Optimising the topological information of the A8-persistence groups

Op imising he opological in o ma ion o he A∞-pe sis ence g oups F ancisco Belch´ı 1 Abs ac Pe sis en homology ypically s udies he e olu ion o homology g oups Hp(X) (wi h coe icien s in a ield) along a il a ion o opological spaces. A∞-pe sis ence ex ends his heo y by analysing he e olu ion o subspaces such as V:= Ke ∆n|Hp(X)⊆Hp(X), whe e {∆m}m≥1deno es a s uc u e o A∞-coalgeb a on H∗(X). In his pape we illus a e how A∞-pe sis ence can be use ul beyond pe sis en homology by discussing he opological meaning o V, which is he mos basic o m o A∞-pe sis ence g oup. In addi ion, we explo e how o choose A∞-coalgeb as along a il a ion o make he A∞-pe sis ence g oups ca y mo e ai h ul in o ma ion. Con en s In oduc ion 2 1. A∞-s uc u es 4 2. Ba codes 7 3. Topological meaning o A∞-pe sis ence g oups 10 4. Compa ible A∞-s uc u es o a oid zigzag ambigui ies 16 5. Appendix 18 6. Conclusions 24 Acknowledgemen s 25 Re e ences 25 1F ancisco Belch´ı-Guillam´on, Ma hema ical Sciences, Uni e si y o Sou hamp on, Building 54, High ield, Sou hamp on SO17 1BJ, UK. [email p o ec ed] ORCID: 0000-0001-5863-3343 This wo k has been suppo ed by he Spanish MINECO g an s MTM2010-18089 and MTM2013-41762-P, by he Jun a de Andaluc´ıa g an FQM-213 and by he UK’s EPSRC g an Joining he do s: om da a o insigh , EP/N014189/1. Key wo ds: pe sis en homology, zigzag pe sis ence, A∞-pe sis ence, opological da a analysis; A∞- (co)algeb as, Massey p oduc s, kno heo y, a ional homo opy heo y, spec al sequences. 2010 Ma hema ics subjec classi ica ion: 16E45, 18G55, 55S30, 57M25, 57Q45, 18G40, 55P62, 55U99. All igu es ha e been d awn by he au ho using mainly he so wa e Inkscape. 1 a Xi :1706.06019 1 [ma h.AT] 19 Jun 2017 F ancisco Belch´ı In oduc ion Pe sis en homology [5,6] compu es he (pe sis en ) Be i numbe s o a sequence o opo- logical spaces and con inuous maps K:K0//K1//. . . //KN c ea ed by a ying a pa ame e such as ime, hickness, in ensi y, heigh , e c. Depending on he con ex , his can allow us o disco e highly non-linea s uc u e in da a o o compu e no el geome ic desc ip o s o shapes. Fo ins ance, G. Ca lsson e al. conside ed a poin cloud da ase buil om 3 by 3 high-con as pa ches om g ey-scale na u al images and s udied an unknown space Xon which he poin s accumula ed wi h high densi y. They used pe sis en homology o es ima e some Be i numbe s o X, bu wen u he o ind ha he 2-skele on o X o med a Klein bo le [4]. This ex a knowledge was hen used as s a ing poin o de elop a dic iona y o ex u e ep esen a ion [18]. This is a g ea example o how impo an i can be o ind pe sis en opological in o ma ion beyond he le el o Be i numbe s. A∞-pe sis ence [2] aims a doing so in a semi-au oma ed way by s udying pe sis en opological in o ma ion a he le el o A∞-s uc u es – algeb aic cons uc ions ha can encode a ibu es ela ed o cup p oduc and highe o de Massey p oduc s (see Fig. 1). We plan o wo k wi h A∞-coalgeb as on he homology H∗(X) o a space Xand wi h A∞-algeb as on i s cohomology H∗(X). These s uc u es codi y a good deal o opological in- o ma ion o X, as his pape will illus a e. Fo all concep s below ela ed o A∞-s uc u es, see §1. Gi en wo homo opy equi alen spaces, he ans e ed A∞-coalgeb as hey induce coincide up o isomo phism. Hence, ideally, we would like o s udy he pe sis ence o his whole isomo phism class o A∞-s uc u es. Un o una ely, his class is oo la ge o compu e in gene al, so we end up ha ing o sac i ice some o his s uc u e in e u n o compu abil- i y. Mo e conc e ely, a e choosing a ield o coe icien s, we ix a ans e ed A∞-coalgeb a (H∗(Ki),{∆i n}n) o each space Kiin Kand hink o de ining a ec o subspace Vi⊆Hp(Ki) o e e y i= 0, . . . , N, so ha he ollowing wo condi ions hold: (1) Vicon ains enough in o ma ion om he A∞-coalgeb a {∆i n}n o be use ul. (2) Viis simple enough o allow a easible pe sis ence compu a ion. Pe sis en homology encodes in a ba code he e olu ion along Ko he ec o space Hp(Ki). By (2) we mean ha , simila ly, we should be able o encode in a ba code he e olu ion along Ko he subspace Vi⊆Hp(Ki). In [2], we used zigzag pe sis ence [3] o p o e ha Vi:= Ke ∆n|Hp(Ki)sa is ies (2), whe e he e and hence o h |Wdeno es he es ic ion o he map o he subspace W. In his a icle, we do he ollowing. On he one hand, we d aw a mo e comp ehensi e pic u e o Vi’s ade-o be ween simplici y and e ained in o ma ion om he A∞-s uc u e. 2 Op imising he opological in o ma ion o he A∞-pe sis ence g oups Figu e 1. Pe sis ence applied o poin cloud da ase s: (a) Be i numbe s con ibu e aluable opological in o ma ion abou solid objec s, able o dis inguish e.g., a ci cum e ence om a disc. Pe sis en homology adap s hese in a ian s o he s udy o poin cloud da ase s, so ha a sample om a ci cum e ence can be old apa om ha o a disc. (b) A∞-s uc u es p o ide a much mo e de ailed s uc u al desc ip ion han ha o Be i numbe s. They con ain all he in o ma ion o he cup p oduc , and hence can ell apa e.g., a o us om a wedge o sphe es S1∨S2∨S1. On op o ha , hey con ain in o ma ion ela ed o (highe o de ) Massey p oduc s, allowing us o also dis inguish e.g., 3 unlinked ings om he Bo omean ings (see Fig. 3). (c) In his con ex , he aim o A∞-pe sis ence is o use he s eng hs o A∞-(co)algeb as o sha pen he ool o pe sis en homology, e.g., o allow he dis inc ion o mo e in ol ed poin cloud da ase s. To do so, we show bo h wha we lose by no using he whole isomo phism class o A∞- s uc u es, and wha we s ill gain wi h espec o Be i numbe s. Speci ically, we show ha dim Viis no a homo opy in a ian in he mos gene al se ing (Ex. 3.1), bu his dimension can s ill eco e in some cases in o ma ion no eadily a ailable in he Be i numbe s o e en he cup p oduc (Thm. 3.2, Co . 3.3 , P op. 3.5 and 3.6). On he o he hand, ecall ha he p esence o a homological ea u e in he sequence Kmay some imes be ep esen ed in a zigzag ba code by wo o mo e di e en , o ally independen in e als. This can mislead us o belie e ha hey may be de ec ing mo e han one ea u e, as exempli ied in [23,§5.6.6]. As A∞-pe sis ence uses a weaked e sion o a zigzag ba code (see [2, De . 2.8]), in §4we will show how his ambigui y a ec s A∞-pe sis ence and discuss a way o bypass his issue. This wo k is o ganised as ollows: in §1we p o ide some backg ound on A∞-s uc u es. In §2we ecall, om an algeb aic poin o iew, he o dina y and A∞ e sions o ba codes. These a e ools used o encode pe sis ence in o ma ion. In addi ion, we gi e an al e na i e 3 F ancisco Belch´ı p oo o Lemma 2.4, which has as co olla y he ba code decomposi ion heo em o pe sis en homology (Thm. 2.3). In §3we illus a e when A∞-pe sis ence can be use ul beyond pe sis en homology by discussing he opological meaning o he measu emen s dim Viand hei duals in cohomology (Ex. 3.1, P op. 3.5 and 3.6, Thm. 3.2 and Co . 3.3 ). The esul s in his sec ion can be used o choose he igh n o s udy he pe sis ence o ∆nand hey also sugges a new pe sis ence app oach o links. In §4we exhibi how he a o emen ioned zigzag ambigui y p oblem a ec s A∞-pe sis ence and p opose a way o deal wi h i . This in ol es showing ha we can some imes make cohe en choices o A∞-s uc u es on he sequence Kso ha wha A∞-pe sis ence has o decompose is a pe sis ence module (Thm. 4.2). We hen p o e he ba code decomposi ion heo em o A∞-pe sis ence o he case o such cohe en choices o A∞-s uc u e (Thm. 4.3) o s ess how much hings simpli y by bypassing he need o zigzag machine y. We lea e some o he p oo s o §5and inish wi h some conclusions in §6. The esul s in §3and §4should be o in e es bo h o algeb aic opologis s and o mo e applied scien is s in ol ed in da a analysis in some way. Le us ix some no a ion and assump ions o he es o he pape . Fo he basic no ions o algeb aic opology used (such as cup p oduc ) we e e he eade o classics such as [9]. Th oughou his pape , we will always wo k o e a ield o coe icien s Fwhich we will usually omi om he no a ion. Fo ins ance, H∗(X) will deno e he homology o Xwi h coe icien s in F. E e y opological space Xconside ed will be assumed o ha e ini e dimensional homology g oups in all deg ees, dim Hp(X)<∞. We will deno e by K:K0//K1//. . . //KN a ini e sequence o opological spaces and con inuous maps and, o any in ege s p≥0 and 0≤i≤j≤N, we will deno e by i,j p:Hp(Ki)−→ Hp(Kj) and i,j :H∗(Ki)−→ H∗(Kj) he linea maps induced in homology by he composi ion Ki//Ki+1 //. . . //Kj. 1. A∞-s uc u es Looking o algeb aic s uc u es ha include all he in o ma ion o he homology g oups and e en o he s anda d cohomology algeb a gi en by he cup p oduc , and p o ide s ill mo e de ailed homological in o ma ion, we na u ally bump in o A∞-s uc u es. De ini ion 1.1.An A∞-coalgeb a s uc u e {∆n}n≥1on a g aded ec o space Cis a amily o maps ∆n:C−→ C⊗n 4 Op imising he opological in o ma ion o he A∞-pe sis ence g oups o deg ee n−2 such ha , o all n≥1, he ollowing S ashe iden i y holds: SI(n): n X i=1 n−i X j=0 (−1)i+j+ij 1⊗n−i−j⊗∆i⊗1j∆n−i+1 = 0. The iden i ies SI(n) o n= 1,2,3 s a e ha i (C, {∆n}n≥1) is an A∞-coalgeb a, hen ∆1 is a di e en ial on Cand he comul iplica ion ∆2is coassocia i e up o he chain homo opy ∆3. Mo eo e , any di e en ial g aded coalgeb a (DGC hence o h) (C, ∂, ∆) can be iewed as an A∞-coalgeb a (C, {∆n}n≥1) by se ing ∆1=∂, ∆2= ∆,and ∆n= 0 o all n > 2.An A∞-coalgeb a (C, {∆n}n≥1) is called minimal i ∆1= 0. De ini ion 1.2.Amo phism o A∞-coalgeb as : (C, {∆n}n≥1)→(C0,{∆0 n}n≥1) is a amily o maps (k):C−→ C0⊗k, k ≥1, o deg ee k−1, such ha o each i≥1, he ollowing iden i y holds: MI(i): X p+q+k=i q≥1,p,k≥0 (1⊗p⊗∆0 q⊗1⊗k) (p+k+1) =X k1+···+k`=i l,kj≥1 ( (k1)⊗ · · · ⊗ (k`))∆`. We say ha he mo phism o A∞-coalgeb as is: •an isomo phism i (1) is an isomo phism, •aquasi-isomo phism i (1) induces an isomo phism in homology. The e a e some i ial A∞-coalgeb a s uc u es one can always endow a g aded ec o space Cwi h, such as he one gi en by ∆n= 0 o all n. This s uc u e does no gi e in gene al any new in o ma ion, so ins ead, we will ocus ou a en ion in a special kind o A∞- coalgeb as we will e e o as he ans e ed ones. As we will see be ween his sec ion and §3, ou o all he A∞-coalgeb a s uc u es one could endow H∗(X) wi h, he ans e ed ones can gi e pa icula ly meaning ul in o ma ion abou he homo opy ype o X. Fo ins ance, we men ioned ha ∆3measu es he deg ee o which he coassocia i i y o ∆2can be elaxed in an A∞-coalgeb a (C, {∆n}n≥1). Howe e , e en i ∆2is s ic ly coassocia i e, ∆3may be non- i ial, and in a ans e ed A∞-coalgeb a, his non- i iali y can gi e c ucial in o ma ion (see §3). De ini ion 1.3.We will say ha an A∞-coalgeb a (H∗(X),{∆n}n) on he homology o a space Xis a ans e ed A∞-coalgeb a (induced by X) i i is minimal and quasi-isomo phic o he A∞-coalgeb a (C∗(X),{∂, ∆,0,0, . . .}), 5 F ancisco Belch´ı whe e (C∗(X), ∂) deno es he singula chain complex o Xand ∆ deno es an app oxima ion o he diagonal. We will d op he ’induced by X’ om he no a ion when no con usion is possible. Analogously, in a educed homology se ing, we will e e o an A∞-coalgeb a e H∗(X),{∆n}n as a ans e ed one i i is minimal and quasi-isomo phic o he induced A∞-coalgeb a a he educed chains le el e C∗(X),{e ∂, e ∆,0,0, . . .}. An immedia e consequence o De ini ion 1.3 is ha all ans e ed A∞-coalgeb as on H∗(X) induced by Xa e isomo phic. In he p oo o P op. 3.5 we will use he ollowing ema k. Rema k 1.4.Fo e e y space X, he e always exis ans e ed A∞-coalgeb as on H∗(X) induced by X, which can be compu ed in se e al ways, mos o hem amoun ing o an applica- ion o he Homo opy T ans e Theo em [11,12] (hence he name ans e ed A∞-coalgeb as). This can be used as an algo i hm ha akes as inpu a diag am o he o m φ;;(M, d) π//(N, d) ι oo(1.1) in which (N, d) is a chain complex, (M, d) is a DGC wi h comul iplica ion ∆, and he deg ee 0 chain maps πand ιand he deg ee 1 chain homo opy φmake he ollowing hold: πι = idN, πφ =φι =φ2= 0 and φis a chain homo opy be ween idMand ιπ,i.e., φd +dφ =ιπ −idM. The ou pu o he algo i hm includes an explici minimal A∞-coalgeb a s uc u e {∆n}non Nwi h ∆2=π⊗2∆ιand such ha , i φ= 0, hen ∆n= 0 o all n > 2. The dual no ion o an A∞-coalgeb a is ha o an A∞-algeb a. Depending on he ools mo e sui able o making he compu a ions, i can be mo e con enien o wo k wi h one no ion o he o he . De ini ion 1.5.An A∞-algeb a s uc u e {µn}n≥1on a g aded ec o space Ais a amily o maps µn:A⊗n−→ A o deg ee 2 −nsuch ha , o all n≥1, he ollowing S ashe iden i y holds: SI(n): X n= +s+ s≥1 , ≥0 (−1) +s µ +1+ 1⊗ ⊗µs⊗1⊗ = 0. E e y hing said o A∞-coalgeb as dualizes o A∞-algeb as. In an A∞-algeb a (A, {µn}n≥1), µ3measu es he associa i i y o µ2bu e en i µ2is s ic ly associa i e, µ3may be non- i ial 6 Op imising he opological in o ma ion o he A∞-pe sis ence g oups and gi e use ul in o ma ion (see §3). The Homo opy T ans e Theo em also wo ks o di - e en ial g aded algeb as (DGA hence o h) and A∞-algeb as, and we can de ine ans e ed A∞-algeb as on he cohomology o a space, H∗(X), jus as in De ini ion 1.3, using he cup p oduc ins ead o an app oxima ion o he diagonal. In pa icula , o any ans e ed A∞-algeb a {µn}non H∗(X), µ2coincides wi h he cup p oduc . Hence, ans e ed A∞-s uc u es encode all he in o ma ion in he homology g oups o Xand in i s cohomology algeb a as well, bu he e is mo e. Fo ins ance, in [2, Thm. 1.3] we exhibi ed a way o build pai s o spaces wi h isomo phic homology g oups and isomo phic cohomology algeb as bu non-isomo phic ans e ed A∞-coalgeb as (on hei homology), and T. Kadeish ili p o ed in [10, P op. 2] ha unde mild condi ions on a opological space X, any ans e ed A∞-algeb a on i s cohomology de e mines he cohomology o i s loop space, H∗(ΩX), whe eas he cohomology ing o Xalone does no . 2. Ba codes In pe sis ence, we a e in e es ed in inding ou how he opology e ol es along he sequence o opological spaces and con inuous maps Kc ea ed by a ying a pa ame e such as ime, hickness, in ensi y, heigh , e c. This is done by applying algeb aic- opology cons uc ions o Kand decomposing he esul in o he smalles pieces possible in a ce ain sense, ob aining a g aphical ep esen a ion called a ba code. In his sec ion we ecall, om an algeb aic poin o iew, he no ions o ba code in pe sis en homology [5,6] and A∞-pe sis ence [2]. In addi ion, we gi e an al e na i e p oo o he decomposi ion heo em o pe sis en homology. De ini ion 2.1.Fo e e y 0 ≤i≤j≤Nand p≥0, he p h pe sis en homology g oup o he sequence Kbe ween Kiand Kjis de ined as he ec o space Hi,j p(K):= Im i,j p and i s co esponding pe sis en Be i numbe is de ined as βi,j p(K):= dim Hi,j p(K). De ini ion 2.2.Ape sis ence module Vis a ini e sequence o ini e dimensional ec o spaces and linea maps be ween hem o he o m V0//V1//. . . //VN. The Fundamen al Theo em o Pe sis en Homology can be s a ed as ollows: Theo em 2.3.[5,§3] (Fundamen al Theo em o Pe sis en Homology) Fo any in ege p≥0, he e exis s a unique mul ise Mpo in e als o he o m [i, j) o 0≤i < j≤N, and in e als o he o m [i, ∞) o 0≤i≤N, such ha o all 0≤i≤j≤N, 7 F ancisco Belch´ı he dimension dim Hi,j p(K)equals he numbe o in e als in Mp(coun ed wi h mul iplici ies) which con ain he in e al [i, j]. In pa icula , dim Hp(Ki)equals he numbe o in e als in Mp(coun ed wi h mul iplici ies) which con ain he in ege i. The mul ise Mpin Thm. 2.3 is known as he p h ba code o K. Thm. 2.3 is a consequence o he ollowing esul : Lemma 2.4.[5,§3] Le V0 0,1 //V1 1,2 //. . . N−1,N //VNbe a pe sis ence module and se i,j :=   j−1,j ◦. . . ◦ i,i+1, i + 1 < j, idVi, i =j. Then he e exis s a unique mul ise Mo in e als o he o m [i, j) o 0≤i<j≤N, and in e als o he o m [i, ∞) o 0≤i≤N, such ha o all 0≤i≤j≤N, he dimension dim Im i,j can be compu ed as he sum o he mul iplici y o each in e al in M ha con ains [i, j]. G. Ca lsson and A. Zomo odian i s p o ed Lemma 2.4 and he e o e Thm. 2.3 in i s ull gene ali y in [5,§3] wi h a sligh ly di e en no a ion and ocabula y. They made use o he s uc u e heo em o ini ely gene a ed modules o e P incipal Ideal Domains. Nex we gi e an al e na i e p oo o Lemma 2.4 (and he e o e o Thm. 2.3) making use, ins ead, o he ollowing well-known ac in linea algeb a: Lemma 2.5.(F obenius inequali y on ma ix anks) Le A, B and Cbe ma ices such ha he p oduc s AB, ABC and BC a e de ined. Then ank(B)− ank(AB)− ank(BC) + ank(ABC)≥0. He e is an al e na i e p oo o Lemma 2.4: P oo . Le us se di,j :=   dim Im i,j,0≤i≤j≤N, 0,o he wise. Le us assume ha he e exis ed a mul ise Mconsis ing o : •a numbe Ni,j−1o in e als o he o m [i, j) o all 0 ≤i<j≤N, and •a numbe Ni,N o in e als o he o m [i, ∞) o all 0 ≤i≤N, such ha , o all 0 ≤i≤j≤N, di,j = #{in e als in M ha con ain [i, j]}. 8 Op imising he opological in o ma ion o he A∞-pe sis ence g oups Then, an inclusion-exclusion- ype a gumen shows ha , o all 0 ≤i≤j≤N,Ni,j is o ced o be Ni,j =di,j −di−1,j −di,j+1 +di−1,j+1.(2.1) Hence, i Mexis s, i mus be unique. On he o he hand, he exis ence o Mamoun s o Ni,j ≥0 holding o e e y 0 ≤i≤j≤N. Finally, no ice ha Lemma 2.5 implies ha hese inequali ies Ni,j ≥0 hold, p o ing he exis ence o M. Thm. 2.3 hen ollows om applying Lemma 2.4 o he pe sis ence module Hp(K0) 0,1 p //Hp(K1) 1,2 p //. . . N−1,N p //Hp(KN). Fo he es o his sec ion, choose a ans e ed A∞-coalgeb a s uc u e {∆i n}non he homology o each space Kiin K. De ini ion 2.6.Fo e e y 0 ≤i≤j≤N,p≥0 and n≥1, he p h ∆n-pe sis en g oup be ween Kiand Kjis he ec o space (∆n)i,j p(K):= Im i,j p|∩j k=iKe (∆k n◦ i,k p). These a e he A∞-pe sis ence g oups in e ms o ans e ed A∞-coalgeb as in homology. In A∞-pe sis ence [2], we s udy he e olu ion o ec o subspaces such as Vi:= Ke ∆n|Hp(Ki) ⊆Hp(Ki). The p oblem is ha he maps i,i+1 p:Hp(Ki)−→ Hp(Ki+1) do no es ic , in gene al, o maps Vi//Vi+1 (see o ins ance [2, Thm. 3.1]), he e o e no p oducing a pe sis ence module V0//V1//. . . //VN.Despi e ha , we can s ill comple ely un- de s and he pe sis en g oups in De . 2.6 by using zigzag echniques [3]. Indeed, he e is he A∞coun e pa o Thm. 2.3: Theo em 2.7.[2, Co . 2.9, De . 2.8], Fo any pai o in ege s p≥0and n≥1, he e exis s a unique mul ise Mp,n o in e als o he o m [i, j] o 0≤i<j≤N, such ha o all 0≤i≤j≤N, he dimension dim(∆n)i,j p(K)equals he numbe o in e als in Mp,n (coun ed wi h mul iplici ies) which con ain he in e al [i, j]. In pa icula , dim Ke ∆i n|Hp(Ki)equals he numbe o in e als in Mp,n (coun ed wi h mul- iplici ies) which con ain he in ege i. The mul ise Mp,n in Thm. 2.7 is called he p h ∆n-ba code o K. The cons uc ion o his ba code in ol es zigzag decomposi ions and in §4we will deal wi h a haza d his p esen s. 9 F ancisco Belch´ı Massey p oduc s. The in e es ed eade can explo e his in [22, Thm. V.7(7)] and mo e ecen ly in [1]. Finally, no ice ha esul s such as P op. 3.5 and 3.6 can be exploi ed by A∞-pe sis ence o s udy da ase s wi h an unde lying (low o high-dimensional) link s uc u e. Also, i we know in which pa icula con ex we a e wo king on, we can use he esul s in his sec ion o choose he igh n o ocus on when using A∞-pe sis ence. 4. Compa ible A∞-s uc u es o a oid zigzag ambigui ies Gi en he sequence o opological spaces and con inuous maps K:K0//K1//. . . //KN, le {∆i m}m≥1deno e a ans e ed A∞-coalgeb a on H∗(Ki), o each 0 ≤i≤N, and le n≥1 and p≥0 be wo in ege s. Once all his is ixed, o simpli y no a ion, w i e ∆i o ep esen he es ic ion o ∆i n o Hp(Ki). Le us look a a pa icula example o a momen . Example 4.1.Le us assume ha N= 999, Ke ∆500 ={0}and ha he e exis s a homology class α∈Hp(K0) such ha 06= 0,jα∈Ke ∆j, o all j6= 500,and 0 6= 0,500α /∈Ke ∆500. No ice ha a homology class can indeed beha e like his, as seen in [2,§3]. I is s aigh o wa d o check ha he exis ence o he class α, whose image by 0,j alls in o Ke ∆j− {0}a 99.9% o he e ms Kj o ming K, esul s in he c ea ion, in he p h ∆n-ba code o K, o wo (appa en ly un ela ed) in e als I= [0,499] and J= [501,999], each con aining a mos 500 in ege s, which amoun s o only 50% o he e ms in K. Ex. 4.1 shows ha he ∆n-pe sis en g oups and hei co esponding ba codes a e no as ai h ul as we would like in desc ibing he e olu ion o subspaces such as Ke ∆nalong K. This is due o he need o using zigzag pe sis ence [3] in he cons uc ion o he ∆n-ba codes. One way o a oid his issue is o choose sequences Kand ans e ed A∞-s uc u es along K in a cohe en manne o allow us o compu e A∞-pe sis ence ia pe sis ence modules a he han ia mo e gene al zigzag modules. In e ms o he p h ∆n-ba code in homology, his boils down o being able o compu e a ans e ed A∞-coalgeb a {∆i m}mon H∗(Ki), o each Kiin K, such ha he ollowing will hold o e e y 0 ≤i<N: i,i+1 pKe ∆i⊆Ke ∆i+1.(4.1) In his sec ion we show a way o cons uc il a ions Kand he co esponding ans e ed A∞-coalgeb as so ha assump ion (4.1) holds (see Thm. 4.2), and p o e he ba code decom- posi ion heo em o A∞-pe sis ence o he case o such compa ible choices o A∞-s uc u es 16 Op imising he opological in o ma ion o he A∞-pe sis ence g oups (see Thm. 4.3) o s ess how much hings simpli y, e en a a heo e ical le el, by bypassing he need o zigzag machine y. He e is a cons uc ion exhibi ing an example o a choice o A∞-s uc u es ha make assump ion (4.1) hold. Theo em 4.2.Le K0be a 1-connec ed CW complex and le N > 0be an in ege . Fo all 0< i ≤N, le Kideno e he wedge Ki−1∨Sni, o some ni>1. Then he e exis s a ans e ed A∞-coalgeb a e H∗(Ki;Q),{∆i n}non he educed a ional homology o Ki, o each 0≤i≤N, such ha he ollowing inclusion holds o e e y n≥1,p≥0and 0≤i<N: i,i+1 pKe ∆i⊆Ke ∆i+1, whe e ∆ideno es he es ic ion o ∆i n o e Hp(Ki;Q)and i,i+1 p:e Hp(Ki;Q)−→ e Hp(Ki+1;Q) deno es he map induced by he inclusion Ki,−→ Ki+1. We sa e he p oo o Thm. 4.2 o §5and p oceed o p o e he ollowing pa icula case o he undamen al decomposi ion heo em o A∞-pe sis ence [2, Thm. 2.7], o show how much he assump ion (4.1) simpli ies hings; compa e his o he p oo in [2, Thm. 2.7]. Theo em 4.3.Fix some in ege s p≥0and n≥1. I ∆ideno es he es ic ion o ∆i n o Hp(Ki)and i,i+1 p(Ke ∆i)⊆Ke ∆i+1 holds o all 0≤i<N, hen he e exis s a unique mul ise Mp,n o in e als o he o m [i, j) o 0≤i<j≤N, and in e als o he o m [i, ∞) o 0≤i≤N, such ha o all 0≤i≤j≤N, he dimension dim(∆n)i,j p(K)equals he numbe o in e als in Mp,n (coun ed wi h mul iplici ies) which con ain he in e al [i, j]. In pa icula , dim Ke ∆i n|Hp(Ki)equals he numbe o in e als in Mp,n (coun ed wi h mul- iplici ies) which con ain he in ege i. P oo . Fix a ans e ed A∞-coalgeb a s uc u e {∆i n}non H∗(Ki), o all 0 ≤i≤N, and ix wo in ege s n≥1 and p≥0. I he inclusion i,i+1 p(Ke ∆i)⊆Ke ∆i+1 holds o all 0≤i<N, hen he maps i,j pin he pe sis ence module Hp(K0) 0,1 p //Hp(K1) 1,2 p //. . . N−1,N p //Hp(KN) es ic o maps gi,j p ha o m he pe sis ence (sub)module Ke ∆0g0,1 p //Ke ∆1g1,2 p //. . . gN−1,N p //Ke ∆1.(4.2) Since (∆n)i,j p(K) = Im gi,j p, applying Lemma 2.4 o (4.2) inishes he p oo .  Rema k 4.4.In he e minology in oduced in [2, De . 2.5], Ex. 4.1 shows ha a class ha ∆n- alls asleep and ∆n-wakes up again would be ep esen ed in he ∆n-ba code as appa en ly independen in e als, each co esponding o a di e en pe iod in which he class has been 17 F ancisco Belch´ı ∆n-awake, and assump ion (4.1) would amoun o making su e no class can ∆n-wake up once i has ∆n- allen asleep. 5. Appendix We elega ed o his sec ion he p oo o Ex. 3.1 and Thm. 3.2 and 4.2. We will s a by ecalling some esul s in ol ing a ional homo opy heo y, whose s anda d e e ence is [7], mos ly o explain he ela ion be ween Quillen minimal models and ans e ed A∞-coalgeb as. Rema k 5.1.Le us s a by no ing ha A∞-coalgeb a s uc u es on a g aded ec o space Ca e in one- o-one co espondence wi h di e en ials in he comple e enso algeb a b T(s−1C)=Πn≥1Tn(s−1C) wi h Tn(s−1C)=(s−1C)⊗n. He e, s−1Cdeno es he desuspension o C,i.e., (s−1C)p=Cp+1, he g ading in b T(s−1C) is gi en by b T(s−1C)p= Πn≥1Tn(s−1C)p, whe e Tn(s−1C)p=M p1+...+pn=ps−1Cp1⊗. . . ⊗s−1Cpn and he p oduc is gi en by conca ena ion. Gi ing a di e en ial d:b T(s−1C)−→ b T(s−1C) in his algeb a is equi alen o gi ing i s es ic ion d:s−1C−→ b T(s−1C) and ex ending das a de i a ion. Such d:s−1C−→ b T(s−1C) can be w i en as a sum d=Pn≥1dn, wi h dn(s−1C)⊂Tn(s−1C), o n≥1. Wi h his no a ion, he ope a o s {∆n}n≥1and {dn}n≥1uniquely de e mine each o he ia ∆n=−s⊗n◦dn◦s−1:C→C⊗n, dn=−(−1)n(n−1) 2(s−1)⊗n◦∆n◦s:s−1C→Tn(s−1C). (5.1) De ini ion 5.2.Gi en an A∞-coalgeb a (C, {∆n}n), he comple e enso algeb a b T(s−1C) wi h he di e en ial gi en by (5.1) o ms a DGA ha is called he coba cons uc ion o (C, {∆n}n) and is usually deno ed by ΩC. This cons uc ion beha es nicely wi h espec o mo phisms: gi ing a mo phism o A∞- coalgeb as : (C, {∆n}n≥1)→(C0,{∆0 n}n≥1) is equi alen o gi ing a mo phism o DGAs, which we deno e in he same way, be ween he co esponding coba cons uc ions : ( b T(s−1C), d)−→ (b T(s−1C0), d0). Indeed, w i e |s−1C=Pk≥0 kwi h k:s−1C→Tk(s−1C0) and conside he mo phisms (k):C→C0⊗kinduced by kelimina ing desuspensions. Then, i is s aigh o wa d o check ha he equali y d =d0 ansla es o he iden i ies MI(i) in De ini ion 1.2 sa is ied by he maps { (k)}k≥0.18 Op imising he opological in o ma ion o he A∞-pe sis ence g oups De ini ion 5.3.Adi e en ial g aded Lie algeb a o e Q(DGL hence o h) is a di e en ial g aded Q- ec o space (L, ∂) in which: •L=⊕p∈ZLpin endowed wi h a linea ope a ion, called Lie b acke , [,]: Lp⊗Lq−→ Lp+q, p, q ∈Z, sa is ying an isymme y, [x, y] = (−1)|x||y|+1[y, x], and he Jacobi iden i y x, [y, z]=[x, y], z+ (−1)|x||y|y, [x, z], o any homogeneous elemen s x, y, z ∈L. In o he wo ds, Lis a g aded Lie algeb a. •The di e en ial ∂, o deg ee −1, sa is ies he Leibniz ule, ∂[x, y]=[∂x, y]+(−1)|x|[x, ∂y], o any pai o homogeneous elemen s x, y ∈L. The enso algeb a T(V) = ⊕n≥0Tn(V) gene a ed by he g aded ec o space Vis endowed wi h a g aded Lie algeb a s uc u e wi h he b acke s gi en by commu a o s: [a, b] = a⊗b−(−1)|a||b|b⊗a, o any homogeneous elemen s a, b ∈T(V). Then, he ee Lie algeb a L(V) is he Lie subalgeb a o T(V) gene a ed by V. Obse e ha L(V) is il e ed as ollows, L(V) = ⊕n≥1Ln(V), in which Ln(V) is he ec o space spanned by Lie b acke s o leng h n, ha is Ln(V) = L(V)∩Tn(V). In pa icula , o se a di e en ial in L(V), i is enough o de ine linea maps ∂n:V−→ Ln(V) o each n≥1 in such a way ha ∂=Pn≥1∂nsqua es o ze o and sa is ies he Leibniz ule. No e ha ∂1, he linea pa o ∂, is he e o e a di e en ial in V. Rema k 5.4.Gi en a DGL o he o m (L(V), ∂), he di e en ial ∂can be ex ended as a de i a ion o g aded algeb as o T(V) o u n i in o a DGA. In o he wo ds, conside he unc o U:DGL −→ DGA which associa es o e e y DGL (L, ∂) i s uni e sal en eloping algeb a UL which is he g aded algeb a T(L)/h[x, y]−(x⊗y−(−1)|x||y|y⊗x)i, x, y ∈L, 19 F ancisco Belch´ı wi h he di e en ial induced by ∂. Then, whene e (L, ∂) = (L(V), ∂), one has U(L(V), ∂) = (T(V), ∂). In [19], D. Quillen associa es o e e y 1-connec ed opological space Xo he homo opy ype o a CW complex, a pa icula DGL (L(V), ∂) which is educed,i.e., V=⊕p≥1Vp, o which: H∗(V, ∂1)∼ =s−1e H∗(X;Q) = e H∗+1(X;Q).(5.2) H∗(L(V), ∂)∼ =π∗(ΩX)⊗Q∼ =π∗+1(X)⊗Q.(5.3) This DGL is a Quillen model o Xand i is called minimal whene e ∂1= 0. In his case, (5.2) becomes V∼ =s−1e H∗(X;Q).(5.4) The Quillen minimal model o Xis unique up o isomo phism. Rema ks 5.4 and 5.1 oge he wi h isomo phism (5.4) asse ha any Quillen minimal model (L(V), ∂) o Xinduces a s uc u e o ans e ed A∞-coalgeb a on e H∗(X;Q). Gi en a 1-connec ed CW complex Y, a (no necessa ily minimal) Quillen model o Ycan be desc ibed in e ms o a CW decomposi ion o Yas ollows: Theo em 5.5.[22, III.3.(6)] Le Y=X∪ en+1 be a 1-connec ed space ob ained a - aching an (n+ 1)-cell o X ia a map :Sn→X. Le (L(V), ∂)be a Quillen model o X and le Φ∈L(V)n−1be a cycle ep esen ing he homology class in Hn−1(L(V), ∂)which is iden i ied, ia isomo phism (5.3) wi h he homo opy class [ ]∈πn(X)⊗Q. Then, he ee DGL (L(V⊕Qa), ∂0)de e mined by    ∂0 =∂ , o all ∈V ∂0a= Φ o ms a Quillen model o Y. We now p oceed o p o e he claim in Ex. 3.1, which poin s ou ha , in gene al, he numbe s dim Ke ∆n|Hp(X)depend on he choice o ans e ed A∞-s uc u e. P oo . The space X=CP2∨S7,can be CW-decomposed as e0∪e2∪e4∪e7, whe e he a aching map o he 4-cell is he Hop map η:S3−→ S2and he es o a aching maps a e all i ial. In u n, η=1 2g∈π3(S2)⊗Q, whe e gdeno es he Whi ehead p oduc g= [idS2, idS2].Hence, ia Thm. 5.5, he ee DGL (L(x1, y3, z6), ∂) o ms a Quillen minimal model o X, whe e subsc ip s deno e deg ee and he di e en ial is gi en by ∂x1= 0, ∂y3= 1 2[x1, x1] and ∂z6= 0. 20 Op imising he opological in o ma ion o he A∞-pe sis ence g oups Conside he isomo phism o (non di e en ial) ee Lie algeb as ψ:L(x1, y3, z6)∼ = −→ L(x1, u3, 6) gi en by ψ(x1) = x1, ψ(y3) = 1 2u3and ψ(z6) = 6−[u3, u3].Then, se in he ee Lie al- geb a on he igh he di e en ial ∂=ψ∂ψ−1so ha ψbecomes an isomo phism o DGLs be ween (L(V), ∂) = (L(x1, y3, z6), ∂) and (L(W), ∂) = (L(x1, u3, 6), ∂).A sho compu a ion shows ha in (L(W), ∂), ∂x1= 0, ∂u3= [x1, x1] and ∂ 6= 2[[x1, x1], u3].Since (L(V), ∂) and (L(W), ∂) a e isomo phic DGLs, (L(W), ∂) is also a Quillen minimal model o X, and in pa - icula V∼ =W∼ =s−1e H∗(X;Q).A sho compu a ion shows ha , in he uni e sal en eloping algeb a (T(W), d) = U(L(W), ∂), d3 6= 4x1⊗x1⊗u3−4u3⊗x1⊗x1.In o he wo ds, in he A∞-coalgeb a s uc u e {∆n}non e H∗(X;Q) gi en by (L(W), ∂), ∆36= 0. On he o he hand, in he uni e sal en eloping algeb a (T(V), d) = U(L(V), ∂), dn= 0 o all n6= 2, which means ha in he A∞-coalgeb a s uc u e {∆0 n}non e H∗(X;Q) gi en by (L(V), ∂), ∆0 n= 0 o all n6= 2; in pa icula , ∆0 3= 0.  To p o e Thm. 3.2 we will i s p o e a p elimina y esul . Lemma 5.6.Le (C, {∆n}n)be an A∞-coalgeb a such ha ∆n6= 0 o some n≥1. I we deno e by k he lowes in ege n≥1such ha ∆n6= 0, hen (C, {0,...,0,∆k,0, . . .}) o ms an A∞-coalgeb a oo. P oo . Le (C, {∆n}n) and kbe as in he s a emen and de ine ∆0 n:=   0, n 6=k ∆k, n =k. Le us deno e by {SI(n)}n he S ashe iden i ies (De ini ion 1.1) on {∆n}nand by {SI0(n)}n he S ashe iden i ies on {∆0 n}n. The pai (C, {∆0 n}n) o ms an A∞-coalgeb a i and only i he iden i ies SI0(n) hold o all n≥1. No ice ha all o hem excep o SI0(2k−1) become he i ial iden i y 0 = 0,and ha SI0(2k−1) becomes k+1 X j=0 (−1)k+j+kj 1⊗k−1−j⊗∆k⊗1j∆k= 0, since his is he only SI0(n) in which ∆kcomposes wi h ∆k( enso ed by iden i ies), which is he only non-ze o ope a ion in {∆0 n}. Le us now look a he iden i y SI(2k−1) on {∆n}n. I some ∆nwi h n>kappea s in SI(2k−1), hen ∆n(o ∆n enso ed by iden i ies) is p e o pos composed wi h ∆m(o wi h ∆m enso ed by iden i ies) o some m < k. Since ∆m= 0, all such composi ions anish. The e o e, SI(2k−1) coincides wi h SI0(2k−1). Since we know ha (C, {∆n}n) o ms an A∞-coalgeb a and hence ha SI(2k−1) holds, i ollows ha SI0(2k−1) holds oo and we ha e he e o e checked ha (C, {∆0 n}n) sa is ies all S ashe iden i ies.  21 F ancisco Belch´ı Nex we p o e Thm. 3.2. In his p oo we will make use o he algeb aic Milno -Moo e spec al sequence. The backg ound on spec al sequences needed o ollow his p oo can be ound in [7,§18 & §23(b)] and [8, Ch. 1]. P oo . Le Cbe a g aded ec o space wi h Cp= 0 o all p < 0. Le (C, {∆n}n) be a minimal A∞-coalgeb a and le ΩC= ( b T(s−1C), d) be i s coba cons uc ion (De ini ion 5.2). We can hen w i e das he sum Pn≥1dn,whe e each dn:b T(s−1C)−→ T≥n(s−1C) = Πp≥nTp(s−1C) sa is ies dnTp(s−1C)⊆Tp+n−1(s−1C) o all p∈Nand ac s on T1(s−1C) = s−1Cas he composi ion dn:s−1Cs//C−(−1)n(n−1) 2∆n //C⊗n(s−1)⊗n //Tn(s−1C). Le us se k:=   min{n|∆n6= 0},i {n|∆n6= 0} 6=∅ +∞,o he wise, and le us de ine Ω0C,d0=Pn≥1d0 nand k0analogously o a minimal A∞-coalgeb a (C, {∆0 n}n) isomo phic o (C, {∆n}n). Fi s hing o no ice is ha since he A∞-algeb as a e isomo phic, hei coba cons uc ions ΩCand Ω0Ca e isomo phic DGAs. Le us now di ide he p oo in o wo complemen a y cases: ·Case ∆n= 0 o all n: I ∆n= 0 o all n,dmus be 0. The ac ha ΩCand Ω0Ca e DGA isomo phic o ces d0 o be 0 oo. This implies ha ∆0 n= 0 o all n. Hence, k= +∞=k0and dim Ke ∆m|Cp= dim Cp= dim Ke ∆0 m|Cpholds o e e y m≥1 and p≥0. ·Case ∆n6= 0 o some n: In his case, kbecomes he in ege min{n|∆n6= 0}.Since (C, {∆n}n) is minimal, kmus be a leas 2. No e ha kis ob iously equal o min{n|dn6= 0}.Applying again he a gumen jus used in he p e ious case, i he e exis s some nsuch ha ∆n6= 0, he e mus exis some n0such ha ∆0 n06= 0 as well. Hence, all his holds o k0as well. Mos o he es o he p oo will consis on showing ha k=k0and ha he DGAs ( b T(s−1C), dk) and ( b T(s−1C), d0 k) a e isomo phic. Assume k≤k0and le b T(s−1C) = F0⊇F1⊇. . . ⊇Fp⊇Fp+1 ⊇. . . be he wo d-leng h il a ion o ΩC,i.e., o all p∈N,Fpis he di e en ial ideal gi en by Fp=T≥p(s−1C). 22 Op imising he opological in o ma ion o he A∞-pe sis ence g oups In he algeb aic Milno -Moo e spec al sequence, E0is he g aded algeb a associa ed o his il a ion, i.e., o all p∈N, Ep 0=Fp Fp+1 ∼ =Tp(s−1C) and E0=M p∈N Ep 0∼ =T(s−1C), and he di e en ial ∂0=Lp∈N∂p 0:E0−→ E0is he map induced by d. Thus, o all p∈N, ∂p 0:Ep 0−→ Ep 0mus be d1:Tp(s−1C)−→ Tp(s−1C),which is 0, since (C, {∆n}n) is minimal. Hence, (E0, ∂0)∼ =(b T(s−1C),0). In his spec al sequence, o all p∈N, we ha e Ep 1∼ =Ep 0∼ =Tp(s−1C),and he map ∂p 1:Ep 1−→ Ep+1 1induced by dis hus d2:Tp(s−1C)−→ Tp+1(s−1C).Hence, (E1, ∂1)∼ = (b T(s−1C), d2).Simila ly, we can easily e i y ha (E0, ∂0)∼ =. . . ∼ =(Ek−2, ∂k−2)∼ =(b T(s−1C),0) and (Ek−1, ∂k−1)∼ =(b T(s−1C), dk). Le ϕ: ΩC−→ ΩC0be an isomo phism o DGAs, ha exis s because (C, {∆n}n) and (C, {∆0 n}n) a e isomo phic A∞-coalgeb as. Since ϕis il e ed, i induces a mo phism o spec al sequences, {E (ϕ)} , om he algeb aic Milno -Moo e spec al sequence associa ed o ΩC o ha associa ed o Ω0C. Recall ha he linea pa o ϕ,ϕ1:s−1C−→ s−1C, is de ined by ϕ(s−1c) = ϕ1(s−1c) + Φ, whe e Φ ∈T≥2(s−1C). Since ϕis an isomo phism, ϕ1mus also be an isomo phism as well. Now obse e ha E0(ϕ) = Ek−2(ϕ) is p ecisely he isomo phism b T(ϕ1): ( b T(s−1C),0) −→ (b T(s−1C),0).Applying a e sion o he Compa ison Theo em [8, Thm I in §1.3] in e ms o DGAs, we conclude ha Ek−1(ϕ): (T(s−1C), dk)−→ (T(s−1C), d0 k) is an isomo phism o DGAs. In pa icula , d0 kcanno be ze o. Hence k=k0and he i s claim in Thm. 3.2 holds. Lemma 5.6 gua an ees ha (C, {0,...,0,∆k,0, . . .}) and (C, {0,...,0,∆0 k,0, . . .}) o m wo A∞-coalgeb as. To p o e ha hey a e isomo phic, simply obse e ha he co e- sponding coba cons uc ions a e ( b T(s−1C), dk) and ( b T(s−1C), d0 k),which we ha e jus seen a e isomo phic DGAs ia Ek−1(ϕ). This ells us he e exis s an isomo phism o A∞-coalgeb as : (C, {0,...,0,∆k,0, . . .})−→ (C, {0,...,0,∆0 k,0, . . .}). The iden i y MI(k) om De ini ion 1.2 becomes ∆0 k (1) = (1)⊗k∆k,and since (1) is an iso- mo phism, i ollows ha dim Ke ∆k|Cp= dim Ke ∆0 k|Cp, o all p≥0. Ob iously, by he de ini ion o k, dim Ke ∆m|Cp= dim Cp= dim Ke ∆0 m|Cpholds oo o e e y m < k and p≥0.  23 F ancisco Belch´ı We now p o e Thm. 4.2, which s a es he possibili y, unde ce ain assump ions, o ind- ing compa ible A∞-s uc u es yielding o pe sis ence modules wi h which o compu e A∞- pe sis ence. P oo . Le K0be a 1-connec ed CW complex. I o all 0 < i ≤N,Kideno es he wedge Ki−1∨Sni o some ni>1, hen: •each Kiis a 1-connec ed CW complex as well, and • he maps i,i+1 p:e Hp(Ki;Q)−→ e Hp(Ki+1;Q) and i,i+1 :e H∗(Ki;Q)−→ e H∗(Ki+1;Q) a e inclusions. We s a he cons uc ion by choosing any Quillen minimal model (L(V), ∂) o K0. Since K1can be decomposed as K1=K0∪ eni, whe e he a aching map :Sni→K0is he i ial one, Thm. 5.5 hen ells us ha he ee DGL (L(V⊕Qa), ∂0) de e mined by          ∂0 =∂ , o all ∈V ∂0a= 0 |a|=ni−1 is a Quillen model o K1.Fu he mo e, since ∂0 1|V=∂1= 0 and ∂0a= 0, we ha e ha ∂0 1= 0, so he model (L(V⊕Qa), ∂0) is indeed minimal. Applying his a gumen a each s ep, we ob ain a Quillen minimal model (L(Vi), ∂i) o each Kiso ha ∂i+1 =∂i o any ∈Vi⊂Vi+1 and o all 0 ≤i<N. These Quillen minimal models hen ansla e in o ans e ed A∞-coalgeb as e H∗(Ki;Q),{∆i n}nsuch ha o all n≥1, p≥0 and 0 ≤i<N, i ∆ideno es he es ic ion o ∆i n o e Hp(Ki;Q), hen ∆i+1α= ∆iα o e e y α∈e Hp(Ki;Q)⊂e Hp(Ki+1;Q). Since he maps i,i+1 pand i,i+1 a e inclusions, his means ha he equali y ∆i+1 i,i+1 p= i,i+1⊗n∆i holds and hence so does i,i+1 p(Ke ∆i)⊆Ke ∆i+1. 6. Conclusions Le us use he diag am in Fig. 1 o enume a e he con ibu ions o his wo k. In ela ion o Fig. 1a, we p o ide a new p oo o he Fundamen al Theo em o Pe sis en Homology (see Lemma 2.4 and Thm. 2.3). Ou wo k on Fig. 1b consis s o showing how he pa o A∞-s uc u es we ocus on in A∞-pe sis ence can o m powe ul desc ip o s (Thm. 3.2, Co . 3.3, P op. 3.5 and 3.6 and Ex. 3.1). In ela ion o Fig. 1c, we s udy A∞-pe sis ence decomposi ions o pe sis ence modules (Thm. 4.3 and 4.2). 24 Op imising he opological in o ma ion o he A∞-pe sis ence g oups In he cu en and ollowing pape s we keep de eloping he heo y o A∞-pe sis ence because o i s exci ing g ea po en ial: pe sis en homology has been used success ully in many a eas, such as digi al imaging, senso ne wo ks co e age, ma e ials science, molecula modelling, signal p ocessing, i us e olu ion and diagnosis o hepa ic lesions, and all his has been possible by using pe sis ence a he le el o Be i numbe s. Hence, by enhancing he powe o pe sis ence h ough he use o A∞-s uc u es, who knows whe e we can ge ? Acknowledgemen s I would like o hank P o . Anice o Mu illo and P o . Jim S ashe o hei aluable eedback on his wo k. Re e ences [1] F. Belch´ı, U. Buijs, J. M. Mo eno-Fe n´andez, and A. Mu illo. Highe o de Whi ehead p oduc s and L∞-s uc u es on he homology o a DGL. Linea Algeb a and i s Applica ions, 520:16–31, 2017. 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