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Discrete Morse theory for cellular resolutions

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(21)  beˆSˆƒj^be_qtvd'fit‡d o xSj € t‡d'tv€.b{` R‰ fe_rbeh'jlh z¡_qj^j _aj^k m `‡] o t m d m z  ~9w bed8‚ƒj w^be`‡wl]'`cb o j^h z¡_ m €b{d %œfgtvej^d   ‰ fg_qbeh'j^hiz¡_aj^j9_qjlk m `v] o t m d m z }  j9kax m s o x'b o t‡diw m d'kqj p]'j^d'wlj z m _Œb.wlj^`‡`v]'`cb{A _  R‰ fe_rbeh'jlhµz¡_aj^jœ_aj^k m `‡] o t m d m z§b  R‰ fe_rbeh'jlh € m h']S`‡j „ kq]'ˆSˆ m _ o jlh ‚ %b R‰ fg_rb{h'j^h   ‰ w m € ˆ'`‡/j !  o x'j 4 j oao t¬‰ dp]'€œ‚ j^_ak m z  w bed‚ j w^be`‡wl]'`cb o j^h > z¡_ m € o x'j x m € m ` m 5f % m z o xSj oqm ˆ m ` m fet‡w b{`ƒkaˆybew^j } 

(22) j `‡tvk o8n d m sud j !Sbe€ ˆ'`vj^k m ziw^jl`‡`‡]S`cbe_ Š‰ fg_qbeh'jlhz¡_qjlj

(23) _aj^k m `‡] o t m dSkµtvd'w^`v]'hD‰ tvd'f o xSj W 8b %•` m _P_qjlk m `v] o t m d | ˆS_qj^kaj^d o jlh

(24) >‚ % W 8b %•` m _Œt‡d &´ # ~ o x'j Q w^be_az§_aj^k m `‡] o t m d | 4 8b %gj^_ “†#jljl{b b{d'h Q•o ]'_a€ z¡jl`‡k ( ~ o x'j :A]'`‡`ƒ_qjlk m `v] o t m d | 4 8b %ej^_Abed'h Q•o ]'_q€5z¡j^`vk )' ~5b{d'h _aj^k m `‡] o t m d'kœ•t‡b_ mDmeo j^h w m € ˆ'`vj !Dj^k | ZAm •t n 8* ~ kax m s o xyb o wlj^_ o bet‡d n d m sudbe`vfgj^‚S_rbetvw R‰ fg_rbehSj^h z¡_qj^j _qjlk m `v] o t m d'k `‡t n j ³ %•]'‚ƒjl\^d't =n < k _qj^k m `‡] o t m d | kqj^j +0 ~Œb{d'h o x'j 4 b{_A_aj^k m `‡] o t m dµbe_ajIw^jl`‡`v]'`cbe_ub{d'hKˆ'_aj^kaj^d o d'jCsw^j^`v`‡]'`‡be_ Š‰ fg_qbeh'j^h z¡_aj^jI_aj^k m `‡] o t m dSkŒdyb{€ jl` % o xSjœ³ 9„8‰9b{d'h³ 9„ ? ‰ _aj^k m `‡] o t m dSk bedSh o x'j >x %Dˆ j^_akqtv€5‰ ˆ'`vt‡wltcbe` _qjlk m `v] o t m dHz m _9ˆ m s2j^_a k  m z o xSj € b !Dt‡€ be`ƒt‡h'j^be`    } X `vk m gs2jAkqx m s o xyb o p] meo tvj^d o k  m z0w m ‰ X _ o tvd'tcbedI€ m d m € t‡be`•€ m h']' `vj^kw b{d ‚ j _qj^k m `vej^h8]'kqtvd'f b _aj^`‡b o t gj ej^_qkat m d m z o x'j W +b %•` m _9_qjlk m `v] o t m d0} O d  xybeˆ o j^_ #Ps ju_ajl•t‡jCs h'tvkqwl_qj o jŒ„ m _qkaj o x'j m _ %œbed'h5j !DˆybedSh t oorm   ‰ fg_qbeh'j^h   ‰ w m € ˆ'`‡/j !Dj^k t‡d m _ah'j^_ oqm m ‚ o b{t‡db h't‡kaw^_qj o j „ m _akqj o xSj m _ % z m _Pwlj^`‡`v]'`cb{_P_qjlk m ‰ `v] o t m d'k^3 }  j8z m `‡` m s o xSj beˆ'ˆS_ m bew+x m z  xybe_atŒ]Skqt‡dSf bew %Dwl`‡tvw8€ b o w+x't‡d'fek m d o x'j h'tv_qjlw o jlh w^jl`‡`§fg_rb{ˆ'x m z o x'j   ‰ w m € ˆ'`‡/j ! } O d m _qh'jl_ oqm bew+x'tvjlgj o x'j j !DˆybedD‰ kat m d oqm Š‰ fg_rb{h'j^h   ‰ w m € ˆ'`‡/j !Sjlks2jAfetvgj bedij !Dˆ'`vt‡w^t o h'jlkqwl_qt‡ˆ o t m d m z o x'jA„ m _qkqj w m € ˆS`‡j !#} O d 8# ym _q€ bedib{`‡k m ˆS_qj^kaj^d o k2bPh't 6jl_qj^d o tcbe`Dw m € ˆ'`‡/j ! fgjld'j^_qb o jlh z¡_aj^j^` %œ>‚ % o x'jIwl_qt o t‡w^be` w^jl`‡`‡k m z  sux't‡w+xxSjIkqx m suk orm ‚ jIw+x'bet‡dµx m € meorm ˆSt‡w orm.o x'jIwlj^`v`‡]'`‡be_ w+xyb{t‡d w m € ˆ'`‡/j ! m z o x'j   ‰ w m € ˆS`‡j ! } 

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(26) kax m s o xyb o bek j !Dˆ j^w o j^h o x't‡k h't 6jl_qjld o tcbe` w m € ˆ'`‡/j !bew o ]yb{`‡` % w m t‡d'wlt‡h'jlkHsut o x o x'j w^j^`v`‡]'`‡be_Hw+xyb{t‡dw m € ˆ'`vj ! m z o x'j8„ m _akqj w m € ˆ'`vj !  o x'tvkIs +b %bew+x't‡jC•t‡d'f j !Dˆ'`vt‡wlt o z m _q€œ]'`‡bek z m _ o x'j € beˆ'k tvd w^jl`‡`v]'`cbe_   ‰ fg_qbeh'jlh z¡_qjlj._qjlk m `v] o t m d'k h'j^_atvej^h •tcbKhSt‡kqwl_qj o j „ m _qkaj o x'j m _ %z m _ wlj^`‡`v]'`cb{_ _qjlk m `v] o t m d'kl}  xybeˆ o j^_ ( w m dSkqt‡k o k m zbeˆ'ˆ'`vt‡w^b o t m d'k  e} 

(27) j t‡d o _ m hS]'w^j b5d'jls w^`‡bekqk m z#€ m d m € tcbe` € m h']'`vj^k w^be`‡`vj^h kqxSj^`‡`‡be‚'`vjP€ m d m ‰ € tcb{`'€ m h']'`vj^kl} W x'jlkqjPtvd'w^`v]'h'jAbe`‡`yk o be‚S`‡jŒbed'h.k p]ybe_qjCz¡_qjljPk o be‚'`‡ju€ m d m € tcb{`. .L. !$. . ). J. ! $ . ! $ ,. J ! $ ,. ! $. . . J. ! " #" $. ! $. '. %. G. +.

(28) ´•}. € m 'h ]'`‡jlkVbeks jl`‡`“bek§be`v` Q•o bed'`‡j%­‰ U j^t‡kad'j^_Vtvh'j be`vkw m _a_qjlkqˆ m d'h'tvd'f oqm b kat‡€ ˆ'`‡t¬‰ wltcbe`“w m € ˆS`‡j! sut o x.kax'j^`v`cbe‚S`‡j X `‡j/!Sbed'h'jl_ h']ybe` } 

(29) j ˆ'_aj^kqjld o bed beˆ'ˆS`‡t‡w^b o t m d m zƒhSt‡kqwl_qj o jŒ„ m _akqj o x'j m _ %Iz m _w^jl`‡`v]'`cbe_#_qjlk m `v] o t m d'k orm[o x'ju³9„ ? ‰ _aj^k m `‡] o t m d m zSkax'j^`v`cbe‚'`vjw m ‰ X _ o t‡dStcbed € m d m € tcbe`™€ m h']S`‡j^k  o xyb o %•tvj^`‡hSk#bu€ t‡d'tv€.be`™w^jl` ‰ `v]'`cb{_   ‰ fg_rb{h'j^hz¡_qj^j _qjlk m `v] o t m d m z  } †be_qbe`‡`vj^` oqmKm ]'_[s m _ n

(30) QDnƒm Ž `vh'‚ j^_qf x'bek2w m d'k o _q]Sw o jlh >‚ %.b{`‡fgjl‚'_rbetvwu€ j^bed'k§€ tvd't‡€ be`Dz¡_aj^jA_qjlk m `v] o t m d'kVz m _ b w^`‡bekqk m ztvh'j be`vk o x'b o t‡d'wl`‡]'h'jlk o x'j kax'j^`v`cbe‚'`vj m d'jlk^} 

(31) j ˆ'_ m •t‡h'j bµfej^d'jl_rbe`V€ j o x m h orm beˆ'ˆ'`%h't‡kaw^_aj o jH„ m _akqj o x'j m _ %z m _ w^jl` ‰ `v]'`cb{_[_qj^k m `‡] o t m d'k oqmKo xSj W +b %•` m _ _aj^k m `‡] o t m d6} W x'jH_qjlkq]'` o tvd'fK_aj^k m `‡] o t m dt‡k w^be`‡`vj^h ³ %•]'‚ j^\^dSt n _aj^k m `‡] o t m d | kqt‡dSw^j5t o w^bed ‚ƒj5_qjlf­be_ah'j^h

(32) bek b fgj^dSj^_rb{`‡t‡\^b™‰ o t m d m z0³ %D]S‚ƒjl\^d't n < k _aj^k m `‡] o t m dy~C } 

(33) jAkqx m s o xyb o z m _Vfgj^dSj^_qtvwAbed'h.kax'j^`v`cbe‚S`‡j w m ‰ X _ o t‡d't‡bedK€ m d m € tcb{` € m h']'`‡jlk o x'j^_ajij !Dt‡k o € tvd't‡€ be`:³ %•]'‚ƒjl\^d't n _aj^k m `‡]D‰ o t m dSk o x't‡k2s 8b % ˆ'_aj^kaj^d o tvd'f bœkqj^w m d'h ˆ'_ mDm z:z m _ o x'j[/j !Stvk o j^d'wlj m zbœwlj^`v`‡]'`‡be_ € t‡dSt‡€ be`§z¡_aj^j R‰ fg_rb{h'j^h _qjlk m `v] o t m d m zPkax'j^`v`cbe‚'`vj w m ‰ X _ o t‡dStcbed€ m d m € tcb{` € m h']'`‡jlk^} “m _Pfgj^dSj^_qtvw5€ m d m € t‡be`:t‡h'j^be`‡k o x'j5fgt gj^d _aj^k m `‡] o t m dtvk o x'ji_aj^kR‰ m `‡] o t m d h'j o jl_q€ t‡d'jlhK>‚ % o x'j Q w b{_azc‰ w m € ˆ'`‡/j !8bekAkqx m sud >‚ %.4 8b %gjl_ ƒ†#j^jC{ b Qpo ]'_a€ z¡jl`‡k | kqjlj ( ~C} “m _ k o be‚'`vj.b{d'h k p]yb{_qjlz¡_aj^jª‰ k o be‚'`vjH€ m d m € tcbe`tvh'j b{`‡k o x'ju€ tvd't‡€ be`•_qjlk m `v] o t m d x'beh ‚ j^jld.h'j o jl_q€ t‡d'jlh >‚ % T `‡t‡bex m  ] [j^_R{bet‡_ajA_ajC‰ o X m m kaˆƒjlw tvgjl` % _qbe€ {b 2:Œjl_q\  f :Œtv‚'t6b{d'h †#jljl{bœ>‚ %Hb{`‡fgjl‚'_rbetvwu€ j^bed'k | kqj^j ´ bed'h ´ ~ª} 

(34) j§fgtvej bŒbeˆ'ˆ'`vt‡w b o t m d m zyh't‡kaw^_aj o j „ m _akqj o x'j m _ % z m _:w^jl`‡`‡]S`cbe_ _aj^k m `‡] o t m d'k orm o x'jP>x %•ˆƒjl_qkat‡€ ˆ'`‡tvw^tcb{`y_qj^k m `‡] o t m d z m _2ˆ m s2j^_qk  m z o x'jP€ b !Dt‡€ be`ytvh'j be`    sux'tvw+. x %•t‡jl`‡h'kŒb wlj^`v`‡]'`‡be_ € t‡d'tv€.be` R‰ fg_rb{h'j^h z¡_qjlj _aj^k m `‡] o t m d  m z  “‹Akqtvd'f o x'j z m _q€I]'`cbIz m _ o x'j h't ƒj^_aj^d o t‡be` m z o x'jI„ m _akqj w m € ˆ'`vj ! 's j kax m s o xyb oio xSt‡ki€ t‡d'tv€.be`V_aj^k m `‡] o t m dtvkœt‡k m € m _qˆ'x'tvw orm o x'j m d'j fgt gjld ‚>% T `vtcbex m ] bed'h [j^_a{betv_qj | kqj^j ´ ~C} 

(35) jœkax m s o xyb oPo x'jœw^jl`‡`v]'`cbe_A_qjlk m `v] o t m dµ•t‡b._ m•meo jlhw m € ˆ'`vj!Dj^kPw^bed‚ j m ‚D‰ o bet‡dSj^h z¡_ m € o xSj W +b %•` m _2_aj^k m `‡] o t m d5•tcb h't‡kaw^_aj o j[„ m _qkaj o x'j m _ %5z m _2wlj^`v`‡]'`‡be_ _aj^k m `‡] o t m dSk^} ym _ ˆD‰ 4 m _qjl`2!Sjlh5t‡h'j^be`‡k fgj^d'jl_rb o j^hi‚>%Ib € m d m € tcbe`  | fej^d'jl_rb o j^hit‡d o x'j  kaj^d'kaj o xyb o Ptvk o x'jVka€.be`v`‡jlk o ˆD‰ 4 m _aj^7`  !Dj^h t‡hSj be`gka]'w+x o xyb o  S~6s2jVkqx m s. F! $. #D}. ! $. !$. #. /. . +! "  #" $ . (S}. '•}. '. . ! $. '. 0. &". " ='.

(36) o xyb o bedb{ˆ'ˆ'`‡tvw b o t m d m zIhSt‡kqwl_qj o j„ m _akqj o xSj m _ % z m _Hwlj^`‡`v]'`cb{_._qjlk m `v] o t m d'k %•tvj^`‡hSkub5wlj^`v`‡]'`‡be_ € t‡d'tv€.be`=Š‰ fg_qbeh'jlh z¡_aj^j _qjlk m `v] o t m d m z } 0D} 

(37) j kax m s o xyb o z m _b{dib d'j kaj^€ t‡fg_ m ]'ˆ sut o x o x'j2ˆ'_ m ˆƒjl_ o % o xyb o be`‡` m _ah'j^_ w m € ˆ'`‡/j !Sjlk |  ƒ~   m gj^_ m ˆ j^d t‡d o jl_a{be`‡`vk |   ~ be_qjVkax'j^`v`cbe‚'`vj o xSj^_qj /j !Dt‡k o kœb8w^j^`v`‡]'`‡be_[€ t‡d'tv€.be`   ‰ fg_qbeh'j^hz¡_qjlj._qjlk m `v] o t m d m z o xSj yjl`‡h  m gj^_ o x'j b d'j.kaj^€ t‡fg_ m ]'ˆ_qt‡dSf   } W x'tvk tvkIbew+xSt‡jlej^h ‚>%

(38) bedb{ˆ'ˆ'`‡tvw b o t m d m z hSt‡kqwl_qj o j[„ m _qkaj o xSj m _ % z m _2wlj^`‡`v]'`cb{__qj^k m `‡] o t m d'k oqm o x'j 4 b{_R‰ _aj^k m `‡] o t m d m z

(39)  m ej^_   T gjld o x m ]Sfgxµs2jœxyb gjIkq]'‚'ka]'€ j^h € m k o w b{kqj^k m z€ m d m € tcbe`ƒt‡h'j^be`‡kutvdµsux'tvw+x o x'ju€ tvd't‡€ be`­z¡_qjljA_aj^k m `‡] o t m dœs bek n d m sud.‚ jlz m _ajubed'h w m d'k o _a]'w o j^h.dSjls m dSj^k pk o t‡`v` o x'j^_ajKbe_aj _qj^ka]'` o k5d meo w m gjl_qjlh x'j^_aje} ZŒmeo be‚'` % t‡d 8 ZŒm •t n Q•o ]'_a€ z¡jl`‡k5bedSh † m k o d't n{m s fetvgj b8‚ƒj^be] o t z¡]'`fgj m € j o _at‡wiw m d'k o _q]'w o t m d m z ˆ m ` %•x'j^h'_qbe`w m € ˆ'`vj !Dj^k € t‡d'tv€.b{`‡` %._qjlk m ` Dtvd'f €.b o _ m t‡hbedSh m _qt‡jld o j^h € b o _ m t‡hKt‡h'j^be`‡kl} 

(40) j ‚ j^`vt‡jCgj o xSj^_qj tvk[k o tv`‡``‡be_qfejiˆ meo j^d o t‡be`z m _ w m €œ‚'tvdyb orm _qtcb{`bedSh

(41) fgj m € j o _at‡wiw m d'k o _q]'w o t m d'k t‡d o x'j o x'j m _ % m z#w^jl`‡`‡]S`cbe_ z¡_aj^j[_aj^k m `‡] o t m d'kl}. '. 0. 0 ! $ . 4  , =. . ! $. 1. ! $. 8.

(42) O s bed o orm9o xyb{d n € % behDDtvk m _ m ` n €.b{_  jl` n j^_:z m _ j!Dw^jl`‡`vj^d o s m _ n t‡d'fAkq]'ˆSˆ m _ o  :Aj[ˆ'_ m •t‡h'jlh o x'j m _qt‡fet‡dybe`“tvh'j bIz m _ o x'tvk o x'jlkqtvkubed'hHs bek m ˆƒjldHz m _ h'tvkqwl]'kqkat m d'kub{` ‰ € m k o bed>%I€ t‡dp] o je} W x't‡k s b8% O s bekbe‚S`‡j oqm ‚ j^d'j o z¡_ m €x'tvkfg_aj b o €.b o x'j^€ b o t‡w b{` n d m su`‡jlh'fgj bed'h8t‡d o ]'t o t m dHtvd8b5ˆS`‡j b{kq]'_qbe‚'`‡j b{d'hµ_aj^`‡b !Dj^hµb o € m kqˆSx'j^_aje}. *.

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