Homology and Euler characteristic

In a finite simplicial complex, choose orientations for edges, triangles, and higher simplices. A k-chain is a formal sum of k-simplices with coefficients in a declared ring; use real coefficients first. The boundary of an oriented edge [a,b] is b−a, and the boundary of an oriented triangle [a,b,c] is [b,c]−[a,c]+[a,b]. Applying the boundary operation again cancels each vertex twice with opposite signs. The same alternating-face rule makes the boundary of a boundary zero in every degree.

A cycle is a chain with zero boundary. A boundary is a chain that is itself the boundary of a higher-dimensional chain. Since ∂²=0, every boundary is a cycle. Homology identifies cycles whose difference is a boundary. The perimeter of an unfilled triangle gives a nontrivial one-cycle, while filling the triangle makes that same perimeter a boundary and kills its homology class. Thus the available two-dimensional pieces matter even when the visible edge loop is unchanged.

With field coefficients, the dimension of Hₖ is the kth Betti number. For a finite complex it can be computed as the number of k-simplices minus the ranks of the two adjacent boundary maps. With integer coefficients, homology groups can also contain torsion, such as an element killed by multiplication by two. Passing to real coefficients loses that torsion information. For example, the real projective plane has first integer homology ℤ/2, while its first real homology is zero.

Over the field with two elements, signs disappear because one equals minus one. This simplifies computation but changes some answers: the real projective plane has a nonzero first homology group over that field. Coefficients must therefore accompany a homology claim. Singular homology extends the construction beyond a chosen triangulation, and comparison theorems show agreement with simplicial homology in the appropriate setting. Invariance under homotopy is a theorem, not a consequence of simply counting arbitrary drawing elements.

For a finite complex, rank-nullity makes the boundary-map ranks cancel in the alternating sum, proving the equality between alternating cell counts and alternating Betti numbers. Subdividing a triangulation changes its counts but preserves that sum. A connected circle has one independent component and one independent one-cycle, giving Euler characteristic zero. A connected sphere has one component and one two-dimensional homology class, giving Euler characteristic two. These conclusions use ordinary, unreduced homology.

Equal Euler characteristics do not imply homeomorphism or equal homology. A point and a circle joined to a two-sphere at one point both have Euler characteristic one, although their first and second Betti numbers differ. For an infinite complex, subtracting infinite cell counts is undefined; additional finiteness or convergence structure is necessary. Combine this total with the graded homology information and the declared class of spaces.