aa r X i v : . [ m a t h . K T ] D ec Algorithms in A ∞ -algebras Mikael Vejdemo-JohanssonDecember 3, 2019
To Tornike Kadeishvili
Abstract
Building on Kadeishvili’s original theorem inducing A ∞ -algebra structures on the homologyof dg-algebras, several directions of algorithmic research in A ∞ -algebras have been pursued.In this paper we will survey work done on calculating explicit A ∞ -algebra structures fromhomotopy retractions; in group cohomology; and in persistent homology. In his 1980 paper [18], Kadeishvili proved that the homology of any dg-algebra has an induced A ∞ -algebra structure. The proof itself is by induction on the arity of the operation, writingout explicitly how to create m n ( a , . . . , a n ) in terms of lower order operations. By giving theseexpressions, the proof is basically a formulation of what an algorithm for the computation of an A ∞ algebra structure on the homology of a dg-algebra would look like.Even though this initial setup has a strongly algorithmic flavor, actual algorithms and computer-supported calculations of A ∞ -algebra structures emerged far later. In this paper, we plan to givean overview of work on creating software computing A ∞ -algebra structures and the contexts andtechniques used for these. Merkulov [24] has given Kadeishvili’s construction a more concrete,and combinatorially accessible presentation – but the focus for this paper is in explicit algorithmiccomputation with computer implementations, preferably publically accessible. We fix a ring k . All tensor products are over k unless otherwise noted, and tensor powers aredenoted by A ⊗ n = A ⊗ · · · ⊗ A .A graded k -vector space A is an A ∞ -algebra if one of the following equivalent conditions hold1. There is a family of maps µ i : A ⊗ i → A , called higher multiplications fulfilling the Stasheffidentities St n : X i X j µ i ◦ j µ n − i = 0 .
2. There is a family of chain maps from the cellular chain complex of the associahedra toappropriate higher endomorphisms of Aµ n : C ∗ ( K n ) → Hom( A ⊗ n , A ) . A is a representation of the free dg-operad resolution A ss ∞ of the associative operad.An A ∞ -coalgebra is defined by dualizing this definition.The structure was introduced by Jim Stasheff in [27]. Good introductory surveys have beenwritten by Lu, Palmieri, Wu and Zhang [21, 22] as well as by Bernhard Keller [19, 20].A graded k -vector space A is a differential graded algebra (dg-algebra) if it is equipped with adifferential operator ∂ : A → A of degree − m : A ⊗ A → A ofdegree 0, such that the Leibniz rule holds: ∂m ( x, y ) = m ( ∂x, y ) + ( − | x | m ( x, ∂y )1 module over a dg-algebra A is a graded vectorspace M with a differential operator ∂ : M → M and an associative multiplication m : M ⊗ A → M that obeys the Leibniz rule.Kadeishvili proved in his 1980 paper [18] that the homology of a dg-algebra has an inherited andquasi-isomorphic A ∞ -algebra structure. The proof starts out defining µ = 0 and µ ([ x ] ⊗ [ x ]) =( − | x | +1 [ x · [ x ], as well as starting to define an A ∞ -morphism f by f simply a cycle-choosinghomomorphism.The proof proceeds by induction : if all m j and f j have been defined for j < i , then let U n = n X s =1 m ◦ f s ⊗ f n − s ) + n − X k =0 n − X j =2 f n − j +1 ◦ ⊗ k ⊗ m j ⊗ ⊗ n − j − k +1 The Stasheff axioms can be translated to m ◦ f n = ( f ◦ m n − U n )The right hand side is homological to zero, so we can pick f n to be a bounding element of thisdifference. Extend by linearity.The structure of this argument lends itself excellently well to concrete and algorithmic calcula-tions, and there has been a few approaches to algorithmic and computer-aided A ∞ -algebra work.There are three main topics that emerge, and we will dedicated a chapter to each of them.First up, in Section 3, we will review Ainhoa Berciano’s work on contractions of dg-algebras todg-modules, with implementations in the computer algebra system Kenzo.Next, in Section 4 we will go to the realm of group cohomology. Mikael Vejdemo-Johanssonworked on algorithms to directly calculate A ∞ -algebra structures on the modular group cohomologyof p -groups, and generated a number of ways to recognize feasibility of the calculation as wellas a stopping criterion. Stephan Schmid, later on, used some of Vejdemo-Johansson’s resultsin a concrete calculation of the A ∞ -algebra structure on the modular group cohomology of thesymmetric group S p .Finally, in Section 5, we will describe recent work by Murillo and Belch´ı on using A ∞ -coalgebrastructures on persistent homology rings to create new perspectives on bottleneck distances andstability of persistence barcodes. Ainhoa Berciano’s work [3–6, 8] starts with perturbation theory. This framework has as its coreresult the Basic Perturbation Lemma [10], that describes how a contraction changes under pertur-bation.A contraction connects two dg-modules M and N , abstracting the homotopy concepts of de-formation retracts. A contraction consists of morphisms f : N → M , g : M → N and φ : N → N such that f and g are almost an isomorphism – up to a homotopy operation in N . In other words,we require f g = M gf + φ∂ N + ∂ N φ = N f φ = 0 φg = 0 φφ = 0A contraction preserves homology: H ( N ) is canonically isomorphic to H ( M ), but this isomor-phism tends not to transfer algebraic structures from N to M .If the differential structure on N is perturbed: instead of boundary operator ∂ N , N is equippedwith a new boundary operator ∂ N + δ , then the Basic Perturbation Lemma produces a new con-traction. The requirement for this construction is that φδ is pointwise nilpotent: for any x ∈ N there is an n so that ( φδ ) n ( x ) = 0.Then there is a new contraction f δ , g δ , φ δ between N equipped with the boundary operator ∂ N + δ and M equipped with the boundary operator ∂ M + ∂ δ given by ∂ δ = f δ X i ≥ ( − i ( φδ ) i g f δ = f − δ X i ≥ ( − i ( φδ ) i φ g δ = X i ≥ ( − i ( φδ ) i g φ δ = X i ≥ ( − i ( φδ ) i φ which translates well to recursion: this is how algorithms enter the picture A ∞ -coalgebra structures between DG-modulesusing the tensor trick [16]. The tensor trick starts with a dg-coalgebra C , a dg-module M anda contraction from C to M . Take the tensor module of the desuspension of all components inthis contraction to produce a new contraction. With the cosimplicial differential, we can use theBasic Perturbation Lemma and obtain a new contraction. The tilde cobar differential generatesan induced A ∞ -coalgebra structure, explicitly given by the comultiplication operations∆ i = ( − [ i/ i +1 f ⊗ i ∆ [ i ] φ [ ⊗ ( i − . . . φ [ ⊗ ∆ [2] g ; ∆ [ k ] = k − X i =0 ( − i ⊗ i ⊗ ∆ ⊗ k − i − (1)This derivation allows Berciano to prove [6] that in H ∗ ( K ( π, n ); Z p ) for a finitely generatedabelian group π , the only non-null morphisms in the A ∞ -coalgebra structure have to have order i ( p −
2) + 2 for some non-negative integer i .The final formula in 1 is concrete enough that it has been implemented on the computeralgebra platform Kenzo in the packages ARAIA ( A lgebra R eduction A - I nfinity A lgebra) andCRAIC ( C oalgebra R eduction A - I nfinity C oalgebra). Fix a group G and a field k . The cohomology algebra of the Eilenberg-MacLane space H ∗ ( K ( G, k G ( k , k ) of the group ring and is called the group cohomology H ∗ ( G ). Because of the connection to the Ext-algebra, the group cohomology can be calculatedfrom the composition dg-algebra of Hom( F ∗ , F ∗ ) for a free resolution F ∗ → k in the categoryof G -modules. Several computer algebra systems, including Magma [9] and GAP [14] supportcalculations with G -modules. In such a system, we create a free resolution F ∗ of k . Chain maps F ∗ → F ∗ are then represented by a sequence of maps, one for each degree, each determined bylower-dimensional maps through commutativity of the corresponding squares in the chain mapdiagram. With Hom( F ∗ , F ∗ ) represented, we can compute H ∗ G as H ∗ Hom( F ∗ , F ∗ ).Since the Hom( F ∗ , F ∗ ) is a dg-algebra, by Kadeishvili’s theorem, H ∗ G has an induced A ∞ -algebra structure.Vejdemo-Johansson [28–30] studies this A ∞ -algebra structure from a strictly algorithmic per-spective. A ∞ A cornerstone of Vejdemo-Johansson’s approach to computing A ∞ -algebras is the following theo-rem ([29, Theorem 3]): Theorem 1. If A is a dg-algebra and1. There is an element z ∈ H ∗ A generating a polynomial subalgebra (ie is not a torsion element)2. H ∗ A is a free k [ z ] -module3. H ∗ A has a k [ z ] linear A n − -algebra structure induced by the dg-algebra structure on A , suchthat f ( z ) f k ( a , . . . , a k ) = f k ( a , . . . , a k ) f ( z )
4. We have a chosen k [ z ] -basis b , . . . of H ∗ A and all m k ( v , . . . , v k ) and f k ( v , . . . , v k ) arechosen by Kadeishvili’s algorithm for all combinations of basis elements v j ∈ { b , . . . } Then a choice of m n and f n by Kadeishvili’s algorithm for all input values taken from this k [ z ] -basis extends to a k [ z ] -linear A n -algebra structure on H ∗ A induced by the dg-algebra structureon A . The condition 3 says that for the A ∞ -morphism H ∗ A → A produced by Kadeishvili’s construc-tion, the cycle chosen for z commutes – on a chain level – with each chain map chosen for thehigher operations. This is the key condition for the theorem – and also the one that makes thetheorem most fragile.The theorem tells us we can construct an A ∞ -algebra structure step by step. If there is oneof these non-torsion central elements z , we can reduce the complexity of H ∗ A for the purpose of3alculating its higher operations – if we find a family of central elements z , . . . , z k such that H ∗ A is a finite module over k [ z , . . . , z k ], then it is enough to study the finitely many basis elements b , . . . , b m in a presentation of H ∗ A as a k [ z , . . . , z k ]-module. This makes each calculation a finite(though large) in terms of the number of input combinations that need to be studied. Using thistheorem is easier if – as is the case for cyclic groups – the resolution F ∗ is periodic.Theorem 1 makes it easier to extend from A n − to A n , using a condition that can be checkedfor each extension step. Once the condition – commutativity of the representative chain maps –fails, the structure calculated thus far is valid, but further extensions are obstructed. The key tobring computational effort down to a finite time endeavour lies in [29, Theorem 5]: Theorem 2.
Let A be a dg-algebra. If in an A q − -algebra structure on H ∗ A , f k = 0 and m k = 0 for all q ≤ k ≤ q − then the A q − -structure is already an A ∞ -structure with all higher f n andall higher m n given by zero maps. Through finding central elements, the infinitely many basis elements of H ∗ A can be broughtdown to a finite number of basis elements to check. And by finding a large enough gap, in whichall chain representations and all products vanish, the computation can be terminated producing aresult.This approach was implemented as a module distributed with Magma [9], and was used bothto confirm Madsen’s [23] computation of A ∞ -algebra structures on the group cohomology of cyclicgroups and to conjecture [30] the start of an A ∞ -structure on the cohomology on some dihedralgroups. In [25], Saneblidze and Umble gave an explicit construction for a diagonal on the associahedra.This construction translates directly to a method to combine A ∞ -algebra structures on V and W into an A ∞ -algebra structure on V ⊗ W .Vejdemo-Johansson uses this construction in [28] to prove non-triviality of some operations on H ∗ ( C n × C m ). From results by Berciano and Umble [7], we know that any non-trivial operation onthis group cohomology of arity less than n + m − , n, m or n + m −
2. Bercianoalso shows [3] that any non-zero higher coproduct on H ∗ ( C q × C q ) has arity k ( q −
2) + 2 for some k .In addition to the induced operations in arities 2 , n, m and n + m −
2, Vejdemo-Johansson showsthat there are non-trivial operations of arity 2 n + m − n + 2 m −
4. The original article statesa far more generous claim: that all the arities k ( n −
2) + k ( m −
2) + 2, ( k − n −
2) + k ( m −
2) + 2and k ( n −
2) + ( k − m −
2) + 2 have non-zero operations – this argument turned out to have asubtle flaw, and was retracted. More details are available in [30].Any practical use of the Saneblidze-Umble diagonal would benefit greatly from a computer-facilitated access to the coefficients of the diagonal construction. In an unpublished preprint [31],Vejdemo-Johansson provides a computer implementation of an algorithm to enumerate the Saneblidze-Umble terms.
Schmid [26] studies the group cohomology of the symmetric group S p on p elements, with coeffi-cients in the finite field F p with p elements. For this group cohomology, he presents a basis withwhich he is able to prove that the only non-trivial A ∞ -operations on H ∗ S p are of arity 2 and p .To do this, he goes through large and somewhat onerous explicit calculations to show that thereis a periodic projective resolution of F p over F p S p , and that the resolution has a large enough gapto allow the use of Vejdemo-Johansson’s theorem. Persistent homology and cohomology form the cornerstone of the fast growing field of TopologicalData Analysis. The fundamental idea is to study the homology functor applied to diagrams oftopological spaces V : V ֒ → V ֒ → . . . These spaces are often generated directly from datasets, by constructions such as the ˇCechconstruction: for data points X = { x , . . . , x N } , an abstract simplical complex ˇ C ǫ has as its4ertices X and includes a simplex [ x i , . . . , x i d ] precisely if the intersection of balls T dj =0 B ǫ ( x i j ) isnon-empty. If ǫ increases, no intersections will become empty, and so no simplices will vanish. Sothe ˇCech complexes, as ǫ sweeps from 0 to ∞ , generates a nested sequence of topological spaces.The inclusion maps ι ji : V i → V j lift by functoriality to linear maps on homology: H ∗ ( ι ji ) : H ∗ V i → H ∗ V j . We may define a persistent homology group as the image P H i,j ∗ ( V ) = img H ( ι ji ).For more details on the data analysis side, we recommend the surveys [11, 15, 32] As the homology functor is applied to the diagram of topological spaces, using coefficients froma field k for the homology computation, the result is a diagram of vector spaces. By eitherimbuing the resulting diagram with the structure of a module over the polynomial ring k [ t ], oras representations of a quiver Q of type A n , the corresponding classification theorems produce adecomposition of H ∗ ( V ) into a direct sum of interval modules . These interval modules are N -gradedmodules defined by a pair of indices b, d , and are defined as 0-dimensional for degrees k < b andfor degrees k > d . For degrees from b to d , the interval module is one-dimensional, with identitymaps connecting each space to the next.Thus, the homology of a diagram V of topological spaces with field coefficients can be describedby a multiset Dgm( V ) = { ( b i , d i ) } i ∈ I , called the persistence barcode or persistence diagram of thediagram V . The dimension of the persistent homology group P H i,j ∗ ( V ) is exactly the number ofintervals ( b k , d k ) in Dgm( V ) such that b k ≤ i ≤ j ≤ d k .Between any pair of such diagrams we can create a distance called the bottleneck distance .Setting ∆ = { ( x, x ) : x ∈ R } , this distance is defined by: d B (Dgm( V , W )) = inf γ : V ∪ ∆ ∼ −→ W ∪ ∆ max v ∈ V ∪ ∆ d ( v, γv )This distance measures the largest displacement needed to change Dgm( V ) into Dgm( W ), whileallowing intervals to disappear into and emerge from the infinite set of possible 0-length intervals.First with the bottleneck distance (and in later research more sophisticated), a range of stabilitytheorems have been proven, starting in [13]. Good overviews can be found in [12, 32]. Thesetheorems take the shape of Theorem 3 (Stability meta-theorem) . If d ( V , W ) < ǫ , then d ′ (Dgm( H ( V )) , Dgm( H ( W ))) < ǫ forspecific choices of distances d and d ′ . A ∞ in persistence Murillo and Belch´ı introduced in [1, 2] an A ∞ -coalgebra approach to barcode distances. If eachnew cell introduced in the step from V i to V i +1 , with all the V j chosen to be CW-complexes, andworking over the rationals Q , then there is a set of compatible choices of A ∞ -coalgebras for theentire sequence.They define a ∆ n -persistence group∆ n P H i,j ∗ ( V ) = img( H ∗ ( ι ji ) | T jk = i ker(∆ kn ◦ ι ki In other words, the ∆ n -persistence group retains from the ordinary persistence groups preciselythose elements out of H ∗ ( V i ) whose images in each V k vanish under application of the highercoproduct ∆ n .These ∆ n -persistence groups generate ∆ n -persistence barcodes as multisets Dgm ∆ n ( V ) of in-tervals [ b, d ]. From these barcodes, the dimension of ∆ n P H i,j ∗ ( V ) equals the number of [ b k , d k ] ∈ Dgm ∆ n ( V ) such that b k ≤ i ≤ j ≤ d k . As pointed out by the authors, these higher order barcodesmay “flicker” in a way that classical persistence strictly avoids: the same element can exist overseveral disjoint intervals. This has been carefully avoided in the greater literature on persistenthomology: the flickering behavior invites wild representation theories, where the decompositionthat generates barcodes is no longer available. 5 .3 A ∞ bottleneck distance Herscovich [17] introduces a novel metric on persistent homology. Herscovich constructs a metricon locally finite Adams graded minimal A ∞ -algebras, and then quotients by quasi-isomorphism toestablish a metric on persistent homology barcodes equipped with an A ∞ -algebra structure.The question of stability of this metric is left open by Herscovich, except to note that the 1-arycase coincides with the classical bottleneck distance. References [1] Francisco Belch´ı. “Optimising the topological information of the A ∞ -persistence groups”. 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