KHOA HOC & PHAT TRIEN
^hH-l cf
RdnghepnhotoviTavanca
(GIOI THIEU TOPO HOC)
Nguyen H a u Viet H a n g
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o nhirng vin de cua hinh hoc, nhung lai khong phu thuoc vdo kich ca to nho, rang hep, ddi ngan cua cde dot tuong hen quan. Nhttng van de nhu the thudc ve mdt ITnh vue duge ggi la Topo hoc (Topology), Trong nhiing van de thudc loai nay, chuyen mot manh dat rdng hay hep, vuong hay meo chang quan trgng gi. (The cd la khdng!) Vi the, nhiing ngudi budn dat, budn bat ddng san chd nen hgc Tdpd. NSu nhu vi td mo ma hg cii hge, hg the nao cung ca quyet rang cae nha Tdpd hgc la nhung ke dien, ham hap.Cao dam khoat luan nhu the khdng kheo de dan den tu bien, mu md, roi de sinh ra ndi nham.
De tranh chuyen do, ta hay bat dau bang mdt vai vi du. Ddi khi, vai vi du thuc chat cd the de ra mgt ly thuyet, cd khi cdn de ra ca mdt nganh hgc.
Leonhard Euler (1707 - 1783), nha toan hoc vi dai ngudi Thuy ST, duoc xem la cha de cua nganh Tdpo hgc, vi dng la ngudi dau tien nghien ciiu hai bai toan sau day
Bai toan ve 7 chig'c cau
Kdnigsberg la mot thanh phd cd thude Vuong qudc Phd va nude Diic eho den 1945. Sau Dai chiSn The gidi II, nd thudc Lien Xd (cu) rdi Nga, va duge ggi la Kaliningrad. Chi cd rat it dau tich cua Kdnigsberg cdn sot lai ngay nay d Kalin- ingrad
Ci thanh pho Kdnigsberg, cd 7 ehiee cau.
Chiing ndi hoac la hai bd sdng, hoac mdt bd sdng va mdt trong hai cii lao, hoac ndi hai cii lao do.
(Xem ban dd duoi day).
Tir xua, cu dan d Kdnigsberg da dat eau hoi:
Lieu cd the di mdt lan qua tat ca 7 chiec cau ma khdng ed cau nao phai lap lai hay khdng?
Khdng can de tam nhieu lam den vi tri cu the 32 TiaSong
cua 7 chiee can. Dieu quan trgng nhat raa ngudi ta quan sat duge tir bai toan nay la nhu sau: Ddy Id mot vdn di cua hinh hoc, nhung khong phu thuoc vdo do lan cua cde yeu to tham du (ddng sdng rdng hay hep;
nhung chiec cau dai hay ngan, to hay be; cac cii lao Idn nhd the nao). Van
de chi phu thuoc hinh dang va vi tri tuong ddi ciia cae yeu td.
Dua tren nhan xet dd, sa do 7 eai cau d Kbnigsberg dugc ve lai nhu sau:
Khdng CO bang chirng nao edn lai
chiing td rang Euler da tdi Kdnigsberg. Tuy nhien, nam 1735 ong da chung minh rang mong muon tim mdt each di qua ca 7 chiec cau "mdt lan, khong lap lai" la khdng the thuc hien dugc.
Chiing ta thir tim hieu Idi giai cua Euler cho bai toan 7 cay ciu. Tren ban d6 Kdnigsberg, hay thay moi bd sdng, moi eu lao bang mdt diem, ggi la dinh, thay moi ehiee eau bang mgt dudng ndi cac dinh, ggi la canh. Hinh thu dugc ggi la mdt do thi. Bai toan ve 7 cai ciu d Kdnigsberg thuc chit la chuySn cd gSng "ve bang mot net" do thi sau day:
KHOA HOC & PHAT TRIEN
Bai toan nay rat quen thuge vdi tre em qua trd ehai "ve hinh bang mgt net". Cd ai thud thieu thdi lai chSng da timg dau dku vdi cau dd ve cai phong bi ehi bang mgt net?
Hay bat dau vdi nhan xet don gian sau day: Mdi khi ta di qua mdt dinh, thi ed 2 canh (2 eay eau) xuat phat tir dinh dd da duge di qua: Canh di tdi, va canh di ra khdi dinh dd. Nhu the, mdi lan di qua mdt dinh, sd eanh ndi vdi dinh dd ma ta chua di qua giam di 2. Cho nen, nSu mgt dinh cd sd canh ndi tdi la mdt sd ehan (ggi tat la dinh chan) thi mdi lan di tdi dd ta ludn con dudng de thoat ra ngoai. Cdn tai mdi dinh le, ehang han cd (2k+l) dudng ndi vdi dinh do, thi sau k lan di qua, tdi Ian (k+1) ta se het dudng de di khdi dinh do.
Nhw vfty, cac dinh le chinh Id cde cdn trd cho viec "di qua"
md khong phdi diirng lai. Chien thuat eiia ta la khdng xuit phat tir cac dinh ehan (nSu van edn dinh le), vi nSu xuat phat tir mgt dinh chan, khi di khdi dinh dd, ehiing ta se bien dinh chan nay thanh mgt dinh le trong phSn tiep theo cua trd choi.
Ta chi can xet eae do thi lien thdng, nghia la dd thi ma giiia 2 dinh bat ky eiia nd deu ed it nhat S606.
mdt dudng ndi. (Viee ve mpt do thi khdng lien thdng hien nhien qui ve ve tiing thanh phan lien thong eiia nd.) Dua tren nhiing nhan xet ve dinh ehan va dinh le ndi tren, ta ed the chung minh:
Trong mdi do thi, sd eac dinh le luon la mgt sd chan,
Mdt dd thi lien thdng khdng cd dinh le nao, can tdi thieu I net ve.
Mdt do thi lien thdng ed 2n dinh le (n>0), can tdi thieu n net ve.
Cach ve nhu sau: Xuat phat tir mgt dinh le bat ky (neu ed), ve tuy y cho den khi khdng ve duge niia. Khi dd ta gap mgt dinh le khac. Net ve vira rdi khir bdt 2 dinh le (la diem dau va diem eudi eua net ve). Lap lai qua tiinh tren cho den khi khdng edn dirih le nao. Trudng hgp khdng co dinh le nao, hay xuat phat tir mdt dinh ehan bat ky, ve tuy y eho den khi khdng ve duge niia. Khi do, ta gap lai dinh xuat phat.
Chiing tdi khdng di sau vao ehi tiet ehiing minh nhiing khang dinh tren.
Cai phong bi cd 5 dinh, trong dd 4 gdc la 4 dinh le. Vi the, khdng the ve phong bi bang 1 net. can it nhat 4/2 = 2 net dB ve duge phong bi.
Trong bai toan 7 cay can d Kdnigsberg, ed bdn dinh, deu la cac dinh le. Do dd, khdng the ve dd thi dd bang 1 net. Tdi thieu cin 4/2 = 2 net. Do la ly do trong sudt chieu dai lich sir, khdng ngudi dan nao d Kdnigsberg cd the di mdt lan qua tat ca cac cay cau ma khong chiec cau nao bi lap lai.
Bai toan ve so m^t, s6' canh, va so dinh cua mdt da dien
L. Euler da ehiing minh dinh ly sau day, thoat nhin tudng nhu trd choi tre con: Trong bdt cuda
dien loi ndo, so mat Irir di so canh cong vai so dinh deu bdng 2.
Hay lay vai vi du.
Trong mdt tii dien, sd mat m=4, s6 canh c=(5, sd dinh d=4;
Taco m-c+ d= 4-6+ 4=
2.
Trong mdt hmh hop chu nhat, sd mat m=6, sd canh c=12, sd dinhrf=5;
Ta Cling c6m-c+d=6- J2+ 8= 2.
Vi sao lai cd chuyen liie thi lay dau "cpng", Iiie lai lay dau
"trir" trong dinh ly tren? Xin thua: Mat la mdt ySu td 2 chieu, dinh thi 0 chieu, nhirng yeu to chin chieu thi duoc mang dau cpng; cdn canh la mdt yeu td 1 chieu, tic sd chieu le, nen nd mang dau tni.
Gidng nhu bai toan ve 7 chiec cau, bai toan nay ciing lit mdt van de hinh hgc, nhung khdng phii thupc vao dp Idn cac yeu to.
ThSt vay, mdt da dien du be nhu hat dau hay to nhu Trai dat thi sd mat, sd canh, va sd dinh ciia nd ciing khdng thay ddi. Nhan xet www.liasang.com vn T K j S o n g 3 3
KHOA HOG & PHAT TRIEN
tren ggi y cho suy luan sau day:
Hay tudng tugng da dien loi dugc lam bang eao su. Ta hay thdi phdng da dien loi do thanh mgt qua bdng hinh cau.
Bay gid, thay cho mat cau da ndi d tren, hay iky mat xuySn (cai sam dtd) lam thi nghiem. Cd the phan chia cai sam bang 2 dudng (c=2), mdt dudng eat
Cae mat, canh, va dinh cua da dien biSn thanh cac mat (cong), e?nh (eong), va dinh tren mat eiu. Nhu th6, dinh ly tren eua Euler v8 ban chat la mdt dinh ly ve m^t cau: Trong moi cdch phdn mat cdu thdnh cde hinh da gidc cong, so mat trir so canh cong so dinh deu bdng 2. Han niia, mpi hinh thu duge tu mat cau bang mdt phep bien ddi lien tiic (tuong tu nhu eo dan mang cao su) deu nghiem diing dinh ly nay.
Chung ta vira dat dugc mgt bude tien quan trgng trong each nghi: Bai toan ciia Euler ban dau xet rat nhieu ddi tugng, la bat cii da dien loi nao. Rut cupc, nd la mgt bai toan ve ehi mdt doi tugng duy nhat, dd la mat cau.
Dat dugc budc tien dd la do chiing ta sit dung lap luan ve eae bi^n ddi kieu "co dan cao su".
Ngudi ta ggi dd la cac phep bifen ddi tdpd.
theo vet mang-xdn^, dudng kia cat dgc toan bp chieu dai sam.
Hai dudng nay cat nhau tai mgt diem duy nhat {d=l). Bi eat hai dudng dd, sam trd thanh mpt mat hinh chii nhat (m=l). Vay, sd ma Euler quan tam ciia mat xuyen (sam) lam- c + d= 1 - 2 + 1 = 0.
Tiep theo, hay lay mdt mat
"xuyen kep", cd dugc bang each dinh 2 chiec sam dtd vao nhau.
Ban hay tu ehgn mpt each chia mat "xuyen kep" thanh cac mat gidng nhu hinh vuong (cong), cac canh (cong), va cac dinh.
Chang han, ta chpn each ehia mo ta bang hinh ve tren. Cae dudng do va vang cat nhau d 2 diem (1 diem nhin thay, 1 diem la hinh chieu thang diiug cua diem nhin thiy), vay d= 2. Cac dudng dd va vang dugc 2 dinh ay chia lam 4 canh (4 nua dudng trdn), cdng them 2 dudng mau xanh, vay c=
6. Sau khi eat theo cae dudng ay mat xuyen kep bi chia thanh ra 2 hinh ehii nhat, vay m= 2. Ta cd m-c+ d=2 -6+ 2= -2.
Nhu the, sd Euler eua xuyen kep la-2.
Ta ed the dinh nhieu sam dtd
vdi nhau de tao thanh mgt mat xuyen ed g lo. Sd ma Enter quan tam doi vdi mat dd bang m~c + d=-2(g-I).
Trong tdpd hien dai, dinh ly Euler duge tdng quat hoa nhu sau: Neu chia bdt cu mgt vat the n chieu ndo thdnh cde phdn
"giong nhu da dien ", thi tong so cac phdn v&i chieu chdn trie di tong so cde phdn vai chieu le luon Id mot hdng so, du(rc goi la dac so Euler, cua vgt the do.
Nhu the, mdi vat the deu la su tdng hoa nhip nhang ciia hai phan am va duong, chin va le ndi tai ciia nd, khdng th8 thay ddi. Dac sd Euler, cung con dugc gpi la dae sd Euler - Poincare (bdi viPoincare (1854-1912) chinh la ngudi dau tiSn y thitc duge chuyen nay d trudng hgp sd ehien tuy y), eiia mpt vat the ehinh la mpt loai '''ban thi", mpt loai '"chiing minh thu", mpt "ID Card"' cua vat th6 iy.
He qud Id, nSu hai vkt the co dac sd Euler khae nhau, thi ehiing khong the cai nay bien thanh cai kia sau mdt phep bien ddi thuan nghich lien tuc (kieu nhu CO dan cao su). Ngudi ta noi hai vSt thS dd khdng cUng kieu tops.
Nhu the, mat eau, mat xuyen, va mat xuyen kep khdng ciing kieu tdpd, vi chiing cd dac sd Euler khac nhau (tuong ling bang 2, 0, va -2). V6 m^t true giae, vi sao chiing khdng ciing kieu tdpd? Ly do that don gian:
Mat cau khdng ed Id nao; mat xuyen cd 1 Id (la eai chd ngudi ta vin ehui vao de bi6n sam thanh phao boi); cdn mat xuyen kep cd 2 Id. Cac nha tdpd bao mat xuyen cd I Id, nen cd gidng (genus) bang I; mat xuySn kep ed 2 Id, nen cd gidng bang 2;
mat can khdng cd Id nao, nen c6 gidng bang 0. A ra the, phdi co Id thi gidng mM khdng bi triet 3 4 n9^!^!?9 www.liasang com vn
KHOA HOC & PHAT TRIEN
tieu. Cac nha tdpd that gidi dm d.
Riemann cdn chung minh mdt dinh ly that tham thuy: Mgi mat 2 chieu tran (tiic min mang), bi ch|n (cd the giu trong mdt can phdng), va cd hudng (tire la phan biet duge phia nao la ngoai da, phia nao la trong thit) dhu xae dinh dugc ve mat tdpd ehi bing each dem sd Id tren nd. Cha cha, phai mdi Picasso d8n day mdi dugc.
Nhimg chuyen ke tren cd the dan chiing ta den nhiing gi? Sau day la mdt k&t lu|n that khd tin.
Khang dinh: Du co nhdo nan mot ci^c bdt, hinh cdi hdnh mi, ky den muc ndo, mien Id hinh cua cue bot luc ddu vd khi thoi nan vdn Id cdi hdnh mi, nan xong lai de cdi hdnh vdo dung cho cu. khi do luon ludn co mol hat bdt mi khdng thay doi vi tri.
Th§t vay, sau mdt phep bien ddi lien tuc 2 chieu, cue bdt hinh cai banh rai duge bien thanh mdt hinh cSu S. Ggi .S la mat cau bao quanh B. Khang dinh tren dugc chiing minh bang eac bude suy lu3n sau day:
If Khdng ed phep bien doi nao bien B thanh S va van giu nguyen mgi diem tren S. (Y chiing minh:
Gia sii ton tai mdt phep bien ddi nhu the. Trudc phep bien ddi, S la bien eua B, sau phep bien ddi S phai la bien cua chinh S. Dieu nay vd ly.)
2/ Gia sir phan chirng, sau nhao n|n khdng cd hat bgt mi nao giii nguyen vi tri. Gia sir hat bdt mi JC duge bien thanh/^jt) sau nhao nan. Nita dudng thang ndi f(x) vdi X (keo dai) eat mat eau S
tai mgt diem duy nhat, ky hieu g(x). Phep bi8n ddi x thanh g(x) ehinh la mpt phep bien hinh lien tue, bien B thanh S va giu nguyen mgi diem tren S. (Neu x nam tren S, thi nita dudng thang n6if(x) vai x cat S tai ehinh x.) Dieu nay mau thuan vdi diem I). Mau
thuan nay bac bd gia thigt phan chiing.
Hai bai toan dirtfc Euler nghi§n cmi noi tren la nhiing vi du don gian cua cac van de hinh hge, trong do kich co khdng quan trgng, chi co hinh dang va vi tri tuong doi dong vai trd quyet dinh. Nganh toan hpe nghien ciiu nhirng van de nhu vay ngay nay duge ggi la Tdpd hgc (Topology).
Ngam cho ky thi chuyen kich cd khdng quan trgng da duge tao hoa duy tri nhu mgt trong nhiing nguyen ly hang ddu, ddng vai trd '^ddm bdo an ninh" khdng chi eho xa hdi loai ngudi, raa eho toan bd cac gidng loai trong tu nhien. Neu mot ngudi mua nham ddi giay, chat qua hay rpng qua, tiie la ngudi ay gap mdt van de ve kich cd, thi anh ta dem ddi.
The nhung, neu ngudi ay lay vg, va neu nhu anh ta cQng gap rapt van de ve kich ed, rdi ddi ddi, thi nguy hiem vd ciing. Va neu rat nhieu ngudi sau khi lay vg ciing gap van de ve kich ed nhu the, deu can phai ddi, thi xa hdi chac chan sinh loan.
Ba chiia tha Ndm Hd Xuan Huang(1772-1822) chinh la nha Tdpo hge dau tien eiia Viet Nam, ngudi bang true cam tuyet vdi da phat bieu tudng minh nhiing y tudng tao bao ciia tdpd tit han 200 nam trudc. Khdng nghien eiiu bai toan ve 7 cai cau hay bai toan ve sd raat sd canh sd dinh trong da dien, nhung bang mpt
tiep can day man cam, ba da nhan ra chuyen nay tir xua. Ba viet that nhan van:
'''Rang hep nho to viea vdn cd Ngdn ddi khudn khd cUng nhu nhau".
Hai cau tha do trich trong bai
"Det eiri" eiia ba:
"Thap ngon den len thdy trdng phau
Con CO* mdp mdy sudt canh thdu
Hai chdn dap xuong nang ndng nhdc,
Mdt suot ddm ngang thich thich mau.
Rang hep nho to vica van cd Ngdn ddi khudn khd cUng nhu nhau
Cd ndo mudn tot ngdm cho ky Cha den ba thu mai dai mdu. "
Nhu the, Hd Xuan Huong (1772-1822) dpc lap va gin nhu ddng thdi vdi L. Euler (1707- 1783), da phat bieu tudng rainh quan diem co ban ciia tdpd hgc.
Nir sT hg Hd qua la da khdi dSu day sinh khi cho dam hau sinh lam tdpd ciia Viet Nam, trong do ed ke hgc trd viet bai nay:
'^Mdt mat anh hiing khi tdt gid
Che ddu qudn tu luc sa mua".
Theo dugc Hd nii sT qua la kho. May sao,
"Moi gdi chdn chdn vdn mudn treo".
Vay ma lai bao cac nha Tdpd la ham thi nghe th6 quai nao duge, hd gidi.
Vi thanh
Lao C6-nha-dit doc xong bai nay cuoi pha len, ma rang: "Cd manh dat ciing khdng biet no to hay be, vudng hay meo, lai bao rang nhu nhau tudt. Th^ thi ngheo sudt ddi la phai. Bgn tdpo nay xem ra chi lo chuyen gidng thdi. n
vtww.tiasang com v TioSmg 35
