Defining discrete Morse functions on infinite surfaces

De ining disc e e Mo se unc ions on in ini e su aces R. Ayala a, L.M. Fe n´andez a, J.A. Vilches a aDp o. de Geome ´ıa y Topolog´ıa, Uni e sidad de Se illa, 41080, SPAIN Abs ac We p esen an algo i hm which de ines a disc e e Mo se unc ion in Fo man’s sense on an in ini e su ace including a s udy o he minimali y o his unc ion. Key wo ds: disc e e Mo se unc ion, in ini e su ace, c i ical poin , Mo se inequali ies 1. In oduc ion Unde a classical o smoo h poin o iew, Mo se heo y looks o links be ween global p ope ies o a smoo h mani old and c i ical poin s o a unc ion de ined on i . In [2], Fo man in oduced he no- ion o disc e e Mo se unc ion de ined on a ini e cw-complex and, in his combina o ial con ex , he de eloped a disc e e Mo se heo y as a ool o s udying he homo opy ype and homology g oups o hese complexes. Once a Mo se unc ion has been de ined on a complex, hen i s opological in o ma ion can be deduced om he c i ical simplices o his unc- ion. Since we s udied he p oblem conce ning he de ini ion o disc e e Mo se unc ions on in ini e 1- complexes in o he wo ks [1,4], he goal o his pa- pe is o con inue wi h he ollowing na u al s ep: o de elop a way o cons uc ing disc e e Mo se unc ions on in ini e 2-complexes, in pa icula on connec ed and non compac su aces. Ou me hods a e based on algo i hms de eloped by T. Lewine [3] o he ini e case. Gi en a simplicial complex M, R. Fo man [2] in oduces he no ion o disc e e Mo se unc ion as a unc ion :M−→ Rsuch ha , o any p- simplex σ∈M: (M1) ca d{τ(p+1) > σ/ (τ)≤ (σ)} ≤ 1. Email add esses: lm e @us.es (L.M. Fe n´andez ), [email protected] (J.A. Vilches ). 1This wo k is pa ially suppo ed by P.A.I. (M2) ca d{υ(p−1) < σ/ (υ)≥ (σ)} ≤ 1. Ap-simplex σ∈Mis said o be c i ical wi h espec o i : (C1) ca d{τ(p+1) > σ/ (τ)≤ (σ)}= 0. (C2) ca d{υ(p−1) < σ/ (υ)≥ (σ)}= 0. 2. Cons uc ing disc e e Mo se unc ions In o de o de ine a disc e e Mo se unc ion on an in ini e su ace Swe ecall ha i can be ex- p essed as a coun able union o ini e subcomplexes S=∪n∈NKn, wi h Kn⊆Kn+1 o any n∈N. Indeed, le 0be any e ex o a iangula ion o Sand le K1be he closed s a o 0, ha is, he smalles closed subcomplex o Swhich con ains all edges and iangles including 0. The ollowing i- gu e shows he closed s a o 0in con inous lines. 0 Fig. 1. S a o 0. 20 h EWCG Se ille, Spain (2004) 20 h Eu opean Wo kshop on Compu a ional Geome y To de ine he es o subcomplexes Kn, we can do a successi e hickening o K1:K2will be he closed s a o K1and, in he gene al case, Knwill be he closed s a o Kn−1, whe e he closed s a o a subcomplex means he smalles closed subcom- plex ha con ains all edges and iangles which con ain some simplex o he gi en subcomplex. Nex , we shall use a special g aph which con ains in o ma ion o pa o any Kn, in which all ian- gles and some edges o Ka e ep esen ed. I T1 is a spanning ee in K1, hen he complemen a y g aph o T1in K1, deno ed by D1, is he g aph con- s uc ed as ollows: each e ex o D1co esponds o a iangle o K1o o a bounding edge o K1 ( ha is, an edge which is in a unique iangle o K1 and no in T1) and he e is an edge be ween wo e ices o D1i hese ones co espond o wo ian- gles sha ing an edge o o a iangle and a bounding edge which is in his iangle bu no in T1. Consi- de ing he p eceding igu e, D1is he g aph d awed wi h discon inous lines in he ollowing igu e: 0 1 Fig. 2. D1. Now we enla ge he spanning ee T1un il we ge a spanning ee T2in K2. Then, we ob ain D2as an enla gemen o D1and i is he com- plemen a y g aph o T2in K2. We can con inue his p ocess in successi e s eps. I is in e es ing o poin ou ha his cons uc ion is only possible o 2-dimensional complexes because he union o Tnand Dnco e s whole Kn, o any n∈N. The de ini ion o a disc e e Mo se unc ion on Ss a s wi h he de ini ion o in K1. This p ocess is di ided in wo s eps: i s , we de ine on a spanning ee T1in K1s a ing a 0. This assignmen is made by an inc easing way and such ha we do no in oduce any c i ical e ex o edge bu he e ex 0, which is a global minimum and hence i is a c i ical e ex [4]. On he o he hand, in o de o comple e he de - ini ion o on he whole K1, we shall need o de- ine on D1. To his end, we de ine he deg ee o a e ex co esponding o a iangle as he num- be o edges which a e in his iangle such ha we ha e no assigned hem any alue. Since Sis a su ace, he deg ee o any e ex is a numbe be- ween 0 and 3. Now we s a he de ini ion o on e ices o deg ee 1 assigning hem he g ea es alue o on T1plus one uni . This means ha we ha e assigned ha alue o he iangles o K1 co esponding o such e ices o deg ee one. Nex , we assign he same alue o he ee (no alues as- signed) edges o such iangles. Then, we e-w i e he deg ees o he e ices o D1because hey could ha e changed and con inue assigning alues o e - ices o deg ee one by inc easing in one uni un il inishing wi h all e ices o D1. See he ollowing igu e as an example. 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 Fig. 3. on K1. Now, o ex end o he subcomplex K2, we e- pea he inc easing p ocedu e o assign alues o on T2−T1and on D2−D1. Since his p ocedu e is well de ined o any n∈N, i gi es us a unc ion de ined on whole su ace S. I ’s impo an o poin ou ha in his p ocess may appea si ua ions in which he e a e no deg ee one e ices in a Dn. I implies ha he co esponding edges a e in a cycle o non assigned edges. 3. S udy o The ollowing esul s a es ha he abo e de- ined unc ion is a disc e e Mo se unc ion in Ma ch 25-26, 2004 Se ille (Spain) Fo man’s sense and gi e us in o ma ion abou i s c i ical elemen s. P oposi ion. The unc ion gi en by he p ece- ding p ocedu e is a disc e e Mo se unc ion de ined on he in ini e su ace Sand has a unique c i ical e ex, he ini ial e ex 0, as many c i ical edges as many independen cycles o non assigned edges a e ound and do no ha e any c i ical iangle. Re e ences [1] R. Ayala, L.M. Fe n´andez and J.A. Vilches, Desigualdades de Mo se gene alizadas sob e g a os, Ac as de las III jo nadas de Ma em´a ica Disc e a y Algo ´ı mica (Uni e sidad de Se illa, Spain, 2002) 159– 164. [2] R. Fo man, Mo se Theo y o cell complexes, Ad . in Ma h. 134 (1998) 90-145. [3] T. Lewine , G. Ta a es and H. Lopes. Op imal Mo se-Fo man unc ions o combina o ial 2-mani olds, o appea in Compu a ional Geome y: Theo y and Applica ions (2003). [4] J.A. Vilches, Funciones de Mo se disc e as sob e complejos in ini os, book (Edici´on Digi al @ es, Se illa, 2003).