Computational complexity of topological invariants

COMPUT A TIONAL COMPLEXITY OF TOPOL OGICAL INV ARIANTS MANUEL AMANN Abstra ct. W e answ er the follo wing q uestion p osed by Lech uga: Give n a simply-connected space X with b oth H ∗ ( X, Q ) and π ∗ ( X ) ⊗ Q being finite-dimensional, what is the computational complexity o f an algo- rithm computing t he cup-length and the rational Lusternik–Schnirelma nn category of X ? Basicall y , by a reduction from the decision problem whether a given graph is k -colourable (for k ≥ 3) w e show that (eve n stricter versions of the) p roblems ab ove are NP -h ard. Introduction The theory of compu tational complexit y has dev elop ed a p ow erfu l ma- c hinery of d escribing h o w “difficult”, i. e. h o w time-consuming, it is to an- sw er certain p osed questio ns algo rithmically . Most classically , th is asks for the follo w ing catego rification of problems: The complexity class P describ es all the pr ob lems for whic h there is a p olynomial-ti me solving algorithm; the class NP is formed by those problems whic h m ay at least b e v erified in p olynomial time. Clearly P ⊆ NP , ho w ev er, it is the common b elie f that sev eral problems in NP are m uc h h arder to solve than the problems in P . Kno wn algorithms t ypically run at exp onenti al costs. A whole v ariet y of problems stemming fr om completely d ifferen t areas of mathematics and computer science hav e b een f ound to b e hard er than all the problems in NP , i.e. to b e NP -hard. Just to name a f ew most prominent ones we mentio n the knapsack pr oblem and the subse t sum pr ob- lem , the H amilton cir cuit pr oblem and the tr avel ling salesma n pr oblem , the satisfiability pr oblem and the gr aph c olouring pr oblem . Also in the fi eld of algebraic top olo gy it is easy to imagine sev eral p roblems for whic h it seems difficult to find efficien t solving algorithms. In particular, Rational Homotop y Th eory has the app eal of p ro viding “computable pr ob- lems”, which certainly ask f or algorithmic treatment. I n deed, Rational Ho- motop y Theory p ermits a categorical translation from topology/homotop y theory to algebra at the exp ense of losing torsion inf ormation. Y et, it tu rns out that the algebraic side allo ws for concrete compu tations. Using this appr oac h sev eral top ological problems w ere sho wn to b e N P - hard. In [ 1 ] it is shown that computing the r ational h omotop y groups Date : December 4th, 2011. 2010 Mathematics Subje ct Classific ation. 68Q17 (Primary), 55P62 ( Secondary). Key wor ds and phr ase s. computational complexit y , cup-length, LS- category , p ure el- liptic sp ace. The author was sup p orted by a Research Grant of the German Research F oundation. 2 MANUEL AMANN π ∗ ( X ) ⊗ Q of a simply-connected CW-complex X is NP -hard. S o is the problem of whether a simp ly-connected s p ace X with d im π ∗ ( X ) ⊗ Q < ∞ also has fin ite-dimensional rational cohomology (cf. [ 4 ]). In the same article it wa s sh o wn that for formal sp aces, i.e. for s paces for wh ic h th e rational homotop y typ e can b e formally derive d from the r ational cohomology alge- bra, the computation of Betti num b ers, of cup -length and of the rational Lusternik–Sc hnirelmann category are N P -hard problems. In [ 4 ] it is shown that the computation of Betti n um b ers of a simply-connected space with b oth finite-dimensional rational homotop y and finite-dimensional r ational homotop y , a (r ational ly) el liptic sp ac e , is NP -hard. Ho w ever, already in the article [ 5 ] and then explicitly in [ 4 ] the follo w in g question is p osed: Question (Lec huga) . Given an e l liptic sp ac e , what is the c omputational c omplexity of c omputing its r ational cup-length or i ts r ational Lusternik– Schnir elmann c ate gory? Note that the metho ds from the results present ed ab ov e do not answe r this question, as th ey do not app ly to the case of elliptic spaces. (F or a definition of th e top ological inv ariants w e refer the reader to section 2 .) In this article we shall answer Lec h u ga’s question by rev ealing these prob- lems as NP -hard —w e do so already for the same question p osed on th e sub cl ass of pu re elliptic sp aces. F or this we sp ecify the f ollo w in g pr oblems P : L et X b e a simply-connected top ological space with fin ite-dimensio- nal rational homology H ∗ ( X, Q ) and with fi nite-dimensional rational homotop y π ∗ ( X ) ⊗ Q . What is its cup -length? Q : Let X b e a simply-connected top olo gical with fin ite-dimensional r a- tional homology H ∗ ( X, Q ) and with fin ite-dimensional rational ho- motop y π ∗ ( X ) ⊗ Q . What is its rational Lusternik–Schnirelmann catego ry? W e are inte rested in th e computational complexity of the pr oblems P and Q . The co dification of a simply-connected space X will b e given as the data con tained in its minimal Sul livan mo del (Λ V X , d), i.e. X will b e repr esen ted b y the degrees of the h omogeneous generators x 1 , . . . , x l of V X and the co- efficien ts of th e p olynomials in the x i whic h r epresen t the differen tial. Recall th at a minimal Su lliv an mo del of a simp ly-connected space is a free (graded) comm utativ e graded algebra Λ V o v er the Z -graded rational v ecto r space V = V ≥ 2 together with a d ifferen tial d defined by d : V ∗ → (Λ V ) ∗ +1 and extended to Λ V as a deriv ation. The d ifferential satisfies that its image lies in th e su balgebra of elemen ts of wordlength at least t w o in V , i.e. im d ⊆ Λ ≥ 2 V . One then requires the existence of a quasi-isomorphism (Λ V , d) → A PL ( X ), i.e. a morphism of differentia l graded algebras to the p olynomial different ial forms A PL ( X ) on X ind ucing an isomorphism on homology . Thus (Λ V , d) enco d es th e r ational homotop y type of X . (See [ 2 ].3, COMPUT A TIONAL COMPLEXITY O F TOP O LOGICAL INV ARIANTS 3 [ 2 ].10 and [ 2 ].12 for the missing definitions.) In particular, the homology algebra H (Λ V , d) of the m in imal mo del is the cohomology algebra of X . Th us p roblem P translates to problem P ′ : Let (Λ V , d ) b e a simply-connected elliptic Sulliv an algebra. What is its cup -length? Since th e rational Lustern ik–Sc hnirelmann categ ory of a simp ly-connected space with rational homology of fi n ite t yp e equ als the category of its minimal Sulliv an mo d el (cf. [ 2 ].29.4, p. 386), problem Q b ecomes Q ′ : Let (Λ V , d ) b e a simply-connected elliptic Sulliv an algebra. What is its Lu sternik–Sc hnirelmann category? By the same theorem the rational T o omer inv ariant e 0 ( X ) of a simply- connected space with rational homology of fi n ite typ e equals the T o omer in v arian t e (Λ V , d ) of its minimal m o del. The rational cohomology algebra of a simp ly-connected elliptic space satisfies P oincar ´ e d ualit y . On simply- connected spaces with cohomo logy satisfying P oincar’e dualit y the ratio- nal T o omer in v arian t equals the r ational Lusternik–Sc hnirelmann catego ry (cf. [ 2 ].38, p. 511). T hus pr oblems Q a nd Q ′ ha v e the obvious analogue ˜ Q : Let (Λ V , d) b e a simply-connected elliptic Sulliv an algebra; (resp ec- tiv ely let X b e a simply-connected elliptic space). What is its (ra- tional) T o omer inv arian t? A Sulliv an algebra (Λ V , d ) is pur e if V = P ⊕ Q with Q = V ev en and P = V odd and the differenti al d satisfies d | Q = 0 and d( P ) ∈ Λ Q Classical examples of spaces admitting pur e mo d els are biquotien ts; resp ec- tiv ely , in particular, their sub cla ss of h omogeneous sp aces. W e shall determine the computational complexities of the follo w ing stricter problems, i.e. we shall show that they are NP -hard. In particular, this will answ er th e o riginal question by Lec h uga. P ′′ : Let (Λ V , d) b e a simply-connected pure elliptic S u lliv an algebra. What is its c up-length? Q ′′ : Let (Λ V , d) b e a simply-connected pure elliptic S u lliv an algebra. What is its rati onal Lustern ik–Sc hnirelmann category? This leads us to our main theorems. Theorem A. The p r oblem P ′′ is NP -har d. Theorem B. The pr oblem Q ′′ is NP -har d. Since pure elliptic space s form a sub class of e lliptic spaces, we obtain 4 MANUEL AMANN Corollary C. The pr oblems P and P ′ ar e NP -har d. So ar e the pr oblems Q , Q ′ and ˜ Q . Structure of the article. In section 1 w e briefly review some basic con- cepts from the theory of computational complexit y . W e recall the defin itions of the top ological inv arian ts in section 2 b efore we p ro v e theorem A in section 3 . Section 4 is devo ted to the pro of of theorem B . 1. Basic not ions fr om complexity theor y Let us r ecall some definitions f rom complexit y theory . Definition 1.1 (p r oblem, solution, complexit y) . A pr oblem is a set X of ordered pairs ( I , A ) of bitcod ed strings with the pr op erty that for eac h instance I there exists an answer A such that ( I , A ) ∈ P . A de cision p r oblem is a f unction X with v al ues in { 0 , 1 } . A solution of a problem is an algorithm wh ic h compu tes for eac h in put I an ou tp ut A suc h that ( I , A ) ∈ X in fi nitely many steps. The c omplexity of a problem is the infi mum of the (asymptotic) run times of all solutio n algorithms. Definition 1.2 ( P , NP ) . Sup p ose give n an in s tance I and a suggested pro of A I for the fact th at ( I , 1) ∈ X for the decision p roblem X . A p oly nomial verifier of X is an algorithm whic h chec ks in p olynomial time whether A I really pro v es that I is true. The class of all decision problems for which there exists a p olynomial time solution algo rithm is called the class P . The class of all decision problems for whic h th er e e xists a p olynomial v erifier form the class NP . A decision problem Y is NP -c omplete , if all X ∈ NP can b e redu ced to Y in p olynomial time. An arb itrary p roblem X whic h is harder th an all the problems Y ∈ NP is called NP -har d , i.e. for eac h problem Y ∈ NP an algorithm solving X can b e translated in p olynomial time to an algorithm solving Y . In order to sho w that a problem is NP -h ard one tends to use a redu ction principle: If one can redu ce an N P -complete p roblem A 1 to a problem A 2 , then th e latter has to be NP -hard. Indeed, sin ce A 1 is NP -complete, it is maximally h ard in NP , i.e . ev ery problem A ∈ NP can be reduced to A 1 . Consequent ly , ev ery problem A ∈ NP can b e reduced t o A 2 in p olynomia l time. Thus also A 2 is harder than all the problems in NP . 2. The topological inv ariant s In this section w e intend to briefly r ecall the d efinitions of the top ol ogical in v arian ts which app ear in abu n dance in this article. T hey w ill partly b e defined using Rational Homotop y Th eory . W e recommend the textb o ok [ 2 ] for an int ro duction to this field. W e sh all follo w the notation and defin itions pro vided there. Let us start w ith the simplest in v arian t which is pro vided b y Definition 2.1 ((rational) cup -length) . Th e (r ation al) cu p-length c 0 ( X ) of a path-c onnected to p ological space X is the greatest n um b er n ∈ Z ∪ {∞} COMPUT A TIONAL COMPLEXITY O F TOP O LOGICAL INV ARIANTS 5 suc h that th ere are cohomology classes [ x 1 ] , . . . , [ x n ] ∈ H > 0 ( X, Q ) satisfying [ x 1 ] ∪ . . . ∪ [ x n ] 6 = 0, i.e. c 0 ( X ) = sup { n ∈ N 0 | ∃ [ x 1 ] , . . . , [ x n ] ∈ H > 0 ( X, Q ) : [ x 1 ] ∪ . . . ∪ [ x n ] 6 = 0 } Let us now in tro duce the notion of Lusternik–Schnirelmann category and its rational analogue. A subs et U ⊆ X of a top ologica l sp ace X is called c ontr actible in X if its inclusion i : U ֒ → X is h omotopic to a constan t map. Definition 2.2 ((rational) Lusternik–Sc hnirelmann category of a space) . The Lusternik–Schnir elmann c ate gory cat X of a top ologica l space X is the least num b er m ∈ Z ∪ {∞} su c h that X is the union of m + 1 op en subsets U i , eac h contract ible in X . The r ational Lusternik–Schnir elmann c ate gory of X is the least n um b er m ∈ Z ∪ {∞} suc h th at X ≃ Q Y and cat Y = m . W e sh all mainly dra w on the d efinition of ca tegory in the setting of Sul- liv an algebras. In order to pro vide a d efinition in this case we supp ose that (Λ V , d) is a Sulliv an algebra a nd th at m ≥ 1. T aking the quotien t of (Λ V , d) b y all the elemen ts Λ >m V of wo rdlength larger than m induces the structur e of a commutat iv e co c hain algebra for (Λ V / Λ >m V , d ). T his is du e to the fact that th e differentia l d is a deriv at ion. The surjection f m : (Λ V , d ) → (Λ V / Λ V >m , d) extends to a mo del ϕ m of f m . (Λ V , d) f m ( ( P P P P P P P P P P P P   i m / / (Λ V ⊗ Λ Z ( m ) , d ) ϕ m ≃   (Λ V / Λ >m V , d ) (1) With this nota tion w e mak e Definition 2.3 (Lu s ternik–Sc hnirelmann cate gory o f an alge bra) . The Lusternik–Schnir elmann c ate gory cat(Λ V , d) of a S ulliv an algebra (Λ V , d) is the least num b er m ∈ Z ∪ {∞} such that there is a co c hain algebra morphism p m : (Λ V ⊗ Z ( m ) , d ) → (Λ V , d) suc h th at p m ◦ i m = id Λ V . The r ational category of a simply-connected topological space with r a- tional h omology of finite t yp e equ als the category of a resp ectiv e S ulliv an mo del (Λ V , d )—cf. pr op ositio n [ 2 ].29.4 , p. 386. Clearly , one alw a ys has cat 0 X ≤ cat X . F or simply-connected CW- complexes one ob tains that cat 0 X = cat X Q (cf. prop osition [ 2 ].28.(i), p . 371). Let us even tually briefly comment on T o omer’s in v arian t. 6 MANUEL AMANN Definition 2.4 ((rational) T o omer inv arian t) . The T o omer invariant e( X, Q ) of a top ol ogical space is the least n um ber m for wh ic h there is a con tin u ous map f : Z → X from an n -cone Z (cf. the definition on [ 2 ], p. 359) suc h th at H ∗ ( f , Q ) is injectiv e. The r atio nal T o omer i nvariant e 0 ( X ) is the least num b er m such that X ≃ Q Y and e( Y , Q ) = m . In the notation of diagram 1 we defin e the T o om er invariant of a Sul livan algebr a e(Λ V , d ) to b e the least v alue m ∈ Z ∪ {∞} suc h that H ( f m , Q ) is injectiv e. Again, the rational T o omer inv ariant of a simply-connected top ological space with rational homology o f finite type equals the T o omer inv arian t of a resp ectiv e Sulliv an mo d el (Λ V , d )—cf. prop osition [ 2 ].29.4, p. 386. W e remark that in the light of the cited resu lts w e may use results for- m ulated for simply-connected top ological spaces with r ational homology of finite t yp e and translate them to resp ect iv e S ulliv an mo dels. 3. Proof of th eorem A Let G = ( V , E ) b e a (non-directed) fi nite connected simple graph with v ertices V = { v 1 , . . . , v n } and edges E = { ( e i , e j ) | ( i, j ) ∈ J } f or some ind ex set J . F ollo wing [ 6 ], p . 91, we asso ciate to G and a give n in teger k ≥ 2 a finitely generated pur e Sulliv an algebra by V ev en G,k = h x 1 , . . . , x n i with deg x i = 2 and d x i = 0 for all 1 ≤ i ≤ n , by V odd G,k = h y i,j i ( i,j ) ∈ J with deg y i = 2 k − 3 and b y d y i,j = k X l =1 x k − 1 r x l − 1 s for all ( i, j ) ∈ J . Out of the data giv en by the graph G = ( V , E ) and the constan t k we compute the follo w ing integ ral constant s d G,k :=  (2 k − 3) · | E | − | V | 2  and (with n = | V | ) d ′ n,k :=  n ( n − 1)(2 k − 3) − n 2  (whic h we may incorp orate in the co difi cation of (Λ V G,k , d)). Let us thus asso ciate to G and k , r esp ectiv ely to (Λ V G,k , d) and k , ye t another Sulliv a n algebra (Λ W G,k , d), which extends (Λ V G,k , d). W e set W G,k := V G,k ⊕ h z 1 , . . . z n i COMPUT A TIONAL COMPLEXITY O F TOP O LOGICAL INV ARIANTS 7 with deg z i = 4( d ′ n,k + n ) + 3 and with d z i = x 4( d ′ n,k + n +1) i for all 1 ≤ i ≤ n . Ob viously , this algebra ca n b e constructed out of the algebra (Λ V G,k , d) and the v alue k in p olynomia l time. Remark 3.1. The fin itely-generate d algebra (Λ W G,k , d) is pure an d even elliptic, i.e. its cohomology is fin ite-dimensional. T his easily follo ws from the fact that, b y construction, eac h form in W ev en G,k —whic h necessarily de- fines a cohomolog y class, since (Λ W G,k , d) is pur e—represent s a nilp otent cohomology class [ w ] ∈ H (Λ W G,k , d). Indeed, w e hav e [ x i ] 4( d ′ n,k + n +1) = 0 for all 1 ≤ i ≤ n .  W e enco d e the graph G as th e num b er n of its v ertices together w ith the adjacency matrix representing the edges—an ( n × n )-matrix. Recall that w e enco ded sp aces b y their minimal Sulliv an mo dels. F or fixed k , the spaces |h (Λ V G,k , d) i| and |h (Λ W G,k , d) i| , i.e. the spatial realisations (cf. [ 2 ].17) of the constructed minimal Su lliv an algebras, thus ha v e a codification the length of whic h is b ound ed b y a p olynomial in the length of the instance given by the graph. This m eans th at our trans lations from g raphs to algebras can b e done in p olynomial time. As w e remarke d, also the translation from (Λ V G,k , d) to (Λ W G,k , d) can b e ac hiev ed with p olynomial effort. Lemma 3.2. The fol lowing assertio ns ar e e quivalent (i) The Sul livan algebr a (Λ V G,k , d) is el liptic. (ii) The elements [ x i ] ∈ H (Λ V G,k , d) ar e nilp otent for al l 1 ≤ i ≤ n . (iii) F or al l 1 ≤ i ≤ n i t holds th at [ x i ] d G,k +1 = 0 for al l 1 ≤ i ≤ n . Pr oof . W e shall pr o v e that the ellipticit y of the algebra is equiv alent to the n ilp otence of th e [ x i ]. The assertion on th e ord er of the [ x i ] th en can b e deduced as foll o ws: If (Λ V G,k , d) is elliptic, the formal d imension of (Λ V G,k , d) is giv en b y X ( i,j ) ∈ J deg y i,j − X 1 ≤ i ≤ n (deg x i − 1) =(2 k − 3) · | E | − | V | ≤ 2 d G,k b y [ 2 ].32, p. 434. Consequ en tly , b y degree reasons, w e obtain that [ x i ] d G,k +1 = 0 for all 1 ≤ i ≤ n . 8 MANUEL AMANN If (Λ V G,k , d) is elliptic, the classes [ x i ] are necessarily nilp oten t, since H (Λ V G,k , d) is finite-dimensional by d efi nition. Hence it only remains to pro v e the r ev erse im p lication in order to show the lemma. Supp ose that all the x i are nilp ote n t. W e need to sho w that H (Λ V G,k , d) is finite-dimensional. Ho w ev er, this algebra is fi n ite-dimensional o v er Q if and only if it is finite-dimensional o ver C . Thus we ma y assume that we are u sing complex co efficien ts. Since th e co efficien t field then is algebraically closed and the algebra (Λ V G,k , d) is simply-connected with V fin ite-dimensional, w e ma y use the criterion from p r op osition [ 2 ].32.5, p. 439, sa ying that (Λ V G,k , d) is elliptic if and only if ev ery morphism ϕ : (Λ V G,k , d) → ( C [ z ] , 0) is tr ivial—here deg z = 2. Suc h a morp h ism, ho w ever, is trivial on all degrees V > 2 G,k = ( V > 2 G,k ) odd . Supp ose it is not trivial in degree t w o, then it is giv en on a non-zero x ∈ V 2 G,k b y x 7→ αz with α ∈ C . Since th e [ x i ] are nilp ote n t element s, there is a certain p ow er of [ x ] whic h v anishes. In other words, there exists an elemen t ˜ x in Λ V with 0 6 = d ˜ x ∈ C [ x ] ⊆ Λ V G,k This element has o dd degree and ϕ ( ˜ x ) = 0. Thus ϕ do es not commute with differen tials; a con tradiction.  Lemma 3.3. If [ x i ] d G,k +1 = 0 in H (Λ W G,k , d) for al l 1 ≤ i ≤ n , then we obtain an isomorph ism of Sul livan algebr as (Λ W G,k , d) ∼ = (Λ V G,k , d) ⊗ (Λ h z ′ 1 , . . . , z ′ n i , 0) with d eg z ′ i = 4( d ′ n,k + n ) + 3 . Pr oof . Since G is a simple (und irected) graph, there are at most  n 2  edges in E . W e deriv e that d G,k =  (2 k − 3) | E | − n 2  ≤  (2 k − 3) · n ( n − 1) / 2 − n 2  = d ′ n,k and infer the inequalit y 2( d G,k + 1) ≤ 4( d ′ n,k + n ) + 3 (for n ≥ 1). Due to the fact that (Λ W G,k , d) is id entical to (Λ V G,k , d) in degrees b elo w degree 4( d ′ n,k + n ) + 3, we obtain that H ≤ 4( d ′ n,k + n )+3 (Λ V G,k , d) ⊆ H ≤ 4( d ′ n,k + n )+3 (Λ W G,k , d) is a grad ed su balgebra. Consequent ly , since [ x i ] d G,k +1 = 0 in H (Λ W G,k , d) and since deg[ x i ] d G,k +1 = 2( d G,k + 1) ≤ 4( d ′ n,k + n ) + 3 COMPUT A TIONAL COMPLEXITY O F TOP O LOGICAL INV ARIANTS 9 w e conclude that there is an elemen t ˜ x i ∈ (Λ V G,k , d) with deg ˜ x i = 2 d G,k + 1 and with d ˜ x i = x d G,k +1 i for eac h 1 ≤ i ≤ n . It f ollo ws th at for e ac h 1 ≤ i ≤ n the element z ′ i := z i − ˜ x i · x 2 d ′ n,k +2 n − d G,k +1 i ∈ (Λ W G,k , d) is closed and not exact. Th e asserted splitting of differen tial graded algebras is a d irect consequence.  The main tool for pro ving theorem A will b e the follo wing Prop osition 3.4. Supp ose that n, k ≥ 1 . The fol lowing two sta tements ar e e quivalent: (i) It holds th at c(Λ W G,k , d) ≤ d ′ n,k + n (ii) The Sul livan algebr a ( V G,k , d) is el liptic. Pr oof . W e use th e characte risation for the ellipticit y of (Λ V G,k , d) pro vided in lemma 3.2 . If (Λ V G,k , d) is elliptic, i.e. if [ x i ] d G,k +1 = 0 for all [ x i ] ∈ H (Λ V G,k , d) (with 1 ≤ i ≤ n ), then lemma 3.3 yields the isomorphism (Λ W G,k , d) ∼ = (Λ V G,k , d) ⊗ (Λ h z ′ 1 , . . . , z ′ n i , 0) from whic h w e d irectly see that the cup -length of (Λ W , d ) satisfies c(Λ W G,k , d) = c(Λ V G,k , d) + n since deg z ′ i is o dd for 1 ≤ i ≤ n . Since (Λ V G,k , d) is elliptic with an elemen t of minimal degree sitting in de- gree tw o, its cup-length can b e estimated f rom ab o ve by its form al d im en sion divided by t w o, i.e. in particular by c(Λ V G,k , d) ≤ 2 d ′ n,k / 2 = d ′ n,k It f ollo ws th at c(Λ W G,k , d)) ≤ d ′ n,k + n . Con v ersely , if c(Λ W G,k , d) ≤ d ′ n,k + n then, in particular, the elemen ts [ x i ] ∈ H (Λ W G,k , d) satisfy that H (Λ W G,k , d) ∋ [ x i ] d ′ n,k + n +1 = 0 Again w e use that (Λ W G,k , d) is identical to (Λ V G,k , d) in degrees b elo w degree 4( d ′ n,k + n ) + 3 and that H ≤ 4( d ′ n,k + n )+3 (Λ V G,k , d) ⊆ H ≤ 4( d ′ n,k + n )+3 (Λ W G,k , d) is a grad ed su balgebra, therefore. Since deg x d ′ n,k + n +1 i = 2( d ′ n,k + n ) + 2 ≤ 4( d ′ n,k + n ) + 3 for n, k ≥ 1, we conclude that H (Λ V G,k , d) ∋ [ x i ] d ′ n,k + n +1 = 0 10 MANUEL AMANN (with x i no w considered an elemen t in V G,k and with the giv en p o wer of its cohomology class already v anish in g in H (Λ V G,k , d)). Th us all th e elemen ts [ x i ] (for 1 ≤ i ≤ n ) are nilp oten t elemen ts in H (Λ V G,k , d). Due to lemma 3.2 it follo ws that the algebra (Λ V G,k , d) is elliptic.  The pr oblem of k -colouring a graph, i.e. attribu ting one of k differen t colours to a v ertex s u c h that adj acent vertices ha v e differen t colours, P ′′ 2 : Supp ose that k ≥ 3. Is th e graph G k -colourable? is kno wn to b e N P -complete (for k ≥ 3)—for example cf. [ 3 ]. In the pro of of co rollary [ 6 ].4, p. 92, it is sho wn t hat there is a p olynomial reduction of P ′ 2 to the problem P ′ 2 : Giv en a s im p ly-connected Sulliv an algebra (Λ V , d) with dim V < ∞ . Is it elliptic? More precisely , the pr oblem is redu ced to P 2 : Let (Λ V G,k , d) b e as constructed ab o ve. Do es it constitute an elliptic algebra? W e are no w ready to giv e the Pr oof of theorem A . W e consider the follo wing decision p r oblem. P 3 : Let (Λ W G,k , d) b e an algebra as constru cted ab ov e. Is the cup-length of (Λ W G,k , d) smaller than or equal to d ′ n,k + n ? There is a p olynomial reduction of problem P ′′ 2 to p roblem P 2 . Problem P ′′ 2 is NP -complete (for k ≥ 3). By pr op osition 3.4 we can reduce pr oblem P 2 to problem P 3 in p olynomia l time. Hence we see that P 3 is NP -hard . Ho w ever, the original problem P ′′ is obvio usly harder than P 3 ; thus it is NP -hard.  4. Proof of th eorem B The pro of of theorem B will pro ce ed along the lines of the pro of of theorem A . Prop osition 4.1. Supp ose that n, k ≥ 2 . The fol lowing two sta tements ar e e quivalent: (i) It holds th at cat 0 (Λ W G,k , d) ≤ d ′ n,k + n (ii) The Sul livan algebr a ( V G,k , d) is el liptic. Pr oof . Again, we use the c haracterisation for the ellipticit y of (Λ V G,k , d) pro vided in lemma 3.2 . COMPUT A TIONAL COMPLEXITY O F TOP O LOGICAL INV ARIANTS 11 If (Λ V G,k , d) is elliptic, i.e. if [ x i ] d G,k +1 = 0 for all [ x i ] ∈ H (Λ V G,k , d) (with 1 ≤ i ≤ n ), then lemma 3.3 yields the isomorphism (Λ W G,k , d) ∼ = (Λ V G,k , d) ⊗ (Λ h z ′ 1 , . . . , z ′ n i , 0) Using theorem [ 2 ].30.2.(ii ) we compute the category of the tensor pro d u ct as the sum of th e categorie s cat(Λ W G,k , d) = cat (Λ V G,k , d) + cat(Λ h z ′ 1 , . . . , z ′ n i , 0) Since (Λ h z ′ 1 , . . . , z ′ n i , 0) is a form al algebra of cup-length n , its category also equals n by example [ 2 ].29.4, p. 388. (Since this example is formulat ed for sp aces, we observ e that the rational catego ry of a simply-connected space with rational homology of finite type is the category of its minimal Su lliv an mo del—cf. prop osition [ 2 ].29.4, p. 386.) Since (Λ V G,k , d) is elliptic with an elemen t of minimal degree sitting in de- gree t w o, its Lusternik–Sc hnirelmann category can b e estimated fr om ab o v e b y its f ormal dimension divided b y tw o—cf. coroll ary [ 2 ].29.1, p.385. In p articular, w e ob tain cat(Λ V G,k , d) ≤ 2 d ′ n,k / 2 = d ′ n,k It f ollo ws th at cat(Λ W G,k , d)) ≤ d ′ n,k + n . Con v ersely , we assume that cat(Λ W G,k , d)) ≤ d ′ n,k + n . Either a straigh t- forw ard direct c hec k or a quote of prop osition [ 2 ].30.8 .(ii), p. 410, for the trivial fibration with F = X and Y = {∗} yields t hat d ′ n,k + n ≥ cat(Λ W G,k , d) ≥ c 0 (Λ W G,k , d) Th us p rop osition 3.4 yields that (Λ W G,k , d)) is elliptic.  Hence w e m a y establish the Pr oof of theorem B . W e consider the decision p roblem Q 3 : Let (Λ W G,k , d) b e an algebra as constructed ab o ve. Is th e Lustern ik– Sc hnirelmann catego ry of (Λ W G,k , d) sm aller than or equ al to d ′ n,k + n ? Again one uses the p olynomial reduction of P ′′ 2 to P 2 and the fact that P ′′ 2 is NP -c omplete. Due to prop ositio n 4.1 w e reduce P 2 to Q 3 . Thus Q 3 is NP -hard. Again, the original p roblem Q ′′ is hard er than Q 3 , i.e. it is NP -hard, in particular.  Referen ces [1] D. J. Anick. The computation of rational homotop y groups is # P -hard. In Com puters in ge ometry and top olo gy (Chic ago, IL, 1986) , volume 114 of L e ctur e Notes in Pur e and Appl. Math. , pages 1–56. Dekker, New Y ork, 1989. [2] Y. F´ elix, S. Halperin, and J.-C. Thomas. R ational homoto py the ory , volume 205 of Gr aduate T exts in Mathematics . Sp ringer-V erlag, New Y ork, 2001. [3] M. R. Garey and D. S. Johnson. Computers and intr actabili ty . W. H . F reeman and Co., San F rancisco, Calif., 1979. A guide to the th eory of NP-completeness, A S eries of Books in t h e Mathematical Sciences. [4] A. Garv ´ ın and L. L echuga. The compu tation of th e Betti num bers of an elliptic space is a NP-hard p roblem. T op olo gy A ppl . , 131(3):235– 238, 2003. 12 MANUEL AMANN [5] L. Lech uga. A Groeb n er basis algorithm for comput ing t h e rational L.-S. category of elliptic pure spaces. Bul l. B el g. Math. So c. Sim on Stevin , 9(4):533–544, 2002. [6] L. Lech uga and A . Murillo. Complexity in rational homotopy . T op olo gy , 39(1):89–94, 2000. Manuel Amann Dep ar tment o f Ma thema tics University of Tor onto Ear th Sciences 2146 Toronto, Ont ario M5S 2E4 Canada mamann@uni-muens ter.de http://indi vidual.utoronto.ca/mamann/