q -Analog Singular Homology of Convex Spaces
aa r X i v : . [ m a t h . A T ] A ug q -ANALOG SINGULAR HOMOLOGY OF CONVEX SPACES M. ANGEL AND G. PADILLA
Dedicated to Professor A. Reyes,in the occasion of his 76th birthday.
Abstract.
In this article we study some interesting properties of the q -Analog singular homology, which isa generalization of the usual singular homology, suitably adapted to the context of N -complex and amplitudehomology [6]. We calculate the q -Analog singular homology of a convex space. Although it is a local matter;this is an important step in order to understand the presheaf of q -chains and its algebraic properties. Our resultis consistent with those of Dubois-Viol`ette & Henneaux [4]. Some of these results were presented for the XVIIICongreso Colombiano de Matem´aticas in Bucaramanga, 2011. Introduction
The fact that singular homology satisfies the homotopy axiom is a well known result of topological algebra.It can be understood in several ways. From the topological scope it asserts that any topological space that ishomotopic to a single point, must have no topological holes. More than this, homotopic spaces have the samesingular homology and homotopic maps induce the same maps between the respective homology groups. Acustomary proof can be carried out by means of these mathematical facts,(1) The cone construction [2, p.33].(2) A Leibnitz rule for the convex product of singular chains [1, p.220].(3) The double composition of border map ∂ vanishes, i.e. ∂ = 0, which means that singular chainsconstitute a usual chain complex.On the other hand, the theory of N -complexes has raised in the last years as a new homology theory with abroad field of applications in quantum physics [4]. Let N ≥ N -complex is a gradedmodule whose border map ∂ vanishes in the N -th composition, i.e. ∂ N = 0. The m -amplitude homologiesare defined for 1 ≤ m ≤ N −
1; see [3, 6]. For instance, take a complex N -th root of the identity, q ∈ C ;i.e. q N = 1. Then there can be defined q -simplicial chains, as singular chains that are linear combinations ofsingular simplexes where the constants are taken on the ring Z [ q ] and the border map is adequately adapted.Several examples will be treated here below.The main result in this article is that any convex Euclidean space has the same q -Analog singular homologyof a singleton. This is a consequence of the algebraic structure induced by the border map and the combinatorialproperties of q -numbers. In order to prove this,(1) We use the fact that ∂ N = 0, i.e. q -Analog singular chains are a graded N -differential module.(2) We extend the cone construction to a convex product for the q-Analog singular homology.(3) We obtain a q -Leibnitz rule for the convex product and a formula for the Newton’s polynomials.(4) We construct a geometric N -homotopy operator by means of the convex product.An open question we hope to answer in the future is to demonstrate that q -Analog singular chains satisfy theMayer-Vietoris property.The article has been organized as follows. In the sections § § q -numbersand N -complexes. Section § q -singular chains and more examples. In section § Date : 15/02/2011. convex product and show the Leibnitz rule. The last section is devoted to prove the homotopy axiom for q -Analog singular homology, which is our main result.1. q -numbers Recall the definition of q -numbers and some of their properties [6].1.1. q -numbers. Let q ∈ C be a complex non trivial N -th root of the identity i.e. q N = 1 and q = 1. In theclassical literature N is assumed to be a prime integer and q = exp (2 πi/N ), see [6]. The basic q -numbers are(1) [ k ] q = 1 − q k − q = 1 + q + · · · + q ( k − ∀ k ∈ N Notice that [ N ] q = 0. The q -factorial numbers are(2) [ k ] q ! = [1] q · [2] q · · · [ k ] q ≤ k ≤ N − q -combinatorial numbers are(3) (cid:20) kl (cid:21) q = [ k ] q ![ l ] q ! [( k − l )] q ! ∀ ≤ l ≤ k ≤ N − N ≥ q N − R , so R [ q ] = C is the field of complex numbers. Inparticular, q -numbers [ k ] q = 0 have multiplicative inverse in R [ q ]. The following properties follow from thedefinition of q -numbers, we leave the details to the reader. Lemma 1.1.1.
Let ≤ k ≤ n, m ≤ N − . Then, (1) [ m + n ] q = [ m ] q + q m [ n ] q . (2) If n is prime relative to N , then [ n ] q is a unit in Z [ q ] ; and its multiplicative inverse is [ a ] qn where an + bN = 1 for some integers a, b . (3) (cid:20) nk (cid:21) q + q k +1 (cid:20) nk + 1 (cid:21) q = (cid:20) n + 1 k + 1 (cid:21) q = (cid:20) nk + 1 (cid:21) q + q n − k (cid:20) nk (cid:21) q . (4) [ n ] q ! = P σ ∈ Sn q sgn( σ ) where S n is the n th symmetric group and σ runs over all permutations of n elements. N -complexes Let us fix a positive integer N ≥ R, + , · ,
1) as the underlying ring of constants(usually we will take R = Z [ q ]). A N -complex is a generalization of usual chain complexes, and presents a similarbehavior taking into account the integer N , which is called the amplitude of the complex [3, 6].2.1. N -complexes. A N -complex is a pair ( M, ∂ ) such that M is a module and M ∂ ✲ M is a linearendomorphism such that the N -th composition ∂ N = 0 vanishes. We call ∂ the border map . For any integer1 ≤ m ≤ N −
1, we consider the submodules M ∂ ( N − m ) ✲ M ∂m ✲ M B m ( M ) = Im (cid:16) ∂ ( N − m ) (cid:17) ⊂ ker (cid:16) ∂ m (cid:17) = Z m ( M )An element of Z m ( M ) (resp. B m ( M )) is a m - amplitude cycle (resp. border ) . The homology of M withamplitude m is the quotient module H m ( M ) = Z m ( M ) B m ( M )The total homology of M is the graded module H ( M ) = { H m ( M ) : 1 ≤ m ≤ N − } -ANALOG SINGULAR HOMOLOGY OF CONVEX SPACES 3 A morphism of N -complexes ( M, ∂ ) f ✲ ( M ′ , ∂ ′ ) is a linear morphism f such that f ∂ = ∂ ′ f . The inducedarrow is well defined on each amplitude homology H m ( M ) f ✲ H m ( M ′ ), and passes to the total homology H ( M ) f ✲ H ( M ′ ).For any short exact sequence of N -complexes0 ✲ M α ✲ M ′ β ✲ M ′′ ✲ H m ( M ) ∂ ✲ H N − m ( M ) from which arisesan exact hexagon, H m ( M ) H m ( M ′ ) H m ( M ′′ ) H N − m ( M ) H N − m ( M ′ ) H N − m ( M ′′ ) ✏✏✏✏✏✏✶ ✲ PPPPPPq ✏✏✏✏✏✏✮✛PPPPPP✐ α β ∂αβ∂ Graded N -complexes. A graded N -differential module is a pair ( M ∗ , ∂ ) such that M ∗ = { M k : k ∈ Z } is a graded module and ∂ is a ( − M k ∂ ✲ M k − i ∈ Z such that ∂ N = 0. The properties of N -complexes can be extended to the graded case. The amplitude homologyis now a bigraded module H ( M ) = n H m,k ( M ) : 1 ≤ m ≤ N − , k ∈ Z o depending on the amplitude m and the degree k . The inclusion i and the border map ∂ induce, respectively,well defined maps in the bigraded homology.2.3. Examples. (1) Any finite sequence of modules and morphisms0 ✛ M ∂ ✛ M ∂ ✛ · · · ∂N − ✛ M N − ∂N − ✛ M N ✛ N -complex.(2) With a little abuse of notation let us write Z [ q ] [ n ] q ✛ Z [ q ]for the linear function that maps any element α ∈ Z [ q ] to [ n ] q · α . According to § N isprime, [ n ] q = 0 has a multiplicative inverse in Z [ q ] for 1 ≤ n ≤ N −
1. The above map is a moduleisomorphism between free Z [ q ]-modules. ✛ Z [ q ] [2] q ✛ Z [ q ] [3] q ✛ Z [ q ] ✛ · · · [ N − q ✛ Z [ q ] ✛ ✛ ✛ · · · is a N -complex; we use to denote it by (cid:16) Z [ q ] , [ ∗ ] q (cid:17) . A straightforward calculation shows that H m,n (cid:16) Z [ q ] , [ ∗ ] q (cid:17) = (cid:26) Z [ q ] 1 ≤ n = m ≤ N −
20 else
M. ANGEL AND G. PADILLA (3) One can construct N -differential modules with smooth differential forms on R n ; see [3, 6]. There arealso N -complexes with geometric singular chains on any topological space. For more details see thenext sections.2.4. Homotopy of N -complexes. Given any two morphisms of N -differential modules M f,g ✲ M ′ , we saythat they are homotopic and write f ∼ g iff there is a sequence of morphisms of modules M Km ✲ M ′ , for0 ≤ m ≤ N −
1, satisfying(4) N − X m =0 ( ∂ ′ ) m K m ∂ N − m − = ( f − g )The sequence of morphisms K = { K m } m is a homotopy from g to f . The existence of homotopies is anequivalence relation between morphisms of N -complexes; homotopic morphisms induce the same maps in theamplitude homologies. An alternative way to see that this is suitable definition of homotopy between morphismsof differential N -modules is to to follow [7][p.4-5]. Consider, for any pair of N -differential graded modules ( M, ∂ )and (
N, δ ), the graded module Hom (
M, N ) with the N -differential operator given by(5) D ( f ) = N − X i =0 q i (deg( f )+1) δ i f ∂ N − i − A morphism M f ✲ N is compatible with the differentials iff it is a D -cycle, and then it induces awell defined morphism on the k -amplitude homologies H k ( M ) f ✲ H k ( N ) for 1 ≤ k ≤ N −
1. Then, twodifferential morphisms f, g (with deg( f ) = deg( g ) = 0 as above) are homotopic iff their difference f − g is a D -border in Hom ( M, N ). This happens iff there exists a morphism M K ✲ N such that deg( k ) = ( N − f − g ) = D ( k ). Notice that the morphism K has degree deg( K ) = N − q -Chains The N -complex of q -chains on a simplicial set. Recall the construction of simplicial q -chains [3, 6].A simplicial set is a family of non-empty sets and maps X n +1 ∂i ✲ X n ≤ i ≤ n, n ∈ N such that their compositions ( ) satisfy ∂ i ∂ j = ∂ j ∂ i +1 ∀ j ≤ i An element of X n is a basic chain of dimension n . Let N and q be as in § Z [ q ] as the ring of constants. The ( N, q ) -complex generated by X is the graded free Z [ q ]-module that oneach degree n is spanned by X n as a linear basis. q C n ( X ) = Z [ q ] h X n i = ⊕ x ∈ Xn Z [ q ] · x n ∈ N As usual we assume the convention q C n ( X ) = 0 for n <
0. The border map is the graded linear morphism q C n ( X ) ∂ ✲ q C n − ( X ) ∂ = n X i =0 q i ∂ i We must check that our definition makes sense. We write fg for the composition f ( g ( x )) on each x where it makes sense. -ANALOG SINGULAR HOMOLOGY OF CONVEX SPACES 5 Lemma 3.1.1. [Iteration rule for the border map]
The following equality holds ∂ k = [ k ] q ! · X i ≤···≤ ik q i ··· + ik ∂ ik · · · ∂ i ; 0 ≤ k ≤ N Therefore, (cid:0) q C ∗ ( X ) , ∂ (cid:1) is a graded N -complex. [ Proof ] Apply the definition of the border map ∂ and property § (cid:3) In particular, since [ N ] q = 0 we get ∂ N = 0, so q C ∗ ( X ) is a N -complex.3.2. Singular q -chains. A geometric realization is given by the N -complex of Singular q -chains . For eachinteger n ∈ N we write ∆ n for the standard n -simplex, i. e. the convex hull generated on R n +1 with thestandard basis { e , . . . , e n } . A linear map ∆ n L ✲ ∆ m is determined by its values on e , . . . , e n ; we write L = h x , . . . , x n i to mean that x i = L ( e i ) for i = 0 , . . . , n . Take∆ n λj ✲ ∆ n +1 λ i = h e , . . . , b e j , . . . , e n +1 i j = 0 , . . . , n where b e j means to omit the element e j . Given a topological space X = ∅ we define X n as the set of allcontinuous maps ∆ n σ ✲ X . An element of X n is a simplex on X . For each 0 ≤ j ≤ n the j th-face map X n ∂j ✲ X n − is given by the composition ∂ j ( σ ) = σλ j . This family is a simplicial set in our previous sense.The N -complex of singular q -chains on a topological space X q SC n ( X ) = q C n { X n : n ∈ N } ∪ ( ∂ j : X n ✲ X n − : 0 ≤ j ≤ n, n ∈ N )! is the ( N, q )-complex generated by the singular q -simplexes and face maps. An element ξ ∈ q SC n ( X ) is a singular q -chain of dimension n ; it can be written a linear combination ξ = a σ + · · · + a r σ r where each a i ∈ Z [ q ] is a polynomial and each σ i ∈ X n is a simplex of dimension n on X . We also write n = dim( ξ ). Thestandard singular ( N, q ) -homology of X is the homology of this N -complex q H m,n ( X ) = q H m (cid:0) q SC n ( X ) (cid:1) ≤ m ≤ N − , n ∈ N Example: q -homology of a point. If P = { p } is a single point; then P n = { σ n } where ∆ n σn ✲ P isthe constant map. The module q SC n ( P ) = Z [ q ] · σ n ∼ = Z [ q ]is isomorphic to the ring of constants Z [ q ] through the change of basis σ n
1. All face maps ∂ = · · · = ∂ n coincide. The border operator q SC n ( P ) ∂ ✲ q SC n − ( P ) is the zero map for n = 0. For n ≥ ∂ ( aσ n ) = (cid:16) ∂ + q∂ + q ∂ + · · · + q n ∂ n (cid:17) ( aσ n ) = (cid:16) · · · + q n (cid:17) ∂ ( aσ n ) = [ n + 1] q aσ n − can be seen as the multiplication by the element [ n + 1] q ; Z [ q ] ∂ ✲ Z [ q ] ∂ ( a ) = [ n + 1] q · a M. ANGEL AND G. PADILLA
It vanishes when n + 1 a positive multiple of N . In any other case [ n + 1] q = 0 is a unit in Z [ q ]; see § ∂ is a module isomorphism (though not a ring isomorphism). Therefore,(6) q H m,n ( P ) = Z [ q ] 0 ≤ n = m − ≤ N −
20 elsecoincides with the amplitude homology of the N -complex given in the first examples § Exact sequence of a pair.
Given a topological space X and a subspace A ⊂ X ; we consider as usualthe short exact sequence0 ✲ q SC n ( A ) ✲ q SC n ( X ) ✲ q SC n ( X, A ) ✲ § · · · q H m,n ( A ) q H m,n ( X ) q H m,n ( X, A ) q H N − m,n − m ( A ) q H N − m,n − m ( X ) q H N − m,n − m ( X, A ) q H m,n − m ( A ) · · · ✲ ✲ ✲ PPPPPPq ✏✏✏✏✏✮✛✛✛ r δ ∂rδ∂ In the sequel, given an exact sequence from a splitted hexagon as above we will just write(7) · · · ✲ q H m,n ( A ) ✲ q H m,n ( X ) ✲ q H m,n ( X, A ) ∂ ✲ q H N − m,n − m ( A ) ✲ · · · for short. In particular, this one is the ( N, q ) -homology sequence of the pair ( X, A ). There is also a(
N, q )-homology sequence of a triple (
X, A, B )(8) · · · ✲ q H m,n ( A, B ) ✲ q H m,n ( X, B ) ✲ q H m,n ( X, A ) ∂ ✲ q H N − m,n − m ( A, B ) ✲ · · · As usual, the connecting morphism is obtained by chasing in the diagram.4.
Convex product
Now we extend the usual cone construction [2, p. 38] to a convex product, this will be the operation between q -chains in order to have a geometric N -homotopy. Our goal is to construct a homotopy operator K as in § R N − . Since the cone constructions increases the dimension in 1, afirst attempt should be to iterate the conification from N − Convex product.
Suppose that X ⊂ R d is a convex subspace. Given two simplexes∆ m τ ✲ X σ ✛ ∆ n and a point ( α ; β ) = ( α , . . . , α m ; β , . . . , β n ) ∈ ∆ m + n +1 -ANALOG SINGULAR HOMOLOGY OF CONVEX SPACES 7 write | α | = α + · · · + α m and | β | = β + · · · + β n ; so | α | + | β | = 1. Consider τ ∗ σ : ∆ m + n +1 ✲ X τ ∗ σ ( α ; β ) = τ ( α ) | β | = 0 σ ( β ) | α | = 0 | α | · τ (cid:16) α | α | (cid:17) + | β | · σ (cid:16) β | β | (cid:17) elseThe simplex τ ∗ σ above is unique for each pair ( τ, σ ) so the map can be extended to a bilinear operation q SC m ( X ) × q SC n ( X ) ∗ ✲ q SC m + n +1 ( X )For m = 0, τ ( e ) = P is a single point and τ ∗ σ = P ( σ ) is the conification of σ to the vertex P . In general, τ ∗ σ can be thought as a convex combination of τ and σ . The convex product satisfies nice properties withrespect to the border map. Lemma 4.1.1. [Leibnitz rule]
Let τ ∈ q SC m ( X ) and σ ∈ q SC n ( X ) . If mn > then ∂ ( τ ∗ σ ) = ∂ ( τ ) ∗ σ + q m +1 τ ∗ ∂ ( σ )[ Proof ] By the bilinearity of the border map we can suppose that τ, σ are singular simplexes. Apply thedefinition of the border map, see § ∂ i ( τ ∗ σ ) = ( ∂ i τ ) ∗ σ ≤ i ≤ mτ ∗ (cid:0) ∂ i − m − σ (cid:1) m + 1 ≤ i (cid:3) Newton’s terms.
Our main goal on this § is to prove a general formula for ∂ k ( τ ∗ σ ). If τ, σ are 0-dimensional singular simplexes then, by definition of the border map; the border of the 1-simplex τ ∗ σ =[ τ ( e ) , σ ( e )] is(10) ∂ ( τ ∗ σ ) = σ ( e ) + qτ ( e ) = σ + qτ This is the simplest counter-example of the Leibnitz rule since, at § m = dim( τ ) = 0 and n = dim( σ ) >
0, applying the definition of the border map we get(11) ∂ ( τ ∗ σ ) = σ + qτ ∗ ∂ ( σ )This is exactly what happens in the usual case for N = 2 and q = −
1, see [2, p.35 eq.(4.9)]; we will use thisin the sequel. Broadly speaking, since dim( τ ∗ σ ) = m + n + 1, for k ≥ m + n + 2 all terms in a Newton’spolynomial should vanish. One can conjecture that, for min { m, n } ≤ k ≤ m + n + 1 some of the terms vanishand others perhaps not. Given a singular simplex ∆ m τ ✲ X , the Newton’s terms of τ is N i ( τ ) = ∂ i ( τ ) i ≤ m [ m + 1] q ! i = m + 10 i ≥ m + 2We will show the following statement, Proposition 4.2.1. [Newton’s polynomial]
Let τ ∈ q SC m ( X ) and σ ∈ q SC n ( X ) . Then ∂ k ( τ ∗ σ ) = k X i =0 q i ( m +1 − k + i ) · (cid:20) ki (cid:21) q N k − i ( τ ) ∗ N i ( σ ) ∀ k ≥ k > min { m, n } ; thiswill be listed in a sort of lemmas called the tail formulæ . We now carry out the plan. M. ANGEL AND G. PADILLA
Lemma 4.2.2. [Tail formula
Given a simplex ∆ m τ ✲ X let T j = τ ( e j ) for j = 0 , . . . , m . Then ∂ m ( τ ) = [ m ] q ! · m X j =0 q j T m − j [ Proof ] Let us apply twice the iteration rule of the border map § ∂ ( τ ) = m − X i =0 m X j =0 q i + j ∂ i ∂ j ( τ ) = (1 + q ) X ≤ i ≤ j ≤ m − q i + j ∂ i ∂ j ( τ ) Notice that the indexes i, j run over 0 ≤ i ≤ j ≤ m −
1. For ∂ k ( τ ) the indexes i , . . . , i k run over 0 ≤ i ≤· · · ≤ i k ≤ m − k + 1. Finally, for ∂ m ( τ ) the indexes i , . . . , i m run over 0 ≤ i ≤ · · · ≤ i m = m − m + 1 = 1.Therefore, ∂ m ( τ ) = [ m ] q ! · P ≤ i ≤···≤ im ≤ q i ··· + im ∂ im · · · ∂ i ( τ )= [ m ] q ! (cid:16) q ··· +0 ∂ m ( τ ) + q ··· +0+1 ∂ ∂ ( m − ( τ ) + · · · + q ··· +1 ∂ m ( τ ) (cid:17) = [ m ] q ! (cid:16) q ∂ m ( τ ) + q ∂ ∂ ( m − ( τ ) + · · · + q m ∂ m ( τ ) (cid:17) = [ m ] q ! · m P j =0 q j T m − j (cid:3) Lemma 4.2.3. [Tail formula
Given two chains τ ∈ q SC m ( X ) and σ ∈ q SC n ( X ) , ∂ (cid:16) ∂ m ( τ ) ∗ ∂ n ( σ ) (cid:17) = [ m + 1] q ! ∂ n ( σ ) + q [ n + 1] q ! ∂ m ( τ )[ Proof ] By the bilinearity of the convex product and the linearity of the border map, we can assume that∆ m τ ✲ X σ ✛ ∆ n are two simplexes. By lemma § ∂ m ( τ ) = [ m ] q ! m X j =0 q j T m − j ∂ n ( σ ) = [ n ] q ! n X i =0 q i S n − i in a suitable form. Then ∂ (cid:16) ∂ m ( τ ) ∗ ∂ n ( σ ) (cid:17) = ∂ [ m ] q ! m P j =0 q j T m − j ! ∗ [ n ] q ! n P i =0 q i S n − i !! = [ m ] q ! [ n ] q ! m P j =0 n P i =0 q i q j ∂ (cid:0) T m − j ∗ S n − i (cid:1) = [ m ] q ! [ n ] q ! m P j =0 n P i =0 q i q j (cid:0) S n − i + q · T m − j (cid:1) = [ m ] q ! m P j =0 q j ! [ n ] q ! n P i =0 q i S n − i ! + q · [ n ] q ! n P i =0 q i ! [ m ] q ! m P j =0 q j T m − j ! = [ m + 1] q ! ∂ n ( σ ) + q [ n + 1] q ! ∂ m ( τ ) as desired. (cid:3) -ANALOG SINGULAR HOMOLOGY OF CONVEX SPACES 9 Lemma 4.2.4. [Tail formulæ
Let τ ∈ q SC m ( X ) and σ ∈ q SC n ( X ) . If mn > then ∂ k ( ∂ m ( τ ) ∗ σ ) = [ m + 1] q ! [ k ] q ∂ k − ( σ ) + q k ∂ m ( τ ) ∗ ∂ k ( σ ) 1 ≤ k ≤ n [ m + 1] q ! [ n + 1] q ∂ n ( σ ) + [ n + 1] q ! q n +1 ∂ m ( τ ) k = n + 10 else[ Proof ] By the bilinearity of the convex product and linearity of the border map, it is enough to show it onthe generators. Assume that σ, τ are simplexes. Let T j = τ ( e j ) for j = 0 , . . . , m . By § ∂ m ( τ ), equation(11) at § T j ∗ σ for each j , the linearity of ∂ and the bilinearity of the cone-product; ∂ (cid:16) ∂ m ( τ ) ∗ σ (cid:17) = ∂ [ m ] q ! m P j =0 q j T m − j ! ∗ σ ! = [ m ] q ! · m P j =0 q j ∂ (cid:0) T m − j ∗ σ (cid:1) = [ m ] q ! · m P j =0 q j (cid:0) σ + qT m − j (cid:1) = [ m ] q ! · m P j =0 q j ! · σ + [ m ] q ! · q m P j =0 q · j T m − j ! ∗ ∂ ( σ )= [ m + 1] q ! · σ + q · ∂ m ( τ ) ∗ ∂ ( σ ) This proves the equality for k = 1; for 2 ≤ k ≤ n apply this rule and use induction on k . For k = n + 1, bydirect calculations ∂ n +1 (cid:16) ∂ m ( τ ) ∗ σ (cid:17) = ∂ (cid:16) ∂ n ( τ ∗ σ ) (cid:17) = ∂ (cid:16) [ m + 1] q ! [ n ] q ∂ n − ( σ ) + q n ∂ m ( τ ) ∗ ∂ n ( σ ) (cid:17) = [ m + 1] q ! [ n ] q ∂ n ( σ ) + q n ∂ (cid:16) ∂ m ( τ ) ∗ ∂ n ( σ ) (cid:17) For the last term we now apply lemma § ∂ n +1 (cid:16) ∂ m ( τ ) ∗ σ (cid:17) = [ m + 1] q ! [ n ] q ∂ n ( σ ) + q n (cid:16) [ m + 1] q ! ∂ n ( σ ) + q [ n + 1] q ! ∂ m ( τ ) (cid:17) = [ m + 1] q ! [ n + 1] q ∂ n ( σ ) + [ n + 1] q ! q n +1 ∂ m ( τ ) as desired. (cid:3) A similar expression can be obtained for ∂ k ( τ ∗ ∂ n ( σ )), though we will not need it here.4.3. Proof of Proposition § We will proceed by double induction on n + m and k . For n + m = 0 wehave n = m = 0. Consider the following cases: k = 0 which is trivial, k = 1 which gives equation (10) at § k ≥ ∂ k ( τ ∗ σ ) = 0 by a dimension argument. This proves § m + n = 0 and k ≥
0. For m + n > τ, σ with respective dimensions m, n . Let us assume the inductive hypotheses, i.e.that § τ ′ , σ ′ with respective dimensions m ′ , n ′ such that m ′ + n ′ < m + n .For k = 0 there is nothing to prove. For k ≥ dim( τ ∗ σ ) + 1 = m + n + 2, by a dimension argument, the leftside of the Newton’s polynomial at § N k − i ( τ ) ∗ N i ( σ ) in the right side vanishsince, for any i ≤ k , we have i ≤ n ⇒ k − i > m and k − i ≤ m ⇒ i > n , so the statement holds. Hence weonly have to check § ≤ k ≤ m + n + 1. For k = 1 the statement of § § m = deg( τ ) so the power q m +1 in thestatement of § τ ; i.e. ∂ ( τ ∗ σ ) = ∂ ( τ ) ∗ σ + q deg( τ )+1 τ ∗ ∂ ( σ ). Assume the inductive hypothesisfor k ≤ min { m, n } −
1. Then, by the linearity of the border map, ∂ k +1 ( τ ∗ σ ) = ∂ (cid:16) ∂ k ( τ ∗ σ ) (cid:17) = ∂ " k X i =0 q i ( m +1 − k + i ) · (cid:20) ki (cid:21) q ∂ k − i ( τ ) ∗ ∂ i ( σ ) = k X i =0 q i ( m +1 − k + i ) · (cid:20) ki (cid:21) q ∂ (cid:16) ∂ k − i ( τ ) ∗ ∂ i ( σ ) (cid:17) Since k ≤ min { m, n } −
1, all the terms ∂ (cid:16) ∂ k − i ( τ ) ∗ ∂ i ( σ ) (cid:17) in the last sum satisfy the hypothesis of § ∂ k +1 ( τ ∗ σ ) = ∂ k +1 ( τ ) ∗ σ + k P i =1 q ( i +1)( m − k + i +1) · (cid:20) ki (cid:21) q + q ( i +1) · (cid:20) ki + 1 (cid:21) q ! ∂ k − i ( τ ) ∗ ∂ i +1 ( σ ) + τ ∗ ∂ k +1 ( σ ) By property § q -combinatorial numbers can be arranged, so ∂ k +1 ( τ ∗ σ ) = k +1 X i =0 q ( i +1)( m − k + i +1) · (cid:20) k + 1 i + 1 (cid:21) q ∂ k − i ( τ ) ∗ ∂ i +1 ( σ )as desired. We have proved § ≤ k ≤ min { m, n } .For min { m, n } + 1 ≤ k ≤ m + n + 1 consider the following cases. • m < n : We check directly § k = m + 1 ≤ n . Notice that ∂ m +1 ( τ ∗ σ ) = ∂ (cid:16) ∂ m ( τ ∗ σ ) (cid:17) = ∂ " m P i =0 q i (1+ i ) · (cid:20) mi (cid:21) q ∂ m − i ( τ ) ∗ ∂ i ( σ ) § k = m = ∂ (cid:16) ∂ m ( τ ) ∗ σ (cid:17) + m P i =1 q i (1+ i ) · (cid:20) mi (cid:21) q ∂ (cid:20) ∂ m − i ( τ ) ∗ ∂ i ( σ ) (cid:21) linearity of ∂ = h [ m + 1] q ! σ + q∂ m ( τ ) ∗ ∂ ( σ ) i + m P i =1 q i (1+ i ) · (cid:20) mi (cid:21) q (cid:20) ∂ m − i +1 ( τ ) ∗ ∂ i ( σ ) + q (1+ i ) ∂ m − i ( τ ) ∗ ∂ i +1 ( σ ) (cid:21) § § m + 1] q ! σ + m P i =1 q i · (cid:20) mi − (cid:21) q + q i · (cid:20) mi (cid:21) q ! ∂ m − i ( τ ) ∗ ∂ i +1 ( σ ) + τ ∗ ∂ m +1 ( σ ) Group similar terms= [ m + 1] q ! σ + m +1 P j =1 q i · (cid:20) m + 1 j (cid:21) q ∂ m − i ( τ ) ∗ ∂ i +1 ( σ ) § This proves § k = m + 1. Let us assume again, by induction on k , that we have proved it forany integer from 0 to some k such that m + 1 ≤ k ≤ n . Then, by linearity of the border map and theinductive hypothesis, ∂ k +1 ( τ ∗ σ ) = ∂ (cid:16) ∂ k ( τ ∗ σ ) (cid:17) = ∂ " k X i =0 q i ( m − k +1+ i ) · (cid:20) ki (cid:21) q N k − i ( τ ) ∗ N i ( σ ) = k X i =0 q i ( m − k +1+ i ) · (cid:20) ki (cid:21) q ∂ (cid:16) N k − i ( τ ) ∗ N i ( σ ) (cid:17) By definition of the Newton’s terms at § N k − i ( τ ) vanishes for k − i ≥ m + 2. Take only take theterms satisfying 0 ≤ k − i ≤ m + 1; i.e. k − m − ≤ i ≤ k . We get, ∂ k +1 ( τ ∗ σ ) = k P i = k − m − q i ( m − k +1+ i ) · (cid:20) ki (cid:21) q ∂ (cid:18) N k − i ( τ ) ∗ N i ( σ ) (cid:19) = (cid:20) kk − m − (cid:21) q ∂ (cid:18) N m +1 ( τ ) ∗ N k − m − ( σ ) (cid:19) + k P i = k − m q i ( m − k +1+ i ) · (cid:20) ki (cid:21) q ∂ (cid:18) ∂ k − i ( τ ) ∗ ∂ i ( σ ) (cid:19) = [ m + 1] q ! (cid:20) kk − m − (cid:21) q ∂ k − m ( σ ) + q ( k − m ) (cid:20) kk − m (cid:21) q ∂ (cid:18) ∂ m ( τ ) ∗ ∂ k − m ( σ ) (cid:19) + k P i = k − m +1 q i ( m − k +1+ i ) · (cid:20) ki (cid:21) q ∂ (cid:18) ∂ k − i ( τ ) ∗ ∂ i ( σ ) (cid:19) -ANALOG SINGULAR HOMOLOGY OF CONVEX SPACES 11 In the last expression, apply the tail formula § § ∂ k +1 ( τ ∗ σ ) = [ m + 1] q ! (cid:20) kk − m − (cid:21) q ∂ k − m ( σ ) + q ( k − m ) (cid:20) kk − m (cid:21) q (cid:18) [ m + 1] q ! ∂ k − m ( σ ) + q∂ m ( τ ) ∗ ∂ k − m +1 ( σ ) (cid:19) + k P i = k − m +1 q i ( m − k +1+ i ) · (cid:20) ki (cid:21) q (cid:18) ∂ k − i +1 ( τ ) ∗ ∂ i ( σ ) + q m − ( k − i )+1 ∂ k − i ( τ ) ∗ ∂ i +1 ( σ ) (cid:19) Regroup similar terms. Apply property § q -combinatoric numbers; ∂ k +1 ( τ ∗ σ ) = [ m + 1] q ! (cid:20) k + 1 k − m (cid:21) q ∂ k − m ( σ ) + k P i = k − m +1 q i ( m − k + i ) (cid:20) ki − (cid:21) q + q i ( m − k +1+ i ) · (cid:20) ki (cid:21) q ! ∂ k − i +1 ( τ ) ∗ ∂ i ( σ )+ q ( k +1)( m +1) · (cid:20) kk (cid:21) q ∂ ( τ ) ∗ ∂ k ( σ )= (cid:20) k + 1 k − m (cid:21) q N m +1 ( τ ) ∗ N k − m ( σ ) + k P i = k − m +1 q i ( m − k + i ) (cid:20) ki − (cid:21) q + q i · (cid:20) ki (cid:21) q ! ∂ k − i +1 ( τ ) ∗ ∂ i ( σ )+ q ( k +1)( m +1) · (cid:20) kk (cid:21) q ∂ ( τ ) ∗ ∂ k ( σ )= (cid:20) k + 1 k − m (cid:21) q N m +1 ( τ ) ∗ N k − m ( σ ) + k P i = k − m +1 q i ( m − k + i ) (cid:20) k + 1 i (cid:21) q N k − i +1 ( τ ) ∗ N i ( σ )+ q ( k +1)( m +1) · (cid:20) k + 1 k + 1 (cid:21) q N ( τ ) ∗ N k ( σ )= k +1 P i = k − m q i ( m − k + i ) (cid:20) k + 1 i (cid:21) q N k − i +1 ( τ ) ∗ N i ( σ ) Include the vanishing terms of the form N k − i +1 ( τ ) ∗ N i ( σ ) for 0 ≤ i ≤ k − m −
1. We obtain ∂ k +1 ( τ ∗ σ ) = k +1 X i =0 q i ( m − k + i ) (cid:20) k + 1 i (cid:21) q N k − i +1 ( τ ) ∗ N i ( σ )This is the complete expression of the right term in § k + 1. Thus we have proved the statementfor 0 ≤ k ≤ n + 1. Finally, for n + 2 ≤ k ≤ m + n + 1 a similar argumentation can be carried out. Thetail formulæ must be used in both extremes of the sum. • m ≥ n : We leave the details to the reader. (cid:3) q -homology group, augmentation] Since ∆ = { e } is a singleton, each 0-dimensionalsimplex σ in X can be identified to its image point x = σ ( e ) ∈ X . The 0-th module of q chains is then q SC ( X ) = ⊕ σ ∈ X Z [ q ] · σ ∼ = ⊕ x ∈ X Z [ q ] · x Consider the morphism q SC ( X ) ǫ ✲ Z [ q ] X i α i x i X i α i Given a m -simplex ∆ m τ ✲ X the element ∂ m ( τ ) is a 0-dimensional chain. Let us write P j = τ ( e j ) for j = 0 , . . . , m . Applying § ǫ (cid:16) ∂ m ( τ ) (cid:17) = [ m ] q ! · m X j =0 q j = [ m + 1] q ! In particular, for m = N − ǫ (cid:16) ∂ N − ( τ ) (cid:17) = 0 and(12) q H , ( X ) ǫ ✲ Z [ q ] [ τ ] ǫ ( τ )is a well defined linear surjective morphism.The constant map X ✲ P induces a morphism of N -complexes q SC n ( X ) γ ✲ q SC n ( P )called the augmentation . The reduced q -homology q e H m,n ( X ) = ker ( q H m,n ( X ) γ ✲ q H m,n ( P ) ) is the kernel of the corresponding homology morphism. By equation (6) at § q H m,n ( X ) = Z [ q ] ⊕ q e H m,n ( X ) 1 ≤ n = m ≤ N − q e H m,n ( X ) elseA reduced q -homology sequence of the pair · · · ✲ q e H m,n ( A ) ✲ q e H m,n ( X ) ✲ q H m,n ( X, A ) ∂ ✲ q e H N − m,n − m ( A ) ✲ · · · can also be deduced. 5. q -Analog Singular Homology of Convex Spaces We arrive to the main result of this article.5.1.
The index map.
In complete analogy with the usual case ( N = 2, q = − q SC ∗ ( X ) , ∂ ) η ✲ (cid:16) Z [ q ] , [ ∗ ] q (cid:17) that sends each n -simplex to 1 ∈ Z [ q ] in the corresponding degree, for 0 ≤ n ≤ N −
2; and vanishes for n ≥ N − Theorem 5.1.1.
Let X ⊂ R N − be a convex space. Then the index map q SC ∗ ( X ) η ✲ Z [ q ] induces anisomorphism in N -homology. [ Proof ] We follow essentially the same argumentation of [2, p.38]. We will define a map Z [ q ] b P ✲ q SC ∗ ( X )The composition η b P = id must be the identity map on the ( N − Z [ q ] , [ ∗ ]); so b P (1) = ν n will be asingle singular n -simplex for 0 ≤ n ≤ N − n ≥ N −
1. The other composition b P η will be N -homotopic to the identity map id on q SC ∗ ( X ) in the sense of § ν n ’s we will construct a homotopy operator q SC n ( X ) K ✲ q SC n − N +1 ( X )and show how it works. We proceed by steps. -ANALOG SINGULAR HOMOLOGY OF CONVEX SPACES 13 • Definition of K : Fix some singular N − N − ı ✲ X . Since N is a prime integer,[ k ] q is a unit in Z [ q ] for 1 ≤ k ≤ N − N − q ! is also a unit; see § K ( σ ) = 1[ N − q ! · ( ı ∗ σ )Up to the correction by the constant, K is essentially the convex product of ı and σ ; and it can be uniquelyextended to q SC ∗ (cid:16) R N − (cid:17) by linearity. • K is a N -homotopy: We verify that K satisfies § σ ∈ q SC n ( X ). By § ∂ k K∂ N − k − ( σ ) = 1[ N − q ! · ∂ k (cid:16) ı ∗ (cid:16) ∂ N − k − ( σ ) (cid:17)(cid:17) = 1[ N − q ! · k X i =0 q i ( N − − k + i ) · (cid:20) ki (cid:21) q N k − i ( ı ) ∗ N i (cid:16) ∂ N − k − ( σ ) (cid:17) Although ∂ j ( σ ) is a chain and not a simplex, since ∂ i (cid:16) ∂ j ( σ ) (cid:17) = ∂ i + j ( σ ) we will assume the following convention, N i (cid:16) ∂ j ( σ ) (cid:17) = N i + j ( σ ) = ∂ i + j ( σ ) j ≤ n − i [ n + 1] q ! j = n − i + 10 elseTherefore ∂ k K∂ N − k − ( σ ) = 1[ N − q ! · k X i =0 q i ( N − − k + i ) · (cid:20) ki (cid:21) q N k − i ( ı ) ∗ N N − k − i ( σ ) Taking sums in both sides, N − X k =0 ∂ k K∂ N − k − ( σ ) = 1[ N − q ! · N − X k =0 k X i =0 q i ( N − − k + i ) · (cid:20) ki (cid:21) q N k − i ( ı ) ∗ N N − − k + i ( σ ) Let us reorder and group all similar terms taking l = k − i . We arrive to the following expression (14) N − X k =0 ∂ k K∂ N − k − ( σ ) = 1[ N − q ! · N − X l =0 α l N l ( ı ) ∗ N N − l − ( σ ) Let us look for instance the following array of the coefficients α k,l for N = 7. The vertical sums of the entriesin the table correspond to the values of α l . k q q [2] q q q [3] q q [3] q q q [4] q q (cid:20) (cid:21) q q [4] q q q [5] q q (cid:20) (cid:21) q q (cid:20) (cid:21) q q [5] q q q [6] q q (cid:20) (cid:21) q q (cid:20) (cid:21) q q (cid:20) (cid:21) q q [6] q q
66 5 4 3 2 1 0 l Figure 1.
Table of the coefficients α k,i for N = 7. Each horizontal row corresponds to some 0 ≤ k ≤ l = ( k − i ). The powers of q have been simplified with the identity q = 1. These coefficients can be simplified by using the properties of q -numbers. A simple inspection suggests that α l = 0 for 0 ≤ l ≤ N −
2. This is, indeed, the case. Let us write α l = P l = k − i α k,i = P l = k − i q i ( N − − k + i ) · (cid:20) ki (cid:21) q = N − l − P i =0 q i ( N − l − · (cid:20) l + ii (cid:21) q = s P i =0 q is · (cid:20) N − − s + ii (cid:21) q take s = N − l − s P i =0 q is · (cid:20) N − − s + iN − − s (cid:21) q = β s symmetry of combinatorials We check that β s = α N − − s = 0 for 1 ≤ s ≤ N −
1. For s = 1, β = q + q (cid:20) N − N − (cid:21) q = 1 + q [ N − q = [ N ] q = 0 = α N − -ANALOG SINGULAR HOMOLOGY OF CONVEX SPACES 15 Assume that β s = 0 for some s ≤ N −
2. Then, β s +1 = s +1 P i =0 q i ( s +1) (cid:20) N − − s + iN − − s (cid:21) q = s +1 P i =0 q is q i (cid:20) N − − s + iN − − s (cid:21) q by definition= 1 + s +1 P i =1 q is (cid:20) N − − s + iN − − s (cid:21) q − (cid:20) N − − s + iN − − s (cid:21) q ! by § s +1 P i =1 q is (cid:20) N − − s + iN − − s (cid:21) q − s +1 P i =1 q is (cid:20) N − − s + iN − − s (cid:21) q = 1 + ( −
1) + s P i =0 q is (cid:20) N − − s + iN − − s (cid:21) q + q ( s +1) s (cid:20) NN − − s (cid:21) q ! − s +1 P i =1 q is (cid:20) N − − s + iN − − s (cid:21) q split the first sum= s P i =0 q is (cid:20) N − − s + iN − − s (cid:21) q − s P j =0 q ( j +1) s (cid:20) N − − s + jN − − s (cid:21) q [ N ] q = 0 , i = j + 1(2nd sum)= (1 − q s ) β s = 0 by definition By equation (14), the definition of the Newton’s terms § σ , we deduce that N − X k =0 ∂ k K∂ N − k − ( σ ) = α N − [ N − q ! N N − ( ı ) ∗ N ( σ ) = σ whenever n = dim( σ ) ≥ N −
1, and the whole sum in the left term vanishes when n < N −
1. In other words,(15) N − X k =0 ∂ k K∂ N − k − ( σ ) = σ dim( σ ) ≥ N −
10 else (cid:3)
Acknowledgments
G. Padilla would like to thank Professors E. Becerra, V. Tapia and B. Uribe for some helpful conversations,so as A. Barbosa and D. Maya for their remarks on a previous draft manuscript. This article was partiallysupported by the Universidad Nacional de Colombia.
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E-mail address : [email protected] Departamento de Matem´aticas, Edificio 404, Ofic. 315Universidad Nacional de Colombia, Facultad de CienciasCarrera 30, calle 45. Bogot´a - Colombia. Tlf. (+571)3165000. Ext. 13166
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