Homology (mathematics)

In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages.

(This latter notion of homology admits more intuitive descriptions for 1- or 2-dimensional topological spaces, and is sometimes referenced in popular mathematics.)

(Simplices are a generalization of triangles to arbitrary dimension; for example, an edge in a graph is homeomorphic to a one-dimensional simplex, and a triangle-based pyramid is a 3-simplex.)

Replacing simplices with disks of various dimensions results in a related construction called cellular homology.

One of the ideas that led to the development of homology was the observation that certain low-dimensional shapes can be topologically distinguished by examining their "holes."

Since such constructions are somewhat technical, informal discussions of homology sometimes focus instead on topological notions that parallel some of the group-theoretic aspects of cycles and boundaries.

It is this topological notion of no boundary that people generally have in mind when they claim that cycles can intuitively be thought of as detecting holes.

Embedded representatives of 1-cycles, 3-cycles, and oriented 2-cycles all admit manifold-shaped holes, but for example the real projective plane

The general construction begins with an object such as a topological space X, on which one first defines a chain complex C(X) encoding information about X.

is the group of homotopy classes of directed loops starting and ending at a predetermined point (e.g. its center).

from Cn to Cn−1 is called the boundary mapping and sends the simplex to the formal sum which is evaluated as 0 if

is the set of n-simplexes in X and the mi are coefficients from the ring Cn is defined over (usually integers, unless otherwise specified).

) is a chain complex such that all but finitely many An are zero, and the others are finitely generated abelian groups (or finite-dimensional vector spaces), then we can define the Euler characteristic (using the rank in the case of abelian groups and the Hamel dimension in the case of vector spaces).

To understand the global context of this set of local measurements and communication paths, it is useful to compute the homology of the network topology to evaluate, for instance, holes in coverage.

Morse theory relates the dynamics of a gradient flow on a manifold to, for example, its homology.

The KAM theorem established that periodic orbits can follow complex trajectories; in particular, they may form braids that can be investigated using Floer homology.

[13] In one class of finite element methods, boundary-value problems for differential equations involving the Hodge-Laplace operator may need to be solved on topologically nontrivial domains, for example, in electromagnetic simulations.

[14][15] Various software packages have been developed for the purposes of computing homology groups of finite cell complexes.

Linbox is a C++ library for performing fast matrix operations, including Smith normal form; it interfaces with both Gap and Maple.

All three implement pre-processing algorithms based on simple-homotopy equivalence and discrete Morse theory to perform homology-preserving reductions of the input cell complexes before resorting to matrix algebra.

Kenzo is written in Lisp, and in addition to homology it may also be used to generate presentations of homotopy groups of finite simplicial complexes.

[16] This was followed by Riemann's definition of genus and n-fold connectedness numerical invariants in 1857 and Betti's proof in 1871 of the independence of "homology numbers" from the choice of basis.

The square can be twisted to allow edges to meet in the opposite direction, as shown by the arrows in the diagram.

But unlike the torus, following b forwards right round and back reverses left and right, because b happens to cross over the twist given to one join.

However it is wound a second time, which swaps right and left back again; it can be shrunk to a point and is homologous to c. Cycles can be joined or added together, as a and b on the torus were when it was cut open and flattened down.

This is because the Klein bottle is made from a cylinder, whose a-cycle ends are glued together with opposite orientations.

However, following a b-cycle around twice in the Klein bottle gives simply b + b = 2b, since this cycle lives in a torsion-free homology class.

[18] The first recognisable theory of homology was published by Henri Poincaré in his seminal paper "Analysis situs", J. Ecole polytech.

[21][22] Chain complexes (since greatly generalized) form the basis for most modern treatments of homology.

Emmy Noether and, independently, Leopold Vietoris and Walther Mayer further developed the theory of algebraic homology groups in the period 1925–28.