological data analysis including graduate students, faculty, industrial researchers in bioinformatics, biology, business, 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
![Page 2: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/2.jpg)
v2
e2e1
e3
v1 v3
2-simplex = triangle = {v1, v2, v3} Note that the boundary
of this triangle is the cycle
e1 + e2 + e3
= {v1, v2} + {v2, v3} + {v1, v3}
1-simplex = edge = {v1, v2} Note that the boundary
of this edge is v2 + v1 ev1 v2
0-simplex = vertex = v
Building blocks for a simplicial complex
![Page 3: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/3.jpg)
3-simplex = {v1, v2, v3, v4} = tetrahedron
boundary of {v1, v2, v3, v4} ={v1, v2, v3} + {v1, v2, v4} + {v1, v3, v4} + {v2, v3, v4}
n-simplex = {v1, v2, …, vn+1}
v4
v3v1
v2
Building blocks for a simplicial complex
![Page 4: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/4.jpg)
3-simplex = {v1, v2, v3, v4} = tetrahedron
boundary of {v1, v2, v3, v4} ={v1, v2, v3} + {v1, v2, v4} + {v1, v3, v4} + {v2, v3, v4}
n-simplex = {v1, v2, …, vn+1}
v4
v3v1
v2
Building blocks for a simplicial complex
v4
v3v1
v2Fill in
![Page 5: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/5.jpg)
Creating a simplicial complex
0.) Start by adding 0-dimensional vertices (0-simplices)
![Page 6: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/6.jpg)
Creating a simplicial complex
1.) Next add 1-dimensional edges (1-simplices).Note: These edges must connect two vertices.I.e., the boundary of an edge is two vertices
![Page 7: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/7.jpg)
Creating a simplicial complex
1.) Next add 1-dimensional edges (1-simplices).Note: These edges must connect two vertices.I.e., the boundary of an edge is two vertices
![Page 8: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/8.jpg)
Creating a simplicial complex
1.) Next add 1-dimensional edges (1-simplices).Note: These edges must connect two vertices.I.e., the boundary of an edge is two vertices
![Page 9: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/9.jpg)
Creating a simplicial complex
2.) Add 2-dimensional triangles (2-simplices).Boundary of a triangle = a cycle consisting of 3 edges.
![Page 10: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/10.jpg)
Creating a simplicial complex
2.) Add 2-dimensional triangles (2-simplices).Boundary of a triangle = a cycle consisting of 3 edges.
![Page 11: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/11.jpg)
Creating a simplicial complex
3.) Add 3-dimensional tetrahedrons (3-simplices).Boundary of a 3-simplex = a cycle consisting of its four 2-dimensional faces.
![Page 12: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/12.jpg)
Creating a simplicial complex
3.) Add 3-dimensional tetrahedrons (3-simplices).Boundary of a 3-simplex = a cycle consisting of its four 2-dimensional faces.
![Page 13: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/13.jpg)
4.) Add 4-dimensional 4-simplices, {v1, v2, …, v5}.
Boundary of a 4-simplex = a cycle consisting of 3-simplices.= {v2, v3, v4, v5} + {v1, v3, v4, v5} + {v1, v2, v4, v5}
+ {v1, v2, v3, v5} + {v1, v2, v3, v4}
![Page 14: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/14.jpg)
Creating a simplicial complex
n.) Add n-dimensional n-simplices, {v1, v2, …, vn+1}.
Boundary of a n-simplex = a cycle consisting of (n-1)-simplices.
![Page 15: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/15.jpg)
circle = { x in R2 : ||x || = 1 }
Example: Triangulating the circle.
![Page 16: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/16.jpg)
circle = { x in R2 : ||x || = 1 }
Example: Triangulating the circle.
![Page 17: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/17.jpg)
circle = { x in R2 : ||x || = 1 }
Example: Triangulating the circle.
![Page 18: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/18.jpg)
circle = { x in R2 : ||x || = 1 }
Example: Triangulating the circle.
![Page 19: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/19.jpg)
circle = { x in R2 : ||x || = 1 }
Example: Triangulating the circle.
![Page 20: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/20.jpg)
disk = { x in R2 : ||x || ≤ 1 }
Example: Triangulating the disk.
![Page 21: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/21.jpg)
disk = { x in R2 : ||x || ≤ 1 }
Example: Triangulating the disk.
![Page 22: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/22.jpg)
disk = { x in R2 : ||x || ≤ 1 }
Example: Triangulating the disk.
![Page 23: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/23.jpg)
disk = { x in R2 : ||x || ≤ 1 }
Example: Triangulating the disk.
![Page 24: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/24.jpg)
disk = { x in R2 : ||x || ≤ 1 }
=
Example: Triangulating the disk.
![Page 25: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/25.jpg)
sphere = { x in R3 : ||x || = 1 }
Example: Triangulating the sphere.
![Page 26: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/26.jpg)
sphere = { x in R3 : ||x || = 1 }
Example: Triangulating the sphere.
![Page 27: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/27.jpg)
sphere = { x in R3 : ||x || = 1 }
Example: Triangulating the sphere.
![Page 28: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/28.jpg)
sphere = { x in R3 : ||x || = 1 }
Example: Triangulating the sphere.
![Page 29: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/29.jpg)
Example: Triangulating the circle.
disk = { x in R2 : ||x || ≤ 1 }
=
![Page 30: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/30.jpg)
Example: Triangulating the circle.
disk = { x in R2 : ||x || ≤ 1 }
![Page 31: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/31.jpg)
Example: Triangulating the circle.
disk = { x in R2 : ||x || ≤ 1 }
Fist image from http://openclipart.org/detail/1000/a-raised-fist-by-liftarn
![Page 32: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/32.jpg)
sphere = { x in R3 : ||x || = 1 }
Example: Triangulating the sphere.
![Page 33: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/33.jpg)
sphere = { x in R3 : ||x || = 1 }
Example: Triangulating the sphere.
=
![Page 34: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/34.jpg)
Creating a cell complex
Building block: n-cells = { x in Rn : || x || ≤ 1 }
Examples: 0-cell = { x in R0 : ||x || < 1 }
1-cell =open interval ={ x in R : ||x || < 1 } ( )
3-cell = open ball = { x in R3 : ||x || < 1 }
2-cell = open disk = { x in R2 : ||x || < 1 }
![Page 35: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/35.jpg)
v2
e2e1
e3
v1 v3
2-simplex = triangle = {v1, v2, v3} Note that the boundary
of this triangle is the cycle
e1 + e2 + e3
= {v1, v2} + {v2, v3} + {v1, v3}
1-simplex = edge = {v1, v2} Note that the boundary
of this edge is v2 + v1 ev1 v2
0-simplex = vertex = v
Building blocks for a simplicial complex
![Page 36: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/36.jpg)
3-simplex = {v1, v2, v3, v4} = tetrahedron
boundary of {v1, v2, v3, v4} ={v1, v2, v3} + {v1, v2, v4} + {v1, v3, v4} + {v2, v3, v4}
n-simplex = {v1, v2, …, vn+1}
v4
v3v1
v2
v4
v3v1
v2Fill in
Building blocks for a simplicial complex
![Page 37: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/37.jpg)
Creating a cell complex
Building block: n-cells = { x in Rn : || x || ≤ 1 }
2-cell = open disk = { x in R2 : ||x || < 1 }
3-cell = open ball = { x in R3 : ||x || < 1 }
Examples: 0-cell = { x in R0 : ||x || < 1 }
1-cell =open interval ={ x in R : ||x || < 1 } ( )
![Page 38: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/38.jpg)
Creating a cell complex
Building block: n-cells = { x in Rn : || x || ≤ 1 }
2-cell = open disk = { x in R2 : ||x || < 1 }
Examples: 0-cell = { x in R0 : ||x || < 1 }
1-cell =open interval ={ x in R : ||x || < 1 } ( )
3-cell = open ball = { x in R3 : ||x || < 1 }
![Page 39: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/39.jpg)
Example: disk = { x in R2 : ||x || ≤ 1 }
( )U U=
=Simplicial complex
Cell complex
![Page 40: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/40.jpg)
Example: disk = { x in R2 : ||x || ≤ 1 }
( )U U=
=Simplicial complex
Cell complex
![Page 41: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/41.jpg)
Example: disk = { x in R2 : ||x || ≤ 1 }
( )U U=
=Simplicial complex
Cell complex
( )U
![Page 42: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/42.jpg)
Example: disk = { x in R2 : ||x || ≤ 1 }
( )U U=
=Simplicial complex
Cell complex
[ ]U
![Page 43: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/43.jpg)
Example: disk = { x in R2 : ||x || ≤ 1 }
( )U U=
=Simplicial complex
Cell complex
[ ]U =
![Page 44: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/44.jpg)
Example: disk = { x in R2 : ||x || ≤ 1 }
( )U U=
=Simplicial complex
Cell complex
[ ]U =U
![Page 45: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/45.jpg)
Example: disk = { x in R2 : ||x || ≤ 1 }
( )U U=
=Simplicial complex
Cell complex
[ ]U =U =
![Page 46: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/46.jpg)
Euler characteristic (simple form):
= number of vertices – number of edges + number of faces
Or in short-hand,
= |V| - |E| + |F|where V = set of vertices E = set of edges F = set of 2-dimensional faces
& the notation |X| = the number of elements in the set X.
![Page 47: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/47.jpg)
Example: disk = { x in R2 : ||x || ≤ 1 }
( )U U=
=Simplicial complex3 vertices, 3 edges, 1 triangle
Cell complex1 vertex, 1 edge, 1 disk.
[ ]U =U =
![Page 48: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/48.jpg)
Euler characteristic:
Given a simplicial complex C,
let Cn = the set of n-dimensional simplices in C, and
let |Cn| denote the number of elements in Cn. Then
= |C0| - |C1| + |C2| - |C3| + …
= Σ (-1)n |Cn|
![Page 49: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/49.jpg)
Euler characteristic:
Given a cell complex C,
let Cn = the set of n-dimensional cells in C, and
let |Cn| denote the number of elements in Cn. Then
= |C0| - |C1| + |C2| - |C3| + …
= Σ (-1)n |Cn|
![Page 50: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/50.jpg)
Example: sphere = { x in R3 : ||x || = 1 }
Simplicial complex4 vertices, 6 edges, 4 triangles
CellComplex1 vertex, 1 disk
=
U =
![Page 51: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/51.jpg)
Example: sphere = { x in R3 : ||x || = 1 }
U =
Simplicial complex
Cellcomplex
=
![Page 52: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/52.jpg)
Example: sphere = { x in R3 : ||x || = 1 }
U =
Simplicial complex
Cellcomplex
U
=
![Page 53: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/53.jpg)
Example: sphere = { x in R3 : ||x || = 1 }
U =
Simplicial complex
Cellcomplex
U
=
=
![Page 54: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/54.jpg)
Example: sphere = { x in R3 : ||x || = 1 }
U =
Simplicial complex
Cellcomplex
U
=
=Fist image from http://openclipart.org/detail/1000/a-raised-fist-by-liftarn
![Page 55: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/55.jpg)
Example: sphere = { x in R3 : ||x || = 1 }
U =Cellcomplex
U
=
=
![Page 56: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/56.jpg)
Example: sphere = { x in R3 : ||x || = 1 }
U =Cell complex
U
=
=
![Page 57: Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305)](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f9499450d/html5/thumbnails/57.jpg)
Example: sphere = { x in R3 : ||x || = 1 }
U =
Simplicial complexCell complex
Cellcomplex
U
=
=