Ald Mathmatcal Scncs, Vol. 6,, no. 83, 437-446 Foldng of Rgular CW-Comlxs E. M. El-Kholy and S N. Daoud,3. Dartmnt of Mathmatcs, Faculty of Scnc Tanta Unvrsty,Tanta,Egyt. Dartmnt of Mathmatcs, Faculty of Scnc, El-Mnufya Unvrsty Shbn El-Kom, Egyt 3. Dartmnt of Ald Mathmatcs, Faculty of Ald Scnc Tabah Unvrsty, Al-Madnah, Saud Araba sa_na_daoud@yahoo.com Abstract In ths ar w obtand som rlatons btwn th rgular CW-comlx K and ts mag such as homologcal grous, Btt numbrs, Eulr charactrstc and th gnus undr a cllular foldng, a nat cllular foldng, th comoston of two cllular foldngs and fnally th comoston of a cllular foldng and a nat cllular foldng.. Introducton A cllular foldng s a foldng dfnd on rgular CW-comlxs frst dfnd by E. El-Kholy and. Al-Khurasan, [], and varous rorts of ths ty of foldng ar also studd by thm. By a cllular foldng of rgular CWcomlxs, t s mant a cllular ma f : K L whch mas -clls of K to - clls of L and such that f, for ach -cll, s a homomorhsm onto ts mag. Th st of rgular CW-comlxs togthr wth cllular foldngs form a catgory dnotd by C ( K, L. If f C ( K, L ; thn x K s sad to b a sngularty of f ff f s not a local homomorhsm at x. Th st of all sngularts of f s dnotd by f. Ths st corrsonds to th folds of th ma. It s notcd that for a cllular foldng f, th st f of sngularts of f s a ror subst of th unon of clls of dmnsons n. Thus whn w consdr any f C ( K, L,
438 E. M. El-Kholy and S N. Daoud whr K and L ar connctd rgular CW-comlxs of dmnson, th st f wll conssts of -clls, and -clls, ach -cll (vrtx has an vn valncy, [, 3,6]. Of cours, f nd not b connctd. Now a nat cllular n n foldng f : K L s a cllular foldng such that L L conssts of a sngl n- cll, Int L, [4,5]. Th st of rgular CW-comlxs togthr wth nat cllular foldng from a catgory whch s dnotd by N C ( K, L. Ths catgory s a subcatgory of th catgory C ( K, L. Agan f f th st of sngularts of f and ths st corrsonds to th folds of th ma. From now w man by a comlx a rgular CW-comlx.. Man rsults Th followng thorm gvs som rlatons btwn a comlx and ts mag undr a cllular foldng. Thorm.: Lt K and L comlxs of th sam dmnsons n. If ϕ C ( K, L such that ϕ ( K = L K. Thn ( ( K ( L Krϕ, ( ( K = ( L + r k (Kr ϕ, ( χ ( K = χ ( L + ( r k (Kr ϕ, = g ( K = g ( L ( r k ( Kr ϕ (v and = ( = ( ( ( Kr = g K g L r k ϕ comlxs, whr ϕ : ( K ( L s th nducd homomorhsm. n cas of orntabl comlxs,, n cas of non-orntabl Proof: ( Consdr th nducd homomorhsm ϕ : ( K ( L, thr s a short xact squnc ϕ K ϕ Kr ϕ ( Im,
Foldng of rgular CW-comlxs 439 whr s th nducd homomorhsm by th ncluson. Snc ϕ s surjctv w hav Im ϕ ( L and ϕ ([ ] z = ϕ ([ z ] for all z n ( K, hnc th abov squnc wll b tak th form ϕ Kr ϕ ( K ( L. Ths squnc can b slt by th homomorhsm h: ( L ( K such that h( a = ϕ ( a for all a ( L and hnc w hav th rsult. ( From ( w hav ( K ( L Krϕ. Thus rk( ( K = rk( ( L Kr ϕ = rk( ( L + rk(kr ϕ. Thrfor ( K = ( L + rk (Kr ϕ ( Snc ( K = ( L + rk (Kr ϕ for =,,,..., n. Thn ( K = ( L + rk (Kr ϕ ( K = ( L + r k (Kr ϕ M M M n ( K = n ( L + r k (Kr ϕn Thus w hav + + K = + + L + rk φ = χ ( K = χ( L + ( rk (Kr ϕ. = (... ( ( (... ( ( ( (Kr Thrfor (v Snc g = ( x n cas of orntabl comlxs. Thn χ = g. Thus g ( K = g ( L + ( r k (Kr ϕ and hnc = g ( K = g ( L ( r k (Kr ϕ. = In cas of non orantabl comlxs, g = χ.., χ = g Thus g ( x = g ( L + ( r k ( Kr ϕ = Thn g ( K = g ( L ( r k (Kr ϕ. Examl. = (a Consdr th cllular foldng f of a comlx K such that K s a shr nto tslf wth cllular subdvson shown n Fg. (. Th mag of ths ma s a comlx L such that L s a dsc.
44 E. M. El-Kholy and S N. Daoud f K : Z : O : Z Fg. ( Z O O L Kr f Kr f Kr f It s asy to s that th condtons of thorm ( ar satsfd. (b Consdr th cllular foldng g of a comlx K such that K s a tours nto tslf wth cllular subdvson shown n Fg. (. Th mag f ths ma s a comlx L such that L s a cylndr. 3 4 3 K 7 6 g 4 L 6 5 Z : Z : Z Z : Z Kr g Fg.( A gan th condtons of thorm (. ar satsfd. Th followng thorm gvs som rlatons btwn a comlx and ts mag undr a nat cllular foldng.
Foldng of rgular CW-comlxs 44 ( ( Thorm.3: Lt K, L b comlxs of th sam dmnson n. If ϕ NC( K, L such that ϕ ( K = L K. Thn K ( Kr ϕ, ( K = rk (Kr ϕ ( χ ( K = + ( rk (Kr ϕ = (v g K = rk ϕ = and ( ( ( K r =, ncas of orntabl comlxs, g ( K = ( rk (Kr ϕ, ncas of non-orntabl comlxs, whr ϕ : ( K ( L s th nducd homomorhsm. Proof: ( Consdr th nducd homomorhsm ϕ : ( K ( L, thr s a short xact squnc ϕ Kr ϕ ( K Imϕ whr s th nducd homomorhsm by th ncluson. Snc ϕ s surjctv, w hav Im ϕ ( L, but ( L for a nat cllular foldng, hnc th abov squnc wll tak th form Kr ϕ ( K and th roof of ( wll follow thn from th xactnss of th squnc. ( From ( w hav ( K Kr ϕ, Thus rk ( ( K = rk (Kr ϕ, and hnc ( K = rk (Kr ϕ ( Snc ( K = rk (Kr ϕ, =,,..., n Thn ( K = rk (Kr ϕ ( K = rk (Kr ϕ M n ( K = rk (Kr ϕ n
44 E. M. El-Kholy and S N. Daoud Thus ( +... + ( ( K = ( K + ( rk ( Kr ϕ, and hnc χ ( K = + ( rk (K r ϕ =. = (v g = ( χ for orntabl comlxs.., χ = g. Thus w hav g ( K = + ( rk (Kr ϕ = Thn g ( K = ( rk (Kr ϕ, and = g = χ for non-orntabl comlxs..., x = g Thus w hav g ( K = ( rk ( Kr ϕ. = Examl.4 (a Lt K b a comlx such that K s th rojctv lan and th ma f : K L s a nat cllular foldng such that f ( K homomorhc to a dsk. a a K Fg. (3 It s asy to chck that th condtons of thorm ( ar satsfd. (b Consdr a comlx K such that K = T # T, th doubl torus, wth cllular subdvson consstng of tn -clls, twnty -clls and ght -clls. S Fg. (4. Lt f : K K b a nat cllular foldng dfnd by: o o o o o o o f ( 6,..., = ( 4, 5, 3,, and th omttd -clls wll b mad nto tslf. f (, j, k, l, m = (,, 3, 4, 5, whr =,, 3, 4, j =, 9, 5, 6, k = 3, 8, 7, 8, l = 4, 7, 9,, m = 5, 6,, and f ( =, =,..., 8.
Foldng of rgular CW-comlxs 443 5 9 6 7 6 6 5 8 6 7 9 7 8 9 5 f 4 7 K f ( K = L Fg. (4 Th mag f ( K = L, whr L s a dsc wth cllular subdvson consstng of fv -clls, fv -clls and a sngl -cll. Agan th condtons of thorm (.3 ar satsfd. Th followng thorm gvs som rlatons btwn a comlx and ts mag undr th comoston of two cllular foldngs. Thorm.5: Lt K, L and M b comlxs of th sam dmnson n. If f C ( K, L such that f ( K = L K and g C ( L, M such that g ( L = M L. Thn ( g o f C ( M, L and th followng condtons ar satsfd: ( ( K ( M Kr( g o f ( ( K = ( M + rk (Kr( g o f ( χ ( K = χ ( M + ( rk ( Kr ( g o f = g ( K = g ( M ( rk (Kr( g f (v o n cas of orntabl = comlxs and, g ( K = g ( M ( rk (Kr( g o f, n cas of nonorntabl comlxs; = whr ( g of = g o f : ( K ( M s th nducd homomorhsm.
444 E. M. El-Kholy and S N. Daoud Examl.6: Consdr th cllular foldng f of a comlx K such that K s a tours nto tslf wth cllular subdvson shown n Fg. (5. Th mag of ths ma s a comlx L such that L s a cylndr and nto tslf. S Fg. (5. 7 7 9 5 6 7 K 6 4 6 6 f 7 6 L 6 4 g 7 6 6 M 4 Fg. (5 It s asy to chck that th condtons of thorm (.5 ar satsfd. Th followng thorm gvs som rlatons btwn a comlx and ts mag undr comoston of a cllular foldng and a nat cllular foldng. ( ( Thorm.7: Lt K, L and M b comlxs of th sam dmnson n. If f C( K, L such that f ( K = L K, g NC( L, M such that g ( L = M L. Thn g o f N C ( K, M and th followng condtons ar satsfd: K g f ( Kr( o, K = r k g o f ( (Kr( ( x ( K = + ( rk (Kr( g o f = (v g ( K = ( rk (Kr( g o f n cas of orntabl comlxs, and whr = ( = ( (Kr ( = = g K rk g f o n cas of non-orntabl comlxs, ( g of g o f : ( K ( M s th nduc homomorhsm.
Foldng of rgular CW-comlxs 445 Examl.8 : Consdr th comlxs K, L and M such that K, L and M ar tours, cylndr and dsc rsctvly wth cllular subdvson shown n Fg. (6 and lt f C ( K, L, g N C ( L, M 7 3 5 3 4 6 4 f 3 4 6 g K L M Fg. (6 6 It s asy to chck th condtons of thorm (.7 ar satsfd. Rfrncs [] E. M. El-Kholy and Al-Kurasan,. A., Foldng of CW-comlxs, J. Inst. Math & Com. Sc. (Math. Sr., Inda Vol. 4 No.,. 4-48, (99. []. R. Farran, E. El. Kholy and S. A. Robrtson, Foldng a surfac to a olygon, Gomtra Ddcata V. 33,. 55-66, (996. [3] S. A. Robrtson and E. El-Kholy, Toologcal Foldng, commum, Fac. Sc. Unv. Ank. Srs A V. 35,. -7, (986. [4] E. M El-Kholy. and Shahn, R. M., Cllular foldng, J. Inst. Math. & Com. Sc. (Math. Sr. Vol., No. 3,. 77-8, Inda (998.
446 E. M. El-Kholy and S N. Daoud [5] E. M El-Kholy., S.R Lashn and S.N Daoud: Equ- Gauss curvatur foldng, Proc. Indan Acad. Sc.(Math. Sc. Vol.7, No3,.93-3, Inda August (7. [6] A.atchr, Algbrac Toology, Cambrdg Unvrsty rss, London (. Rcvd: March,