Draft:Boundary matrix

A boundary matrix is a sparse matrix that encodes how simplices in an abstract simplicial complex are connected through their boundary relationships. The concept extends to any chain complex where linear maps between chain groups can be represented as matrices. Similar structures arise in cubical complexes, where boundary matrices encode how k-cubes decompose into their (k−1)-dimensional faces.

The term boundary matrix can refer to either of the following:

• The full matrix whose columns and rows are indexed by all simplices in the complex. Each entry is ±1 or 0, depending on whether the row simplex appears as a face of the column simplex, and with what orientation. The column (respectively, row) indices of this matrix will typically include simplices with multiple different dimensions.
• A fixed-dimensional submatrix B_k with columns only indexed by k-simplices and rows only indexed by (k−1)-simplices. This is the portion of the full boundary matrix corresponding to the k-th 'boundary operator' in the chain complex.

Basic construction

Consider a triangle (2-simplex) with vertices A, B, C. Its boundary consists of three edges: AB, BC, CA. In the boundary matrix, the column for triangle ABC would have ±1s in the rows corresponding to edges AB, BC, and CA, and 0s elsewhere.

Computational properties

Sparsity: Boundary matrices are extremely sparse since each simplex has only a small number of faces relative to the total number of simplices in the complex. Block Structure: The matrix naturally decomposes into blocks B_k that map k-simplices to (k−1)-simplices. Most computational topology algorithms work with these individual blocks. Fundamental Property: B² = 0, meaning that applying the boundary operator twice always yields zero. This reflects the topological fact that "the boundary of a boundary is empty."

Applications in computation

Homology Computation: The kernel and image of boundary matrices directly encode homological information. Computing homology involves finding these kernel and image spaces through various matrix algorithms. Cohomology: The transpose of boundary matrices gives coboundary matrices used in cohomology computations. Laplacian Construction: Boundary matrices are used to construct combinatorial Laplacians: L_k = B_k+1B_k+1^T + B_k^TB_k.

Relationship to chain complexes

Chain complexes

The boundary matrix is the matrix representation of the boundary operator ∂_k in a chain complex. A chain complex is an algebraic structure (C, ∂) consisting of:

A sequence of vector spaces (or modules) C₀, C₁, C₂, ... called chain groups Linear maps ∂_k: C_k → C_k−1 called boundary operators The fundamental relation ∂_k−1 ∘ ∂_k = 0 for all k

In matrix terms, this means that when boundary operators are represented as matrices B_k, we have B_k−1 B_k = 0 (the composition of consecutive boundary matrices is the zero matrix). This matrix multiplication property directly encodes the topological fact that "the boundary of a boundary is empty."

From simplicial complexes to chain complexes

Given an abstract simplicial complex K, we construct a chain complex as follows. Before doing so, we must fix an orientation on the complex. Concretely, this means placing the vertices in a linear order, and then representing each simplex as a list of its vertices sorted according to that order. This choice determines the signs in the boundary maps.

Chain Groups: C_k is the vector space (often over ℤ₂ or ℝ) with basis given by the k-simplices of K. Each k-simplex σ corresponds to a basis element, and elements of C_k are formal linear combinations of k-simplices. Boundary Operators: For a k-simplex σ = [v₀, v₁, ..., v_k], the boundary operator is defined as: ∂_k(σ) = Σ_i=0^k (−1)ⁱ [v₀, ..., v̂_i, ..., v_k] where v̂_i indicates that vertex v_i is omitted.

Matrix construction

The boundary matrix B_k is the matrix representation of ∂_k. To compute this matrix one first has to fix an orientation by placing the set of all vertices in a linear order, then representing each simplex as a list of its vertices sorted according to that order. The explicit formula for the matrix entries is: $\partial _{\sigma ,\tau }={\begin{cases}(-1)^{p}&{\text{if }}\sigma =[v_{0},\ldots ,{\hat {v}}_{p},\ldots ,v_{k}]{\text{ and }}\tau =[v_{0},\ldots ,v_{k}]\\0&{\text{otherwise}}\end{cases}}$ where σ_j = [v₀, v₁, ..., v_k] is the j-th k-simplex and τ_i is the i-th (k−1)-simplex. This construction means: the (i,j) entry is ±1 if the i-th (k−1)-simplex is the p-th face of the j-th k-simplex (obtained by deleting the p-th vertex), with sign (−1)^p.

Homology via chain complexes

The k-th homology group H_k is defined as: H_k = ker(∂_k) / im(∂_k+1) This quotient measures "holes" in dimension k:

H₀ counts connected components H₁ counts 1-dimensional holes (loops) H₂ counts 2-dimensional holes (voids)

Orientation and signs

Boundary matrices are always signed, requiring an orientation on each simplex. For simplicial complexes, this is typically done by placing the set of vertices into a linear order [v₀, v₁, ..., v_N], then representing each simplex as the sorted list of vertices it contains, i.e. a list of form [v_i0, v_i1, ..., v_ik] where i₀ < i₁ < ... < i_N.

The sign in the boundary matrix depends on whether the face is obtained by deleting an even or odd-positioned vertex from the sorted simplex representation. For example, if triangle [1,3,5] has boundary consisting of edges [1,3], [1,5], and [3,5], these correspond to deleting vertices at positions 2, 1, and 0 respectively, giving signs (−1)², (−1)¹, and (−1)⁰, which are +1, −1, and +1.

See also

Cubical complex: Alternative to simplicial complexes using hypercubes, with analogous boundary matrix structures Simplicial homology: The homology theory naturally associated with abstract simplicial complexes Chain complex: The general algebraic framework underlying boundary matrices

References

Category:Algebraic topology Category:Computational topology Category:Matrices