altres topònims: (no citats als grups 1 o 1bis):

the union preceding the same step is preserved when moving from a one-critical filtered complex X to a filtered Morse complex M compatible with X .

In this section, we assume that the filtered complex X is one-critical. For example, it might be obtained by sublevel sets with respect to a filtering function φ : X ´Ñ Zn as observed in Remark1.6.5.

5.3.1 Proposition. For each integer n ě 1, each filtration grade u P Zn, and each homology degree k, H_k ´ Xu, Xu1Y ¨ ¨ ¨ Y Xun ¯ – Hk ´ Mu, Mu1Y ¨ ¨ ¨ Y Mun ¯ .

Proof. We claim that, for all 1 ď i ď n, HkpXu

1

Y ¨ ¨ ¨ Y Xuiq – HkpMu

1

Y ¨ ¨ ¨ Y Muiq. For simplifying the notation, we denote by i the grade ui, by XYi the union of Lefschetz complexes X1Y ¨ ¨ ¨ Y Xi, and by MYi _{the union of Lefschetz complexes M}1_{Y ¨ ¨ ¨ Y M}i_.

We proceed inductively on i. The base step is provided by Theorem1.6.8which implies that the inclusion-induced maps from H_kpMiq to HkpXiq are isomorphisms for all integers k. Now, consider

the MV-triples pXYi`1<sub>, X</sub>Yi<sub>, X</sub>i`1_{q and pM}Yi`1<sub>, M</sub>Yi<sub>, M</sub>i`1_{q. Inclusion-induced maps connect the}

chain complexes of the components in pMYi`1<sub>, M</sub>Yi<sub>, M</sub>i`1_{q to those in pX}Yi`1<sub>, X</sub>Yi<sub>, X</sub>i`1_{q and}

commute with the maps in the two short exact sequences. From Theorem1.3.12, we obtain the corresponding long exact Mayer-Vietoris sequences, depicted by rows in the following diagram:

. . .

//

H

k

pX

Yi

q ‘ H

k

pX

i`1

q

// ϕ_k1 

H

k

pX

Yi

Y X

i`1

q

// ϕk 

H

k´1

pX

Yi

X X

i`1

q

// ϕ_k´12 

. . .

. . .

//

H

k

pM

Yi

q ‘ H

k

pM

i`1

q

//

H

k

pM

Yi

Y M

i`1

q

//

H

k´1

pM

Yi

X M

i`1

q

//

. . .

where, by Theorem 6.13 in [164] maps ϕ_k1, ϕk, ϕk´12 are given by naturality property from the

corresponding chain complex maps and, thus, make the diagram commute. By the Five Lemma, if all ϕ_k2, ϕ_k1are isomorphisms, then ϕkis an isomorphism.

Each map ϕ_k1is given as direct sum of two maps. The first map is an isomorphism by inductive assumption. The second one is the isomorphism implied by Theorem 1.6.8. For the maps ϕ_k2, we notice that, once indicated by u ´ 1 the grade with ith-entry defined by ui´ 1, then

Xu´1“ XYiX Xi`1 and Mu´1“ MYiX Mi`1. Indeed, the filtrations are one-critical. This means that there is no cell in the intersection XYi_{X X}i`1_{that might be outside X}u´1_{, and the converse}

inclusion is obvious. Thus, the maps ϕ_k2are the isomorphisms implied by Theorem1.6.8. Hence, the Five Lemma applies and, for all i from 1 to n ´ 1, HkpX1Y ¨ ¨ ¨ Y Xi`1q – HkpM1Y ¨ ¨ ¨ Y Mi`1q

Consider now the inclusions of XYn_{into X}u_{and of M}Yn _{into M}u_{. This allows us to consider the}

relative pairs pXu, XYn_{q and pM}u_{, M}Yn_q.

By Theorem 1.3.10, the two relative pairs induce each a long exact sequence of homology corresponding to the rows in the following diagram:

. . . //H_kpXuq // ψ_k1  H_kpXu, XYn_q // ψk  H_k´1pXYnq // ψ_k´12  . . . . . . //H_kpMuq //H_kpMu, MYn_q //_H k´1pMYnq, //. . .

where, maps ψ_k1, ψ_k, ψ_k2are inclusion-induced and make the diagram commute by applying the same argument as for maps ϕ_k1, ϕ_k2. The maps ψ_k1are the isomorphisms implied by Theorem1.6.8. The maps ψ_k2are the isomorphisms of our claim. Hence, by the Five Lemma, we get that ψ_k are isomorphisms. Thus, HkpXu, XYnq – HkpMu, MYnq which concludes our proof.

A crucial application of Proposition5.3.1is that optimality of V compatible with X can be directly checked on the associated filtered Morse complex M. Moreover, the following Lemma gives a sufficient condition to check, independently for each filtration grade u, that the number mkpuq

of k-cells in any level set Muz ´

Mu1Y ¨ ¨ ¨ Y Mun ¯

is the one expected by considering relative homology.

5.3.2 Lemma. Let u P Zn be any filtration grade of a filtered Morse complex M, and Brel _the

collection of boundary maps for the relative pair pMu, Mu1Y ¨ ¨ ¨ Y Munq. If, for all integers k and all k-cells a P Muz

´

Mu1Y ¨ ¨ ¨ Y Mun ¯

, it holds that B_krela “ 0, then

@k P Z, dim Hk

´

Mu, Mu1Y ¨ ¨ ¨ Y Mun ¯

“ mkpuq.

Proof. Fix an integer k. Notice that m_kpuq is the cardinality of M_kuzpM_ku1Y ¨ ¨ ¨ Y M_kunq and thus, the dimension of the space of relative k-chains in the desired relative pair pMu, Mu1Y ¨ ¨ ¨ Y Munq. If, for all a P M_kuzpM_ku1Y ¨ ¨ ¨ Y M_kunq, we have that B_krela “ 0, then the dimension of the space of relative k-cycles is the same as that of relative k-chains, that is m_kpuq. If, for all a P M_{k`1</sub>u zpM<sub>k`1}u1 Y ¨ ¨ ¨ Y M_{k`1</sub>un q, we have that Brel<sub>k`1}a “ 0, then the dimension of the space of relative k-boundaries is 0. Hence, the dimension of the relative kth-homology of the desired relative pair is mkpuq. This

In document XXIII Jornada d’Antroponímia i Toponímia (página 142-182)