By Toshikazu Sunada

AiPrefaceList of SymbolsTopological crystals-Introduction-1 Quotient objects2 Generalities on graphs3 Homology teams of graphs4 overlaying graphs5 Topological crystals6 Canonical placements7 particular construction8 MiscellanyAppendixBibliographyIndex

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**Sample text**

Ge1 ) is obtained by a cyclic permutation of (e1 , . . , et , es , . . , e1 ). This is a contradiction. This completes the proof of our assertion. 6 Notes (I) As a “dual” of homology, one can introduce the notion of cohomology group. For an additive group A, put C0 (X, A) = { f : V −→ A}, C1 (X, A) = {ω : E −→ A| ω (e) = −ω (e) (e ∈ E)} which we call the group of 0-cochains and the group of 1-cochains, respectively. Define the coboundary operator d : C0 (X, A) −→ C1 (X, A) by (d f )(e) = f t(e) − f o(e) .

D) The argument is straightforward. 5 Homotopy In algebraic topology, the term “homotopy” is used to express “continuous deformations”, by which geometric objects such as curves and surfaces are identified in a loose sense. Here we explain homotopy of paths in a graph from a combinatorial point of view. Let c1 , c2 be two paths in X. We write c2 → c1 when c2 has two successive edges e, f with f = e, ¯ and c1 is obtained from c2 by removing e and f (namely, c1 is obtained by removing a back-tracking part of c2 ).

Equivalence classes of this relation are called homotopy classes. Below are five fundamental properties of homotopy that follow directly from the definition: 1. Suppose c1 , c2 ∈ C(x, y), c1 , c2 ∈ C(y, z). If c1 ∼ c2 and c1 ∼ c2 , then c1 · c1 ∼ c2 · c2 . 2. c · c¯ ∼ 0/ x (x = o(c)). 3. If c1 ∼ c2 , the parity13 of |c1 | and |c2 | is the same. 4. Each homotopy class contains a unique geodesic. 5. For a geodesic path c0 , if c ∼ c0 , then either c = c0 or there exists a sequence of paths c1 , c2 , .