CS 6170: Computational Topology, Spring 2019Lecture 09

Topological Data Analysis for Data Scientists

Dr. Bei Wang

School of ComputingScientific Computing and Imaging Institute (SCI)

University of Utahwww.sci.utah.edu/~beiwang

beiwang@sci.utah.edu

Feb 5, 2019

2-dimensional Manifold

Book Chapter A.II

2-manifold without boundary

A 2-manifold without boundary is a topological space M whose points alllie in open disks.

Intuitively, this means that M locally looks like a plane.

We get a 2-manifold with boundary by removing open disks from a2-manifolds without boundary.

Examples of 2-manifolds

Top: 2-manifold without boundary

Bottom: 2-manifold with boundary

Mobius strip: non-orientable manifold; 2 sides locally, 1 side globally.

Mobius strip: an ant will travel all surface area

Mobius strip: its boundary is a single circle

Quiz: what happens if you cut Mobius strip along its center line?

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TorusDouble torus

T2<latexit sha1_base64="4izDOGMjO3nVFTiDMlNZNKOKsjM=">AAAB9XicbVDLSgMxFL3js9ZX1aWbYCu4KjPd6EoKblxW6AvaacmkmTY0yQxJRilD/8ONC0Xc+i/u/Bsz7Sy09UDgcM693JMTxJxp47rfzsbm1vbObmGvuH9weHRcOjlt6yhRhLZIxCPVDbCmnEnaMsxw2o0VxSLgtBNM7zK/80iVZpFsmllMfYHHkoWMYGOlQaUvsJkEQdqcD2qVYansVt0F0DrxclKGHI1h6as/ikgiqDSEY617nhsbP8XKMMLpvNhPNI0xmeIx7VkqsaDaTxep5+jSKiMURso+adBC/b2RYqH1TAR2MgupV71M/M/rJSa88VMm48RQSZaHwoQjE6GsAjRiihLDZ5ZgopjNisgEK0yMLapoS/BWv7xO2rWq51a9h1q5fpvXUYBzuIAr8OAa6nAPDWgBAQXP8ApvzpPz4rw7H8vRDSffOYM/cD5/AKj7ke4=</latexit><latexit sha1_base64="4izDOGMjO3nVFTiDMlNZNKOKsjM=">AAAB9XicbVDLSgMxFL3js9ZX1aWbYCu4KjPd6EoKblxW6AvaacmkmTY0yQxJRilD/8ONC0Xc+i/u/Bsz7Sy09UDgcM693JMTxJxp47rfzsbm1vbObmGvuH9weHRcOjlt6yhRhLZIxCPVDbCmnEnaMsxw2o0VxSLgtBNM7zK/80iVZpFsmllMfYHHkoWMYGOlQaUvsJkEQdqcD2qVYansVt0F0DrxclKGHI1h6as/ikgiqDSEY617nhsbP8XKMMLpvNhPNI0xmeIx7VkqsaDaTxep5+jSKiMURso+adBC/b2RYqH1TAR2MgupV71M/M/rJSa88VMm48RQSZaHwoQjE6GsAjRiihLDZ5ZgopjNisgEK0yMLapoS/BWv7xO2rWq51a9h1q5fpvXUYBzuIAr8OAa6nAPDWgBAQXP8ApvzpPz4rw7H8vRDSffOYM/cD5/AKj7ke4=</latexit><latexit sha1_base64="4izDOGMjO3nVFTiDMlNZNKOKsjM=">AAAB9XicbVDLSgMxFL3js9ZX1aWbYCu4KjPd6EoKblxW6AvaacmkmTY0yQxJRilD/8ONC0Xc+i/u/Bsz7Sy09UDgcM693JMTxJxp47rfzsbm1vbObmGvuH9weHRcOjlt6yhRhLZIxCPVDbCmnEnaMsxw2o0VxSLgtBNM7zK/80iVZpFsmllMfYHHkoWMYGOlQaUvsJkEQdqcD2qVYansVt0F0DrxclKGHI1h6as/ikgiqDSEY617nhsbP8XKMMLpvNhPNI0xmeIx7VkqsaDaTxep5+jSKiMURso+adBC/b2RYqH1TAR2MgupV71M/M/rJSa88VMm48RQSZaHwoQjE6GsAjRiihLDZ5ZgopjNisgEK0yMLapoS/BWv7xO2rWq51a9h1q5fpvXUYBzuIAr8OAa6nAPDWgBAQXP8ApvzpPz4rw7H8vRDSffOYM/cD5/AKj7ke4=</latexit><latexit sha1_base64="4izDOGMjO3nVFTiDMlNZNKOKsjM=">AAAB9XicbVDLSgMxFL3js9ZX1aWbYCu4KjPd6EoKblxW6AvaacmkmTY0yQxJRilD/8ONC0Xc+i/u/Bsz7Sy09UDgcM693JMTxJxp47rfzsbm1vbObmGvuH9weHRcOjlt6yhRhLZIxCPVDbCmnEnaMsxw2o0VxSLgtBNM7zK/80iVZpFsmllMfYHHkoWMYGOlQaUvsJkEQdqcD2qVYansVt0F0DrxclKGHI1h6as/ikgiqDSEY617nhsbP8XKMMLpvNhPNI0xmeIx7VkqsaDaTxep5+jSKiMURso+adBC/b2RYqH1TAR2MgupV71M/M/rJSa88VMm48RQSZaHwoQjE6GsAjRiihLDZ5ZgopjNisgEK0yMLapoS/BWv7xO2rWq51a9h1q5fpvXUYBzuIAr8OAa6nAPDWgBAQXP8ApvzpPz4rw7H8vRDSffOYM/cD5/AKj7ke4=</latexit>

T2#T2<latexit sha1_base64="6ClmPHrBrZTankIQjs02OTWGU98=">AAACB3icbVDLSsNAFL2pr1pfUZeCDKaCq5J0oyspuHFZoS9oYplMJ+3QyYOZiVBCd278FTcuFHHrL7jzb5y0WWjrgYEz59zLvff4CWdS2fa3UVpb39jcKm9Xdnb39g/Mw6OOjFNBaJvEPBY9H0vKWUTbiilOe4mgOPQ57fqTm9zvPlAhWRy11DShXohHEQsYwUpLA/O06oZYjX0/a83u68i10O9/dWBads2eA60SpyAWFGgOzC93GJM0pJEiHEvZd+xEeRkWihFOZxU3lTTBZIJHtK9phEMqvWx+xwyda2WIgljoFyk0V393ZDiUchr6ujJfUi57ufif109VcOVlLEpSRSOyGBSkHKkY5aGgIROUKD7VBBPB9K6IjLHAROnoKjoEZ/nkVdKp1xy75tzVrcZ1EUcZTuAMLsCBS2jALTShDQQe4Rle4c14Ml6Md+NjUVoyip5j+APj8wcX2pgk</latexit><latexit sha1_base64="6ClmPHrBrZTankIQjs02OTWGU98=">AAACB3icbVDLSsNAFL2pr1pfUZeCDKaCq5J0oyspuHFZoS9oYplMJ+3QyYOZiVBCd278FTcuFHHrL7jzb5y0WWjrgYEz59zLvff4CWdS2fa3UVpb39jcKm9Xdnb39g/Mw6OOjFNBaJvEPBY9H0vKWUTbiilOe4mgOPQ57fqTm9zvPlAhWRy11DShXohHEQsYwUpLA/O06oZYjX0/a83u68i10O9/dWBads2eA60SpyAWFGgOzC93GJM0pJEiHEvZd+xEeRkWihFOZxU3lTTBZIJHtK9phEMqvWx+xwyda2WIgljoFyk0V393ZDiUchr6ujJfUi57ufif109VcOVlLEpSRSOyGBSkHKkY5aGgIROUKD7VBBPB9K6IjLHAROnoKjoEZ/nkVdKp1xy75tzVrcZ1EUcZTuAMLsCBS2jALTShDQQe4Rle4c14Ml6Md+NjUVoyip5j+APj8wcX2pgk</latexit><latexit sha1_base64="6ClmPHrBrZTankIQjs02OTWGU98=">AAACB3icbVDLSsNAFL2pr1pfUZeCDKaCq5J0oyspuHFZoS9oYplMJ+3QyYOZiVBCd278FTcuFHHrL7jzb5y0WWjrgYEz59zLvff4CWdS2fa3UVpb39jcKm9Xdnb39g/Mw6OOjFNBaJvEPBY9H0vKWUTbiilOe4mgOPQ57fqTm9zvPlAhWRy11DShXohHEQsYwUpLA/O06oZYjX0/a83u68i10O9/dWBads2eA60SpyAWFGgOzC93GJM0pJEiHEvZd+xEeRkWihFOZxU3lTTBZIJHtK9phEMqvWx+xwyda2WIgljoFyk0V393ZDiUchr6ujJfUi57ufif109VcOVlLEpSRSOyGBSkHKkY5aGgIROUKD7VBBPB9K6IjLHAROnoKjoEZ/nkVdKp1xy75tzVrcZ1EUcZTuAMLsCBS2jALTShDQQe4Rle4c14Ml6Md+NjUVoyip5j+APj8wcX2pgk</latexit><latexit sha1_base64="6ClmPHrBrZTankIQjs02OTWGU98=">AAACB3icbVDLSsNAFL2pr1pfUZeCDKaCq5J0oyspuHFZoS9oYplMJ+3QyYOZiVBCd278FTcuFHHrL7jzb5y0WWjrgYEz59zLvff4CWdS2fa3UVpb39jcKm9Xdnb39g/Mw6OOjFNBaJvEPBY9H0vKWUTbiilOe4mgOPQ57fqTm9zvPlAhWRy11DShXohHEQsYwUpLA/O06oZYjX0/a83u68i10O9/dWBads2eA60SpyAWFGgOzC93GJM0pJEiHEvZd+xEeRkWihFOZxU3lTTBZIJHtK9phEMqvWx+xwyda2WIgljoFyk0V393ZDiUchr6ujJfUi57ufif109VcOVlLEpSRSOyGBSkHKkY5aGgIROUKD7VBBPB9K6IjLHAROnoKjoEZ/nkVdKp1xy75tzVrcZ1EUcZTuAMLsCBS2jALTShDQQe4Rle4c14Ml6Md+NjUVoyip5j+APj8wcX2pgk</latexit>

Disk Cylinder Mobius strip

https://commons.wikimedia.org/wiki/File:M%C3%B6bius_strip.png

Orientability

If all closed curves in a 2-manifold are orientation-preserving, then the2-manifold is orientable.

Creating compact 2-manifolds using polygonal schema.

M is compact if for every covering of M by open sets, called an opencover, we can find a finite number of the sets that cover M.

A subset of Euclidean space is compact if it is closed and bounded (i.e.,contained in a ball of finite radius).

Classification

Theorem (Classification theorem for compact 2-manifolds)

The two infinite families S2, T2, T2#T2, · · · , and P2, P2#P2, · · · ,exhaust the compact 2-manifolds without boundary.

(Edelsbrunner and Harer, 2010, Page 29)

Classification

Any connected closed surface is homeomorphic to some member of one ofthese three families:

The sphere

The connected sum of g tori, for g ≥ 1

The connected sum of k real projective planes, for k ≥ 1.https://en.wikipedia.org/wiki/Surface_(topology)#Classification_of_closed_surfaces

Polygonal schema

a

a

b b

b

a

a

bTorus Sphere

a

a

b bProjective PlaneP2

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a

a

b bKlein bottle

Projective plane: glue a disk to a Mobius strip

Klein bottle: glue 2 Mobius strips together

Klein bottle: non-orientable surface

https://en.wikipedia.org/wiki/Klein_bottle

Later: show up in data analysis of natural image patches Carlsson et al. (2008).

Betti numbers βi of 2-manifolds

β0 β1 β2circle 1 1 0sphere 1 0 1torus 1 2 1projective plane 1 0 0Klein bottle 1 1 02-hole torus 1 4 1g-hole torus 1 2g 1

Genus

The genus of a connected, orientable surface is an integer representingthe maximum number of cuttings along non-intersecting closed simplecurves without disconnecting the resulting manifold.

https://en.wikipedia.org/wiki/Genus_(mathematics),also for further reading

Computing Homology

Book Chapter B.IV.

Reduction of a boundary matrix

Let ∂p be the p-th boundary matrix

After reduction (row and column operation), ∂p turns out to be a matrixNp in Smith Normal Form

https://en.wikipedia.org/wiki/Smith_normal_form

βp = rank Zp − rank Bp

Example: a triangulation of a circle

∂0 = N0 is a 1× 3 matrix with all 0 entries.

rank C0 = rank Z0 = 3; rank Z1 = 1, rank B0 = 2

β0 = rank Z0 − rank B0 = 3− 2 = 1

β1 = rank Z1 − rank B1 = 1− 0 = 1

Take home exercise

The following simplicial complex contains 4 vertices, 6 edges, 3 triangles.Compute its Betti numbers: β0 = 1, β1 = 0, β2 = 1.

a

b

c

d

References I

Carlsson, G., Ishkhanov, T., De Silva, V., and Zomorodian, A. (2008).On the local behavior of spaces of natural images. Internationaljournal of computer vision, 76(1):1–12.

Edelsbrunner, H. and Harer, J. (2010). Computational Topology: AnIntroduction. American Mathematical Society, Providence, RI, USA.