WebThe boundary operator also allows us to define subgroups of C k: the group of cycles and the group of boundaries. Definition 6.3 (cycle group, boundary group) The kth cycle group is Z k = ker∂ k ={c ∈ C k ∂ kc = ∅}. A chain that is an element of Z k is a k-cycle. The kth boundary group is B k = im∂ k+1 ={c ∈ C k ∃d ∈ C k+1 ... Weband variants in section 3.2, we need the important notions of chains, cycles, and boundaries given in the following section. 3.1 Chains, cycles, boundaries 3.1.1 Algebraic structures First, we recall briefly the definitions of some standard algebraic structures that are used in the book. For details we refer the reader to any standard book on ...
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