By Tomasz Kaczynski

Homology is a robust device utilized by mathematicians to check the houses of areas and maps which are insensitive to small perturbations. This booklet makes use of a working laptop or computer to advance a combinatorial computational method of the topic. The middle of the booklet bargains with homology thought and its computation. Following it is a part containing extensions to additional advancements in algebraic topology, functions to computational dynamics, and functions to photograph processing. incorporated are workouts and software program that may be used to compute homology teams and maps. The publication will entice researchers and graduate scholars in arithmetic, laptop technology, engineering, and nonlinear dynamics.

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11 representing Γ 1 . Then, V = {A, B, C, D}, E = {[A, B], [B, C], [D, C], [A, D]}. The computation of the homology begins just as in the previous example. The basis for the 0-chains, C0 (G; Z2 ), is {A, B, C, D}, while the basis for the 1-chains, C1 (G; Z2 ), is {[A, B], [B, C], [D, C], [A, D]}. Again, using column vectors let ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 0 0 0 ⎢0⎥ ⎢1⎥ ⎢0⎥ ⎢0⎥ ⎢ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ A = ⎣ ⎦, B = ⎣ ⎦, C = ⎣ ⎦, D = ⎣ ⎥ 0 0 1 0⎦ 0 0 0 1 and ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 0 0 1 ⎢1⎥ ⎢0⎥ ⎢0⎥ ⎢0⎥ ⎢ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ . [A, B] = ⎣ ⎦ , [B, C] = ⎣ ⎦ , [D, C] = ⎣ ⎦ , [A, D] = ⎣ ⎥ 0 1 0⎦ 0 0 0 1 0 With this convention, ∂1 becomes the 4 × 4 matrix ⎡ ⎤ 1001 ⎢1 1 0 0⎥ ⎥ ∂1 = ⎢ ⎣0 1 1 0⎦.

5. Keeping track of directions and walking around the edge of Q in a counterclockwise direction, it seems reasonable to deﬁne ∂ Q = [A, B] + [B, C] − [D, C] − [A, D]. 4) indicates that the cycle [A, B] + [B, C] − [D, C] − [A, D] is the interesting algebraic aspect of Γ 1 . In Q it appears as the boundary of an object. Again, the observation that we will make is: Cycles that are boundaries should be considered uninteresting. 5. 6 Mod 2 Homology of Graphs We have done the same example twice using diﬀerent types of arithmetic, but the conclusion is the same.

For example, the unit circle x2 + y 2 = 1 is not a cubical set. In fact, even a simple set such as a point may or may not be a cubical set. In particular, consider the set consisting of one point P = {(x, y, z)} ⊂ R3 . P is a cubical set if and only if x, y, and z are all integers. The following result uses some simple point-set topology (see Chapter 12). 12 If X ⊂ Rd is cubical, then X is closed and bounded. 44 2 Cubical Homology Proof. By deﬁnition a cubical set is the ﬁnite union of elementary cubes.