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 zeroth 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. However homology groups can only rarely be worked out by pure intuition. The homology groups of a shape can be calculated from the homology groups of simpler shapes using the Mayer-Vietoris sequence.

Example

Let T be the torus. The torus comes in one piece, so H₀T = ℤ. It has two "circular holes", so H₁T = ℤ⊕ℤ. It also has a pocket, so H₂T = 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 (glome)

• 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 in square brackets. All groups after the end of the written sequence are implicitly zero.

Examples:

• H T = [1,2,1]
• H S³ = [1,0,0,1]

For shapes whose homology groups are often zero, 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.

Examples:

• 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

Rules for homology groups

The following rules exist for homology groups of min-frame rotatopes.

Cartesian products of two hyperspheres

H S_a x S_b = h₀ + h_a + h_b + h_a+b

Cartesian product of anything with a circle

H_q A x S₁= H_q A ⊕ H_q-1 A

Cartesian products of two hyperspheres and a sphere

H S_a x S_b x S₂ = h₀ + h₂ + h_a + h_a+2 + h_b+2 + h_a+b + h_a+b+2

Min-frame Rotatope Conjecture

H_q S^a₁ x S^a₂ x ... x S^a_k = the number of subsets of {a₁,a₂,...,a_k} that sum to give q, where S^a is an a-net-sphere. This hasn't yet been proven to our knowledge.

Alternatively in Union, H[q] *NetSphere(a#) = #(powerSet(a) & partitionSet(q)); a < N; 0 <= q <= +a where N is the set of natural numbers excluding zero.

External links

• Homology on Wikipedia
• Mayer-Vietoris sequence on Wikipedia