Homology groups (InstanceTopic, 5)

From Hi.gher. Space

Homology groups are a formalisation of the idea of holes in a shape. The homology groups are a list of abelian groups, one for each nonnegative integer. For toratopes, all homology groups are copies of the group of integers ℤ. This is not the case for other shapes, such as the Klein bottle. Intuitively, the 0th homology group counts how many disjoint pieces make up the shape and gives that many copies of ℤ, while the other homology groups count different types of holes.

Example

Let X be the torus. The torus comes in one piece, so H₀X = ℤ. It has two "circular holes", so H₁X = ℤ⊕ℤ. It also has a pocket, so H₂X = ℤ

Notation

The usual notation is to write H_q as an operator that acts on the shape and returns an abelian group. This is the notation used above.

Examples: Let T = torus, S³ = 3-sphere (4D sphere) H₀T = ℤ H₁T = ℤ⊕ℤ = 2ℤ H₂T = ℤ

H₀S³ = ℤ H₁S³ = 0 H₂S³ = 0 H₃S³ = ℤ

Since homology groups of toratopes are always copies of ℤ, we often omit the ℤ and write all the groups together. At the point where all subsequent homology groups are zero we write a .

Examples: H T = 1,2,1. H S³ = 1,0,0,1.

For shapes whose homology groups are often 0, it becomes cumbersome to write out all the zeroes. h_q notation makes some shapes simpler and makes it easier to write general formulas for homology groups.

H T = h₀ + 2 h₁ + h₂ H S³ = h₀ + h₃ H Sⁿ = h₀ + h_n H Tⁿ = sum over k from 0 to n of (n choose k) h_k

H₀ always counts the number of path components, and gives that many copies of ℤ. Intuitively, H₁ counts the number of "circular" holes, H₂ counts "spherical" holes etc. For an object in nD, H_n counts the number of "pockets". However homology groups can only rarely be worked out by pure intuition.

Calculating Homology Groups

[1]

The homology groups of a shape can be calculated from the homology groups of simpler shapes using the Mayer-Vietoris sequence. [2]

Rules For Homology Groups

The following rules exist for homology groups of toratopes.

1. Cartesian products of two hyperspheres HqSa x Sb = h0 + ha + hb + ha+b

2. Cartesian product of anything with a circle Hq A2 = Hq A ⊕ Hq-1 A