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DSC 291๏ฟฝAlgorithmic foundation for ๏ฟฝTopological Data Analysis๏ฟฝ

Topic 3: Simplicial Homology

a.k.a, how we quantify topological features?

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Instructor: Yusu Wang

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Overview

  • (Simplicial) homology groups
    • a way to quantify topological features

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  • Notations
    • Chains, cycles, and homology groups

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  • Matrix view
    • Matrix reduction algorithm

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Part I: ๏ฟฝSimplicial Homology

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Chains

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Chains

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Chains

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Chains

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Chains

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Boundary operator

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Boundary operator

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Cycles and Boundaries

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Cycles and Boundaries

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Cycles and Boundaries

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Homology groups

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Betti numbers

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More examples

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Betti numbers are topological invariants

  • Fact:
    • Two homeomorphic topological spaces have isomorphic homology groups (and thus same Betti numbers).

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Another definition for Euler characteristics

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Examples of triangulations of 2-manifolds.

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Part 2: ๏ฟฝMatrix view and computation

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Boundary Matrix

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Boundary Matrix

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Example.

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Boundary Matrix

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Boundary Matrix

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Boundary Matrix

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Simple Alg to compute Betti numbers

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Another alg: Right-reduction algorithm

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Properties

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  • Lemma:
    • If a matrix is in reduced form, then all its non-zero columns are linearly independent.

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Properties

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Examples.

This is not the only reduction algorithm!! Any elimination via row/column additions to convert a matrix into a reduced form works!

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FIN