Delta Complexes

-Complexes and Chains

Standard -simplex

The standard -simplex, denoted , is a subspace of defined as:

e.g. is a point; is a line segment; is a triangle; is a tetrahedron.

-Complex

A -complex is a topological space constructed by gluing together simplices ( for various ) along their faces. The gluing maps must be linear and preserve the ordering of vertices.

Remark

One can think of it as a more structured and restrictive type of CW complex, where the cells are specifically simplices and the attaching maps have stricter rules.

e.g. We can build a torus using a -complex structure with the following components:

• One 0-simplex: a single vertex, .
• Three 1-simplicies: edges .
• Two 2-cells: simplicies .

Imagine two triangles, and . We glue their corresponding edges to form a rectangle, and then identify the opposite sides of this rectangle to form the torus:

Chain Group

The -th chain group, , is the free abelian group generated by the set of -simplices in the -complex . An element of , called an -chain, is a finite formal sum of -simplices: , where and are -simplices.

Boundary Operator

We define a homomorphism, the boundary operator, . If we denote an -simplex by its ordered vertices , the boundary operator is defined by the alternating sum of its faces: where is the -face opposite the vertex . By convention, we set . i.e., .

e.g.

• Boundary of an edge (1-simplex): . This is the “endpoint minus the start point”. For a loop, where , the boundary is .
• Boundary of a triangle (2-simplex): . This represents the sum of the oriented edges forming the boundary of the triangle.

Simplicial Homology Groups

The boundary operator has a crucial property that allows us to define homology.

Lemma

The boundary of a boundary is zero. That is, the composition of any two successive boundary maps is the zero map:

Corollary

The image of is a subgroup of the kernel of : .

This relationship lets us define the homology groups, which measure the “holes” in our space.

Simplicial Homology Group

The -th simplicial homology group of , denoted , is the quotient group: The elements of are called -cycles, and the elements of are called -boundaries.

Remark

Homology captures the cycles that are not boundaries.

e.g. Recall our -complex structure for the torus : The chain groups are: , , . The boundary maps are:

• , , .
• Since all edges are loops starting and ending at , their boundaries are zero: , , .

:

• .
• .
• . This reflects the single connected component of the torus.

:

• , since the boundary of every 1-chain is 0.
• . This is the subgroup generated by .
• . We can use the relation to eliminate , leaving two independent generators, and . Thus, . These correspond to the two fundamental loops of the torus.

:

• : An element is in the kernel if . This implies . So, the kernel is the subgroup generated by the cycle .
• , since there are no 3-cells in our complex.
• . This represents the 2D “void” enclosed by the torus surface.