By Kaczynsky, Mischaikow, Mrozek.

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**Example text**

3 Connected Components and H0 (X) 55 (b) Determine C(T ) and compute H∗ (T ). bmp). Run the Homology program to find the homology of L. e. remove two pixels) in opposite walls of the labyrinth and again run the program to find the homology of what is left. Make a guess about the solvability of the labyrinth. e. a possibility of passing inside from one gate to another without crossing a wall. cub in the folder Examples of Homology program . Run the program to find the homology of K with respect to (a) coefficients in Z; (b) coefficients in Zp for p = 2, 3, 5, 7.

Create input files for the Homology program for several values of n starting from n = 4. Use the Homology program to compute the dimensions rk of Hk (Gn ; Z2 ). Make a conjecture about the formula for every n ≥ 3. 2 Cubical Homology In Chapter 1 we suggested what were the important elements in Homology. In particular, we used the edges and vertices of a graph to generate algebraic objects that measured the non-triviality of the topology of the graph. In this chapter we shall formally define cubical homology.

The proof will be done by induction on d := emb Q. 32. e. Q = I × P where emb I = 1 and emb P = d − 1. Then, by the definition of the boundary operator ∂(Q Q ) = ∂(I = ∂I Since P P P Q) Q + (−1)dim I I ∂(P Q) Q satisfies the induction assumption, we see that ∂(Q Q ) = ∂ I P Q + (−1)dim I I = ∂I P Q + (−1)dim I I = ∂I P + (−1)dim I I ∂P ∂P ∂P Q + (−1)dim P P ∂Q Q + (−1)dim I+dim P I P ∂Q Q + (−1)dim Q Q ∂ Q = ∂ Q Q + (−1)dim Q Q ∂ Q , where the last equality follows again from the definition of the boundary operator.