1. Chapter 3: Simplicial Homology Instructor: Yusu Wang
CSE 5559 Computational Topology: Theory, algorithms, and applications to data analysis
Chapter 3: Simplicial Homology
Instructor: Yusu Wang

2. Section 1: Simplicial Homology

3. Chains Given a simplicial complex 𝐾, a 𝑝-chain is
A formal sum of 𝑝-simplices 𝑐=∑ 𝑐 𝑖 𝜎 𝑖
Under 𝑍 2 coefficients: a collection of 𝑝-simplices

4. Chains Given a simplicial complex 𝐾, a 𝑝-chain is
A formal sum of 𝑝-simplices 𝑐=∑ 𝑐 𝑖 𝜎 𝑖
Under 𝑍 2 coefficients: a collection of 𝑝-simplices
𝑝-th chain group of 𝐾
𝐶 𝑝 𝐾 : collection of 𝑝-chains with operation +
𝑐 1 = 𝜎 1 + 𝜎 3 ; 𝑐 2 = 𝜎 1 + 𝜎 4 ; ⇒ 𝑐 1 + 𝑐 2 = 𝜎 3 + 𝜎 4
Under 𝑍 2 coefficients,
𝐶 𝑝 𝐾 is a vector space
What is its dimension (rank)?
What is a basis for it?

5. Chains and boundary operator
𝑝-th boundary operator 𝜕 𝑝 : 𝐶 𝑝 → 𝐶 𝑝−1
𝜕 𝑝 𝜎 = set of (𝑝−1)-faces of 𝜎
𝜕 𝑝 𝑐 =∑ 𝑐 𝑖 𝜕 𝑝 ( 𝜎 𝑖 )
Hence 𝜕 𝑝 is a homomorphism (map preserving + operation)
C_p is a free abelian group, which is an abelian group with a basis

6. Chains and boundary operator
𝑝-th boundary operator 𝜕 𝑝 : 𝐶 𝑝 → 𝐶 𝑝−1
𝜕 𝑝 𝜎 = set of (𝑝−1)-faces of 𝜎
𝜕 𝑝 𝑐 =∑ 𝑐 𝑖 𝜕 𝑝 ( 𝜎 𝑖 )
Hence 𝜕 𝑝 is a homomorphism (map preserving + operation)
Chain complex
C_p is a free abelian group, which is an abelian group with a basis
Theorem:
𝜕 𝑝 ∘ 𝜕 𝑝+1 =0

7. Cycles and Boundaries Cycles: 𝑝-cycle: a 𝑝-chain whose boundary is 0
𝑝-th cycle group 𝑍 𝑝 𝐾 =ker⁡( 𝜕 𝑝 )
What is the relation between 𝑍 𝑝 and 𝐶 𝑝 ?

8. Under 𝑍 2 coefficients, 𝐵 𝑝 , 𝑍 𝑝 , 𝐶 𝑝 are all vector spaces.
Cycles and Boundaries
Cycles:
𝑝-cycle: a 𝑝-chain whose boundary is 0
𝑝-th cycle group 𝑍 𝑝 𝐾 =ker⁡( 𝜕 𝑝 )
Boundary cycles:
𝑝-boundary: a 𝑝-cycle which is the boundary of some (𝑝+1)- chain
𝑝-th boundary group 𝐵 𝑝 𝐾 =Im⁡( 𝜕 𝑝+1 )
𝜕 𝑝 ∘ 𝜕 𝑝+1 =0⇒ 𝐵 𝑝 ⊆ 𝑍 𝑝 ⊆ 𝐶 𝑃
Under 𝑍 2 coefficients, 𝐵 𝑝 , 𝑍 𝑝 , 𝐶 𝑝 are all vector spaces.

9. Cycles and Boundaries Cycles: Boundary cycles:
𝑝-cycle: a 𝑝-chain whose boundary is 0
𝑝-th cycle group 𝑍 𝑝 𝐾 =ker⁡( 𝜕 𝑝 )
Boundary cycles:
𝑝-boundary: the boundary of some (𝑝+1)-chain
𝑝-th boundary group 𝐵 𝑝 𝐾 =Im⁡( 𝜕 𝑝+1 )
𝜕 𝑝 ∘ 𝜕 𝑝+1 =0⇒ 𝐵 𝑝 ⊆ 𝑍 𝑝 ⊆ 𝐶 𝑃
𝜕 𝑝+1
𝜕 𝑝

10. Homology groups 𝑝-th cycle group 𝑍 𝑝 𝐾 =ker⁡( 𝜕 𝑝 )
𝑝-th boundary group 𝐵 𝑝 𝐾 =Im⁡( 𝜕 𝑝+1 )
𝑝-th homology group
𝐻 𝑝 𝐾 = 𝑍 𝑝 / 𝐵 𝑝
𝑐 1 is homologous to 𝑐 2 if
𝑐 1 + 𝑐 2 ∈ 𝐵 𝑝 , i.e, 𝑐 1 + 𝑐 2 is a boundary cycle
ℎ=[𝑐]∈ 𝐻 𝑝 :
the family 𝑝-cycles homologous to 𝑐
called a homology class
B forms a normal subgroup of Z, so the right- / left- cosets coincides. => thus can define quotient group
c_1 \in c_2 + B

11. Betti numbers Betti number: 𝛽 𝑝 𝐾 =𝑟𝑎𝑛𝑘 𝐻 𝑝 Theorem:
𝛽 𝑝 𝐾 =𝑟𝑎𝑛𝑘 𝑍 𝑝 −𝑟𝑎𝑛𝑘( 𝐵 𝑝 )
Examples: meaning of 𝛽 0 , 𝛽 1 , 𝛽 2
Rank( 𝐻 0 ) = ?; Rank( 𝐻 1 ) = ?

12. More results Theorem: Theorem:
For a compact orientable 2-manifold with genus 𝑔, we have
𝛽 0 =1, 𝛽 1 =2𝑔, 𝛽 2 =1
Theorem:
Given two simplicial complexes 𝐾 1 and 𝐾 2 such that 𝐾 1 ≅ | 𝐾 2 |, then 𝐻 ∗ 𝐾 1 ≈ 𝐻 ∗ ( 𝐾 2 ).
Hence different triangulations of the same space have the isomorphic homology groups!
Thus homology group is a topological invariant

13. Euler characteristics
Given a topological space 𝑀
its Euler characteristics 𝜒 𝑀 = 𝑝=0 −1 𝑝 𝛽 𝑝 𝑀
Hence Euler characteristics is also independent of the triangulation of a space, and is a topological invariant.
Theorem (Euler-Poincaré formula)
Given a simplicial complexes 𝐾, let 𝑛 𝑝 denote the number of 𝑝-simplices in 𝐾. Then
𝜒 𝐾 ≔𝜒 𝐾 = 𝑝=0 −1 𝑝 𝑛 𝑝
Examples of triangulations of 2-manifolds.

14. Section 2: Matrix view and computation

15. Boundary Matrix 𝐾 𝑝 = 𝛼 1 ,…, 𝛼 𝑛 𝑝 , 𝐾 𝑝−1 = 𝜏 1 ,…, 𝜏 𝑛 𝑝−1
𝐾 𝑝 = 𝛼 1 ,…, 𝛼 𝑛 𝑝 , 𝐾 𝑝−1 = 𝜏 1 ,…, 𝜏 𝑛 𝑝−1
𝐾 𝑝 forms a basis for p-th chain group 𝐶 𝑝
n 𝑝−1 × 𝑛 𝑝 boundary matrix 𝐴 𝑝
𝐴 𝑝 𝑖 𝑗 =1 iff 𝜏 𝑖 ⊆ 𝜎 𝑗
representing 𝜕 𝑝 : 𝐶 𝑝 → 𝐶 𝑝−1

16. Boundary Matrix 𝐾 𝑝 = 𝛼 1 ,…, 𝛼 𝑛 𝑝 , 𝐾 𝑝−1 = 𝜏 1 ,…, 𝜏 𝑛 𝑝−1
𝐾 𝑝 = 𝛼 1 ,…, 𝛼 𝑛 𝑝 , 𝐾 𝑝−1 = 𝜏 1 ,…, 𝜏 𝑛 𝑝−1
𝐾 𝑝 forms a basis for p-th chain group 𝐶 𝑝
n 𝑝−1 × 𝑛 𝑝 boundary matrix 𝐴 𝑝 s.t.
𝐴 𝑝 𝑖 𝑗 =1 iff 𝜏 𝑖 ⊆ 𝜎 𝑗
representing 𝜕 𝑝 : 𝐶 𝑝 → 𝐶 𝑝−1 w.r.t. basis 𝛼 1 ,…, 𝛼 𝑛 𝑝 and 𝜏 1 ,…, 𝜏 𝑛 𝑝−1 𝑎 𝑏 𝑐 𝑑

17. Given a p-chain 𝑐= 𝑖=1 𝑛 𝑝 𝑐 𝑖 𝛼 𝑖
Under basis 𝐾 𝑝 , vector representation of 𝑐 is
𝑐 = 𝑐 1 , 𝑐 2 ,…, 𝑐 𝑛 𝑝 𝑇
Boundary 𝜕 𝑝 𝑐 is a (𝑝−1)-chain with vector representation 𝐴 𝑝 𝑐 w.r.t basis 𝐾 𝑝−1
Example.

18. Boundary Matrix
Let 𝑛 𝑝 , 𝑧 𝑝 , 𝑏 𝑝 denote the rank of 𝐶 𝑝 , 𝑍 𝑝 , and 𝐵 𝑝
𝛽 𝑝 =𝑟𝑎𝑛𝑘 𝐻 𝑝

19. Boundary Matrix
Let 𝑛 𝑝 , 𝑧 𝑝 , 𝑏 𝑝 denote the rank of 𝐶 𝑝 , 𝑍 𝑝 , and 𝐵 𝑝
𝛽 𝑝 =𝑟𝑎𝑛𝑘 𝐻 𝑝
Claim: 𝑖 𝑛 𝑝 = 𝑧 𝑝 + 𝑏 𝑝−1 ;
𝑖𝑖 𝛽 𝑝 = 𝑧 𝑝 − 𝑏 𝑝
Rank-nullity theorem from linear algebra
C is isomorphic to direct sum of B and Z
Splitting lemma

20. Boundary Matrix
Let 𝑛 𝑝 , 𝑧 𝑝 , 𝑏 𝑝 denote the rank of 𝐶 𝑝 , 𝑍 𝑝 , and 𝐵 𝑝
𝛽 𝑝 =𝑟𝑎𝑛𝑘 𝐻 𝑝
Claim: 𝑖 𝑛 𝑝 = 𝑧 𝑝 + 𝑏 𝑝−1 ;
𝑖𝑖 𝛽 𝑝 = 𝑧 𝑝 − 𝑏 𝑝
Consider 𝐴 𝑝
Each columns of 𝐴 𝑝 corresponds to a boundary cycle
Rank of 𝐴 𝑝 gives 𝑏 𝑝 =𝑟𝑎𝑛𝑘( 𝐵 𝑝 )
Why?

21. Boundary Matrix
Let 𝑛 𝑝 , 𝑧 𝑝 , 𝑏 𝑝 denote the rank of 𝐶 𝑝 , 𝑍 𝑝 , and 𝐵 𝑝
𝛽 𝑝 =𝑟𝑎𝑛𝑘 𝐻 𝑝
Claim: 𝑖 𝑛 𝑝 = 𝑧 𝑝 + 𝑏 𝑝−1 ;
𝑖𝑖 𝛽 𝑝 = 𝑧 𝑝 − 𝑏 𝑝
Consider 𝐴 𝑝
Each columns of 𝐴 𝑝 corresponds to a boundary cycle
Rank of 𝐴 𝑝 gives 𝑏 𝑝 =𝑟𝑎𝑛𝑘( 𝐵 𝑝 )
Note, this gives an 𝑂 𝑛 3 time algorithm for computing all 𝛽 𝑝 ’s via Gaussian elimination
Can be improved to matrix multiplication time

22. Right-reduction algorithm
Starting with bnd matrix 𝑀= 𝐴 𝑝
For the 𝑖-th column corresponding to p-simplex 𝜎 𝑖 ,
associate a 𝑝-chain Γ 𝑖 initialized to 𝜎 𝑖
AddColumn(𝑗, 𝑖)
𝐶𝑜 𝑙 𝑀 𝑖 =𝐶𝑜 𝑙 𝑀 𝑖 +𝐶𝑜 𝑙 𝑀 𝑗 ; Γ 𝑖 = Γ 𝑖 + Γ 𝑗

23. Properties Lemma: Lemma:
Each reduction (column addition) step maintains the following invariance: After 𝑘-th stages, 𝑀 (𝑘) has the same rank as 𝐴 𝑝 , and 𝜕 𝑝 Γ 𝑗 (𝑘) =𝑐𝑜 𝑙 𝑀 𝑗 for any 𝑗.
Lemma:
At the end of the reduction algorithm, each non-zero column has a unique low-ID.

24. Reduced form:
A matrix 𝑀 is in reduced form is all non-zero columns are linearly independent.
It is in Smith-Normal form if it has the following structure:
Lemma:
A matrix is in reduced form if each non-zero column has a unique low-ID.

25. Properties Theorem: Examples.
Procedure Left-Reduction(𝑀) terminates in 𝑂( 𝑛 𝑝 2 𝑛 𝑝−1 ) time
The output matrix 𝑀 is in reduced form
The set of non-zero columns in 𝑀 form a basis for 𝐵 𝑝−1
The set { Γ 𝑖 ∣𝑐𝑜 𝑙 𝑀 𝑖 =0 } form a basis for 𝑍 𝑝
Examples.
This is not the only reduction algorithm!! Any elimination via row/column additions to convert a matrix into a reduced form works!