Concept

Homology

An algebraic count of the holes in a space, built from cycles of edges, faces and higher cells modulo those that bound. Its first group is the group of loops made commutative, and it can be computed by linear algebra.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

a loop that dips through and back, and the punctures of the disc. A link drawn with a shaded disc spanning the first loop, seen at an angle, with every place the second loop passes through the disc marked with the direction it was travelling in.

Zero can mean two different things

The linking number counts how often one loop pierces a surface the other one bounds. Two punctures of opposite sign add to nothing, and a loop that never goes through adds to nothing as well — so the answer zero is two pictures wearing one number.

topology · Linking number

A loop that cannot be undone, and the surface it bounds. The loop aba⁻¹b⁻¹ on the figure eight, which cannot be pulled tight, beside a square whose boundary reads the same word, showing that the loop bounds a surface.

What homology forgets about a loop

Let the letters of a loop commute and the loop group of a space becomes its first homology group: a loop now records only how often it went round each hole. What is thrown away is exactly the loops that bound a surface. On the figure eight that is nearly everything — of the loops of sixty letters that homology calls nought, about one in 6,700 is a loop that actually shrinks.

topology · Homotopy

Named alongside it

The objects these essays reach for when they reach for this one.

Abelian group Commutator Covering space Free group Fundamental group Genus Invariant Isotopy Knot Link Linking number Loop

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