Chapter 1 Preliminaries In this chapter we introduce some preliminaries that we will use in the rest of this memoir. The first section is devoted to the main mathematical notions employed in this work. The Kenzo system, a successful Common Lisp program developed by Francis Sergeraert and some coworkers devoted to perform computations in Algebraic Topology, is presented in Section 1.2. Finally, the deduction machinery employed in this memoir, the ACL2 system, is briefly presented in Section 1.3. 1.1 Mathematical preliminaries Algebraic Topology is a vast and complex subject, in particular mixing Algebra and (combinatorial) Topology. Algebraic Topology consists in trying to use as much as possible \algebraic" methods to attack topological problems. For instance, one can define some special groups associated with a topological space, in a way that respects the relation of homeomorphism of spaces. This allows us to study properties about topological spaces by means of statements about groups, which are often easier to prove. 1.1.1 Homological Algebra The following basic definitions can be found, for instance, in [Mac63]. Definition 1.1. Let R be a ring with a unit element 1 6= 0. A left R-module M is an additive abelian group together with a map p : R × M ! M, denoted by p(r; m) ≡ rm, such that for every r; r0 2 R and m; m0 2 M (r + r0)m = rm + r0m r(m + m0) = rm + rm0 (rr0)m = r(r0m) 1m = m 5 6 Chapter 1 Preliminaries A similar definition is given for a right R-module. For R = Z (the integer ring), a Z-module M is simply an abelian group. The map p : Z × M ! M is given by 8 n < m+ ··· +m if n > 0 p(n; m) = 0 if n = 0 n : (−m)+ ··· +(−m) if n < 0 Definition 1.2. Let R be a ring and M and N be R-modules. An R-module morphism α : M ! N is a function from M to N such that for every m; m0 2 M and r 2 R α(m + m0) = α(m) + α(m0) α(rm) = rα(m) α(0M ) = 0N Definition 1.3. Given a ring R, a graded module M is a family of left R-modules (Mn)n2Z. Definition 1.4. Given a pair of graded modules M and M 0, a graded module morphism f of degree k between them is a family of module morphisms (fn)n2Z such that fn : 0 Mn ! Mn+k for all n 2 Z. Definition 1.5. Given a graded module M, a differential (dn)n2Z is a family of module endomorphisms of M of degree −1 such that dn−1 ◦ dn = 0 for all n 2 Z. From the previous definitions, the notion of chain complex can be defined. Chain complexes are the central notion in Homological Algebra and can be seen as an algebraic means to study properties of topological spaces in several dimensions. Definition 1.6. A chain complex C∗ is a family of pairs (Cn; dn)n2Z where (Cn)n2Z is a graded module and (dn)n2Z is a differential of C∗. The module Cn is called the module of n-chains. The image Bn = Im dn+1 ⊆ Cn is the (sub)module of n-boundaries. The kernel Zn = Ker dn ⊆ Cn is the (sub)module of n-cycles. In many situations the ring R is the integer ring, R = Z. In this case, a chain complex C∗ is given by a graded abelian group fCngn2Z and a graded group morphism of degree -1, fdn : Cn ! Cn−1gn2Z, satisfying dn−1 ◦ dn = 0 for all n. From now on in this memoir, we will work with R = Z. Let us present some examples of chain complexes. Example 1.7. • The unit chain complex has a unique non null component, namely a Z-module in degree 0 generated by a unique generator and dn = 0 for all n 2 Z. 1.1 Mathematical preliminaries 7 • A chain complex to model the circle is defined as follows. This chain complex has two non null components, namely a Z-module in degree 0 generated by a unique generator and a Z-module in degree 1 generated by another generator; and the differential is the null map. • The diabolo chain complex has associated three chain groups: { C0, the free Z-module on the set fs0; s1; s2; s3; s4; s5g. { C1, the free Z-module on the set fs01; s02; s12; s23; s34; s35; s45g. { C2, the free Z-module on the set fs345g. and the differential is provided by: { d0(si) = 0, { d1(sij) = sj − si, { d2(sijk) = sjk − sik + sij. Pm and it is extended by linearity to the combinations c = i=1 λixi 2 Cn. We can construct chain complexes from other chain complexes, applying constructors such as the direct sum or the tensor product. Definition 1.8. Let C∗ = (Cn; dCn )n2Z;D∗ = (Dn; dDn )n2Z be chain complexes. The direct sum of C∗ and D∗ is the chain complex C∗ ⊕ D∗ = (Mn; dn)n2Z such that, Mn = (Cn;Dn) and the differential map is defined on the generators (x; y) with x 2 Cn and y 2 Dn by dn((x; y)) = (dCn (x); dDn (y)) for all n 2 Z. The notion of direct sum can be generalized to a collection of chain complexes. Definition 1.9. Let M be a right R-module, and N a left R-module. The tensor product M ⊗R N is the abelian group generated by the symbols m ⊗ n for every m 2 M and n 2 N, subject to the relations (m + m0) ⊗ n = m ⊗ n + m0 ⊗ n m ⊗ (n + n0) = m ⊗ n + m ⊗ n0 mr ⊗ n = m ⊗ rn for all r 2 R; m; m0 2 M, and n; n0 2 N. If R = Z (the integer ring), then M and N are abelian groups and their tensor product will be denoted simply by M ⊗ N. Definition 1.10. Let C∗ = (Cn; dCn )n2Z and D∗ = (Dn; dDn )n2Z be chain complexes of right and left Z-modules respectively. The tensor product C∗ ⊗ D∗ is the chain complex of Z-modules C∗ ⊗ D∗ = ((C∗ ⊗ D∗)n; dn)n2Z with M (C∗ ⊗ D∗)n = (Cp ⊗ Dq) p+q=n 8 Chapter 1 Preliminaries where the differential map is defined on the generators x ⊗ y with x 2 Cp and y 2 Dq, according to the Koszul rule for the signs, by p dn(x ⊗ y) = dCp (x) ⊗ y + (−1) x ⊗ dDq (y) Let us present now, one of the most important invariants used in Homological Al- gebra. Given a chain complex C∗ = (Cn; dn)n2Z, the identities dn−1 ◦ dn = 0 mean the inclusion relations Bn ⊆ Zn: every boundary is a cycle (the converse in general is not true). Thus the next definition makes sense. Definition 1.11. Let C∗ = (Cn; dn)n2Z be a chain complex of R-modules. For each degree n 2 Z, the n-homology module of C∗ is defined as the quotient module Zn Hn(C∗) = Bn It is worth noting that the homology groups of a space X are the ones of its associated chain complex C∗(X); the way of constructing the chain complex associated with a space X is explained, for instance, in [Mau96]. In an intuitive sense, homology groups measure \n-dimensional holes" in topological spaces. A 0-dimensional hole is a pair of points in different path components; and so H0 measures the number of connected components of a space. The homology groups Hn measure higher dimensional connectedness. For instance, the n-sphere, Sn, has exactly one n-dimensional hole and no m-dimensional holes if m 6= n. Moreover, it is worth noting that homology groups are an algebraic invariant, see [Mau96]. That is to say, if two topological spaces are homeomorphic, this means that all their homology groups coincide. Let us finish this section with some additional definitions related to chain complexes. Definition 1.12. A chain complex C∗ = (Cn; dn)n2Z is acyclic if Hn(C∗) = 0 for all n, that is to say, if Zn = Bn for every n 2 Z. Definition 1.13. Let C∗ = (Cn; dCn )n2Z;D∗ = (Dn; dDn )n2Z be chain complexes, a chain complex morphism between them is a family of module morphisms (fn)n2Z of degree 0 0 between (Cn)n2Z and (Dn)n2Z such that dn ◦ fn = fn−1 ◦ dn for each n 2 Z. Definition 1.14. Let C∗ = (Cn; dn)n2Z be a chain complex. A chain complex D∗ = 0 (Dn; dn)n2Z is a chain subcomplex of C∗ if • Dn is a submodule of Cn, for all n 2 Z 0 • dn = dn jD∗ 0 The condition dn = dn jD∗ means that the boundary operator of the chain subcomplex is just the differential operator of the larger chain complex restricted to its domain. We denote D∗ ⊂ C∗ if D∗ is a chain subcomplex of C∗. 1.1 Mathematical preliminaries 9 Definition 1.15. A short exact sequence is a sequence of modules: j 0 C00 − C −i C0 0 which is exact, that is, the map i is injective, the map j is surjective and Im i = Ker j. From now on in this memoir, we will work with non-negative chain complexes, that is to say, fCngn2Z such that Cn = 0 if n < 0. A non-negative chain complex C∗ will be denoted by C∗ = fCngn2N. Moreover, the chain complexes we work with are supposed to be free. Definition 1.16. A chain complex C∗ = (Cn; dn)n2N of Z-modules is said to be free if Cn is a free Z-module (a Z-module which admits a basis) for each n 2 N.