By Kaczynsky, Mischaikow, Mrozek.

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This justifies our first informal definition of k-chains. Utilizing the scalar product we have the following formula. 29 While the notation we are using for chains is consistent, some care must be taken when discussing 0-chains that are generated by elementary cubes in R. Let X ⊂ R be a cubical set. Consider [1] ∈ C0 (X). By definition it is the function [1](Q) = 1 if Q = [1] 0 otherwise. while 2[1](Q) = 2 if Q = [1] 0 otherwise. 2 The Algebra of Cubical Sets [2](Q) = 1 0 43 if Q = [2] otherwise.

5) Observe that k1 = dim(I) and k2 = dim(P ). e. if c = α1 Q1 + α2 Q2 + · · · + αm Qm then ∂k c := α1 ∂k Q1 + α2 ∂k Q2 + · · · + αm ∂k Qm . 33 Let Q = [l] × [k]. Then, ∂0 Q = ∂0 [l] [k] + (−1)dim [l] [l] ∂0 [k] = 0 [k] + [l] 0 = 0 + 0. 6). 2 The Algebra of Cubical Sets 45 where A1 = [l] × [k, k + 1] B1 = [l + 1] × [k, k + 1] A2 = [l, l + 1] × [k] B2 = [l, l + 1] × [k + 1]. (l, k + 1) t✛ A2 (l + 1, k + 1) t ✻ Q −A1 ❄ t (l, k) − B2 ✲ t B1 (l + 1, k) Fig. 6. Boundary of [l, l + 1] × [k, k + 1]. By definition the domain of ∂k consists of the k-chains.

The equivalence classes are elements of the quotient group Zk (X)/Bk (X) (see Chapter 13 for the definition of the quotient group). 43 The k-th cubical homology group or briefly the k-th homology group of X is the quotient group Hk (X) := Zk (X)/Bk (X). The homology of X is the collection of all homology groups of X. The shorthand notation for this is H∗ (X) := {Hk (X)}k∈Z . As was emphasized in the introduction, we will use the homology groups of the cubical set X to gain information about the topological structure of X.