Where does hole collide?

In algebraic topology a hole is a feature of a shape that cannot be filled in and that survives any continuous stretching or bending. Poincaré's 1895 Analysis situs introduced the tools that count such features, assigning numbers, the Betti numbers, that record how many independent loops, cavities, and higher voids a space contains. The modern statement gathers these into homology groups, where each nonzero class certifies a hole of a given dimension: a circle carries one one-dimensional hole, and a hollow sphere encloses one two-dimensional void. Here the hole is an invariant of a space, counted by dimension and rank, and it has nothing to do with a missing charge inside a crystal or a missing tie between groups of people.[1]

Unit

One independent homology class, indexed by its dimension

Recovery means

Not applicable

Measured by

By computing Betti numbers and the ranks of the homology groups of the space

In the physics of solids a hole is the absence of an electron from an otherwise filled band of states, a vacancy that moves and responds to fields as though it were a particle of positive charge. Peierls's 1929 Zur Theorie der galvanomagnetischen Effekte showed that a nearly full band conducts as if it carried positive carriers, and Heisenberg's 1931 Zum Paulischen Ausschließungsprinzip treated these vacancies directly using the exclusion principle. In a semiconductor the electrons and the holes they leave behind act as two kinds of mobile charge, and modern devices are built by controlling how many of each are present. Here the hole is a quasiparticle inside matter, described by its charge, effective mass, and mobility, not a void in an abstract shape or a gap in a web of people.[2][3]

Unit

One vacant electron state that behaves as a positive charge carrier

Recovery means

A hole disappears when an electron drops into the vacant state, a process called recombination

Measured by

By the sign of the Hall coefficient, carrier concentration, effective mass, and mobility

In network sociology a structural hole is the absence of a connection between two people or groups who are each tied to a third party but not to each other. Burt's 1992 Structural holes defined it as the gap in social structure that a well-placed party can span, arguing that the missing tie is itself a source of advantage because information and control flow through whoever bridges it. Burt's 2004 Structural holes and good ideas then gave evidence that people whose contacts sit on opposite banks of such gaps tend to generate more valuable ideas. Here the hole is a missing relationship in a web of people, measured by network constraint and the redundancy of a party's contacts, not a void in a shape or a missing charge in a crystal.[4][5]

Unit

One missing tie between two otherwise disconnected contacts

Recovery means

Not applicable

Measured by

By network constraint and the count of nonredundant contacts that span the gap