On the non-existence of torus actions

Topology and its Applications 209 (2016) 347–366

Contents lists available at ScienceDirect

Topology and its Applications www.elsevier.com/locate/topol

On the non-existence of torus actions José La Luz a , David Allen b,∗ a

Departamento de Matemáticas, Universidad de Puerto Rico, Industrial Minillas 170 Carr 174, Bayamón, PR, 00959-1919, Puerto Rico b Department of Mathematics, Borough of Manhattan Community College, CUNY, 199 Chambers Street, New York, NY 10007, United States

a r t i c l e

i n f o

Article history: Received 1 March 2016 Received in revised form 1 June 2016 Accepted 7 June 2016 Available online 11 June 2016 MSC: primary 14M25 secondary 57N65 Keywords: Algebraic topology Quasitoric manifolds Higher derived functor Cotangent complex Rigidity Torus actions

a b s t r a c t In this paper we compute the higher left derived functors of the indecomposable functor, in certain degrees, for a general class of algebras. The techniques do not depend on the existence of a Projective extension sequence—a common method used to make such computations throughout the literature and as a result, we generalize all known computations of these higher derived functors. As a result of these calculations we have the following applications. First, we prove that certain torus actions on a Quasitoric manifold are restricted due to the combinatorial structure of the orbit implying the non-existence of certain torus actions. Second, we obtain a refinement of the class of equivariantly formal torus actions, splitting them into two classes. In the context of augmented simplicial algebras over a field, the results provide explicit computations of the cotangent complex. We also show very explicitly how one higher derived functor depends on another and in the case of Toric Topology, the generators of these derived functors are linked backed to the combinatorics of the orbit. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Computing the homotopy groups of a Quasitoric manifold and related toric spaces – Borel space and Moment angle complex – is a challenging problem. General results, in particular, high dimensional calculations are sparse within the literature. In earlier work, Buchstaber and his collaborators used cellular-type arguments along with the notion of neighborliness of the orbit space to make a specific high dimensional homotopy calculation [11]. These results gave a particular homotopy group of the Moment angle complex in terms of the combinatorics of the orbit, and this could, by way of a certain fibration, give information on the higher homotopy groups of a Quasitoric manifold. There were also π2 calculations and other miscellaneous computations for the homotopy of the Borel space (which for a Quasitoric manifold M endowed with a torus action is ET ×T M ). In this paper, we are primarily concerned with a Quasitoric manifold M . Specific * Corresponding author. E-mail addresses: [email protected] (J. La Luz), [email protected] (D. Allen). http://dx.doi.org/10.1016/j.topol.2016.06.005 0166-8641/© 2016 Elsevier B.V. All rights reserved.

348

J. La Luz, D. Allen / Topology and its Applications 209 (2016) 347–366

calculations require that we write M (λ) and we will do so when necessary or convenient. We briefly recall, λ is a map of integer lattices that depends on the orbit space [11,16], satisfying certain conditions – what is commonly referred to as condition (∗). There are some interesting observations that arise by way of the long exact sequence of homotopy groups induced from the fibration M → ET ×T M → BT where BT is the classifying space. For instance, by leveraging methods from Toric Topology, there are specific homotopy calculations for CP n CP n and other complicated spaces, at least from a π∗ -computational perspective. Various homotopy-theoretic calculations are strewn throughout the literature, but many of them are collected in [1] and [11]. Before moving on to unstable spectral sequences, we bring to the reader’s attention research in the stable range, specifically, splittings and other decompositions. There is an ongoing and active program being undertaken by Fred Cohen, Martin Bendersky and their collaborators in regards to stable splittings coming from the polyhedral functor. This line of inquiry is described in [8], among others by the said authors. In this setting, many toric spaces can be decomposed into a certain suspension of a wedge which can under certain limited cases, shed insight into homotopy calculations. Another approach to obtaining homotopy information for these Toric spaces and in particular, Quasitoric manifolds, is to set-up and compute, if possible, certain unstable spectral sequences. When referring to these spectral sequences, we are primarily concerned with Toric-Topological applications [1]. For a complex orientable theory E, there is a certain Composite Functor Spectral Sequence (CFSS) [9]. The E∞ -term of this spectral sequence is the associated graded group of the E2 -term of the Unstable Adams Novikov Spectral Sequence [9]. In regards to the CFSS, one part of the input are the higher right derived functors of the primitive element functor Ri P E∗ (M ) where E∗ (M ) is the E-homology of a Quasitoric manifold M thought of as a certain coalgebra. There are the usual complications associated to such objects and the determination of the spectral sequence; namely, computing the E2 -term, computing the differentials and resolving any potential extension issues. However, there are additional difficulties, the least of which is computing Ri P E∗ (M ), which is a very challenging problem. It is also related, as we will elaborate below – in the Main Results section – to a variety of other deep problems, such as computing the cotangent complex, in a certain setting. In the context of Toric Topology there are more complications, such as the complexity of the cohomology ring of quasitoric manifolds—a free algebra modulo a complicated ideal. As a consequence, many of the tools break-down at various stages because most of these spectral sequences work “reasonably” well for spheres and other related spaces such as loops on spheres, but the cohomology ring of these spaces are less complicated than that of a Quasitoric manifold. More specifically, the algebra structure is simpler. Assuming these challenges can be met, there is the task of computing the unstable coaction on the E-homology for these manifolds in the requisite categories—unstable U-comodules and unstable G-coalgebras [9]. We will not stress these comodule structures here, but the interested reader can refer to [1] for coaction formulae. There are methods to compute these higher derived functors, chief among them is the long exact sequence of Ri P E∗ (−) induced from an injective extension sequence, which is, roughly speaking, a short exact sequence in the category of coalgebras over a graded ring [12]. Since coalgebras are often less intuitive, there is the dual notion of projective extension sequence where one speaks of the induced long exact sequence of the higher left derived functors of the indecomposable functor Q i.e., Li QE ∗ (−). If the algebras are of finite type, then one computation can be determined from another. Once again, there is the problem that the conditions required to be projective extension sequence fail almost always, except in the most limited circumstances. That is, the projective or more generally, the freeness condition does not hold. The current theory does not address this issue in a manner that makes explicit computations feasible in our setting. However, the results in this paper do not require any such conditions to hold and we are able to make very explicit computations in a very general setting. We briefly summarize our approach. The chain complex used to compute the higher left derived functors Li QE ∗ (−) is written out explicitly and we develop a bookkeeping mechanism to keep track of cycles as

J. La Luz, D. Allen / Topology and its Applications 209 (2016) 347–366

349

the filtration increases. These explicit calculations, especially in low degrees, allow for direct computations to be made without reference to a projective extension sequence. The cycles representing generators of the higher left derived functors of Q are written down explicitly and in the special case that our algebra is the E-cohomology of a Quasitoric manifold M (λ) the generators are related to the combinatorics of the orbit – (M (λ)/T ). In this way, it is possible to track how the combinatorics manifests in the chain complex and to record how one derived functor depends on another in terms of how the underlying combinatorics is stacked from level to level in the complex. Direct calculations are made in degrees zero, one and two and the machinery can be pushed to make additional computations. However, there is a trade-off; while complete calculations are made in certain degrees, the bookkeeping becomes very difficult and recording classes in higher filtration becomes exceedingly tedious. This is evident when trying to compute, say, the third higher derived functor. One reason for this is the lack of structure inherent in the bookkeeping. The authors are optimistic that additional methods can be used to push the calculations and methods even further. Surprisingly, it turns out that the higher left second derived functor – L2 QE ∗ (M (λ)) – can be used to make statements about torus actions. First a bit of context to explain how this line of questioning came up. During his thesis work under the supervision of Martin Bendersky in 2005, the second author was trying to understand the differences, from a spectral sequence perspective, between the Quasitoric manifold over the square and the Quasitoric manifold over the pentagon, the latter being the connect sum of the Hirzebruch surface with CP 2 . In the first case, the Composite Functor Spectral Sequence was computable, in a sense, since Ri P (M (λ)) = 0 when i > 1 because M (λ) = S 2 × S 2 . Whereas, in the second case, there is not a vanishing of these higher derived functors. In hindsight there are many reasons for this; however, Bendersky asked whether or not one can find a lambda map that would give the cohomology ring of the Quasitoric manifold the structure of an algebra dual to a nice homology coalgebra [12]. This makes sense since the cohomology ring of M (λ) is a quotient of the Stanley–Reisner ring by an ideal generated by linear forms essentially coming from the torus action on the manifold. Simply put, he wanted to know if it was possible to find a lambda to make E∗ (M ) nice as a coalgebra in the sense of [12]? One application of the results in this paper is to address this question and to determine combinatorial conditions for which we can answer in the negative. It is likely that Bendersky’s question was motivated by a visit to Princeton University in 2005 where certain results contained in [1] were presented and Bill Browder asked whether or not the given homotopy calculations can say anything interesting about the torus action. This question still motivates a line of inquiry from the authors. As a second application we are able to split the collection of equivariantly formal torus actions on Quasitoric manifolds into two groups; those that are nice and those that are not using our determination of Li QE ∗ (M (λ)). Finally, the results in this paper are related to cotangent complex calculations. Since we are working within the context of augmented simplicial algebras over a commutative ring, we reduce to the case when the ring is a field k, then the results presented here are explicit structural theorems for the Dn (A|k; k) for n = 0, 1, 2, [21] (p. 295, Definition 8.8.2), [6]. Both authors are grateful to Pete Bousfield for pointing out this explicit relation in a private communication. 1.1. Main results Before listing the main results we set some notation and recall a few key definitions which can be found in later sections of the paper – §2, §3. Let A be a commutative ring with unit and suppose B is a graded augmented algebra which is free as an A-module. By A we mean the category of graded, augmented A-algebras that are free as A-modules. The module of indecomposables is: Q(B) = A ⊗B B η (B = Coker(A → B) where η is the augmentation map) see [19] and §3. We note that Q defines a functor from A to the category of positively graded free modules. The higher left derived functors of the

350

J. La Luz, D. Allen / Topology and its Applications 209 (2016) 347–366

indecomposable functor Li Q(−) are obtained by taking the homology of the simplicial object that arises from the bar resolution of the free algebra functor F . More specifically, there is a canonical resolution over the category of A-algebras that gives rise to a simplicial object where the maps satisfy the usual simplicial identities [21]: s0

s0 d0

· · · F 3 (B)

d1 d2

d0

F 2 (B)

d1

F (B)

d0

B

s1

The simplicial object gives rise to a chain complex: (chu (QF• (B)), δ) where the differential is given by the alternating sum of the face maps. That is, δ = homology we obtain the higher left derived functors of the indecomposable functor



i i (−1) di .

By taking

Li Q(B; A) := Hi (chu (QF• (B))). The reader can refer to [4] for more details concerning the coefficients A. It is these higher derived functors that we will compute in various dimensions. Let B = Z[v1 , . . . , vm ] such that |v1 | = · · · = |vm | = 2k with r1 , . . . , rn ∈ B and C = B/(r1 , . . . , rn ). Theorems 4.9 and 5.3 referenced below hold for these general algebras. However, for the applications geared toward Toric Topology we assume that |vi | = 2 and C will be the Stanley–Reisner ring (we sometimes refer to this as the Face ring and when coefficients are in a complex orientable theory E, we say the E-Face ring) associated to some simplicial complex or its dual polytope. In this setting, the ideal (r1 , . . . , rn ) is generated by square free monomials coming from trivial intersections of facets of P (see Definition 2.3). Concerning the first derived functor we have: Corollary 4.9. L1 Q(C) is a free Z-module and is isomorphic to span{r1 , . . . rn }. Compare this result to Theorem 6.15 [1] where the first higher right derived functor of the primitive element functor of the E-Face ring was computed through the range where a relation among relations of minimal degree did not appear. In addition, the second higher right derived functor was computed in very specific cases—Face rings containing only two square free monomials [1]. The result in that paper depended on the interplay between the Face ring and the Moment angle complex. However, by direct examination, we are able to determine the second higher derived functor and prove that the generators depend on the relations ri with certain degree restrictions on non-trivial intersections of those relations. Theorem 5.3. L2 Q(C) is a free Z-module and is generated by the Z-span of the set of relations among relations in r1 , . . . , rn of minimal degree with non-empty intersection. In a different setting these theorems have other implications in regards to cotangent complex calculations. We are working with augmented simplicial algebras over a commutative ring, so if we reduce to the case when the ring is a field k, then by the Andre–Quillen Homology exact sequence induced from the augmentation, the results presented here are explicit structural theorems for the Dn(A|k; k) for n = 0, 1, 2. Both authors are grateful to Pete Bousfield for pointing out this explicit relation in a private communication. We remark that these calculations partially address a comment made by [6] within the introduction in regards to the lack of explicit structural theorems for the cotangent complex. We will not stress applications

J. La Luz, D. Allen / Topology and its Applications 209 (2016) 347–366

351

outside of torus actions in this paper, but we wanted to alert the interested reader to these computations since they fill a gap in the literature concerning the cotangent complex. In terms of explicit classes or calculations, the interested reader can refer to the proofs of Theorems 4.9 and 5.3 where the bracketed classes provide the required information. Since certain applications are geared toward Toric Topology we need to briefly recall some terminology; additional details can be found in §2. For brevity, we write the topological torus T n as T and although not specified, we assume that the underlying polytope, P , is a fixed n-dimensional, q ≥ 1 neighborly simple convex polytope [11]. When reference is made to a Quasitoric manifold we simply write M (λ); dropping the dimension when the context is clear. By this we are referring to a 2n dimensional manifold with a torus action that is equivariant with respect to the automorphisms of the torus such that the orbit is polytope in the sense described above. The “P” denoting the orbit (of the T -action on M (λ)) is not to be confused with the “P” in the term Ri P (C) which refers to the primitive element functor of a coalgebra C. We remark that the calculations of the derived functors mentioned above do not depend on the existence of a Quasitoric manifold. Given a Quasitoric manifold M (λ) an equivariantly formal T-action on M (λ) is a torus action on the manifold such that the Serre Spectral Sequence of the associated fibration M (λ) → BT P → BT collapses [18]. It was proven in [16] that such a collapse occurs and as a result, H ∗ (BT P )—the equivariant cohomology of M (λ) (also written as HT∗ (M (λ))) is a free H ∗ (BT )-module. We note that the face ring depends only on the polytope P and not λ. For this fibration there are multiplicative extensions in the Serre Spectral Sequence and many of those products show up in the cohomology of the base. These results imply that torus actions on Quasitoric manifolds M (λ) are equivariantly formal. Nice equivariantly formal torus actions are those torus actions for which Ri P (E∗ (M (λ)) vanish for i > 1 (or dually, for which Li QE ∗ (M (λ)) vanish in the same degrees). Manifolds with this property are deemed nice. Theorem 6.2. A Quasitoric manifold M (λ) is nice if and only if for all i = j, ri ∩ rj = ∅ for all relations ri , rj in the E-face ring. The next result provides an explicit condition for determining when a certain class of Quasitoric manifolds exists (or the corresponding λ). We take a moment to recall that for a polytope P as above and λ satisfying condition (∗) described in [16] (it is a map of lattices Zm → Zn where the determinant of the minors of the associated matrix is ±1), then for every pair (P, λ) there is a Quasitoric manifold M (λ) [16]. In the work of Buchstaber and his collaborators they refer to (P, λ) as a dicharacteristic pair [13]. The torus action is intimately related to λ which itself exists only if the manifold exists. This is the relationship we exploit to refine certain classes of torus actions. Corollary 6.3. Let P be a fixed simple convex polytope. If for some indices i and j, there exist relations ri and rj such that ri ∩ rj = ∅ in E ∗ (BT P ), then there does not exist a nice torus action on M (λ). Theorem 6.2 explicitly relates torus actions to unstable methods. Namely, they split the class of equivariantly formal torus actions into two groups: those that are nice and those that are not by linking them to the vanishing of the higher derived functors – both Li Q(−) and Ri P (−) respectively. This opens the door to further investigation into the relation between torus actions and the cotangent complex, specifically, the underlying geometry and deformation theory. For the purposes of this paper, there are a few other interpretations worth detailing. First, from an Unstable Homotopy theoretic perspective, we can address a question posed by Bendersky in 2006. He asked whether or not it was possible to use the torus action to produce manifolds that have the property that they were nice as homology coalgebras. That is, Ri P E∗ (M (λ)) = 0 for i > 1 (or dually, Li QE ∗ (M (λ)) = 0 for

352

J. La Luz, D. Allen / Topology and its Applications 209 (2016) 347–366

i > 1). Namely, can λ be used to control, in some sense, these higher derived functors. The answer depends on the combinatorics of the orbit. Second, how far can the refinement of certain torus actions be pushed using these methods? The paper is organized as follows: in §2 the necessary background from Toric Topology is highlighted. In §3 the left and right (resp.) higher derived functors of the indecomposable and primitive element functor (resp.) are discussed as well as notions related to duality outlined in Milnor’s paper [19]. In §4 we compute the higher derived functors in the general setting. The second higher derived functor is computed in §5 and applications geared toward Toric Topology are addressed in §6. Specifically, the non-existence of torus actions and a refinement of Equivariant formal torus actions. 2. Toric topology In this section we will highlight some well-known constructions in Toric Topology. Some excellent references for this material would include the conference proceedings [14] and the AMS book [11]. For the applications needed in this paper, it is assumed that P is a simple, q ≥ 1 neighborly convex polytope, with m facets F1 , .., Fm . Furthermore, T n will denote the n dimensional topological torus and BT n its classifying space. When there is no chance of confusion we simply write T instead of T n . Using the perspective of derived forms the notion of a quasitoric manifold and related spaces can be described. In fact, derived forms are formulated in a much more general setting, where sets more general than polyhedra are used [14]. Following Buchstaber and Ray’s exposition, there is a map λ

P −−−−→ T (T n ) where T (T n ) is the lattice of subtori of the torus ordered by inclusion and the topological structure is induced from the lower limit topology. λ sends q ∈ P to a certain subtorus, λ(q). The derived space is by definition the following quotient space D(λ) = (T n × P )/ ∼ where (g, q) ∼ (h, q) if and only if g −1 h ∈ λ(q). It can be shown that ∼ is an equivalence relation. The elements in D(λ) are equivalence classes [g, q] for which there is a canonical action of the torus on D(λ) via multiplication on the first coordinate. The orbit space of this action is P ; using the atlas {Uv } given by [11] p. 63, Construction 5.8, it can be shown that the torus action is locally standard [16] p. 420 (here locally standard is referred to as locally isomorphic to the standard representation). To obtain a quasitoric manifold (or Toric manifold in the language of [16]), one must impose conditions on λ and a smooth structure on D(λ) cf. [16]. First, λ associates a circle to the interior of a facet. Hence, if Fj is a facet, then Tλ (Fj ) is a circle. For additional details, see p. 64, (5.3) [11]. Second, if F is a codimension k-face, that is, the intersection of k facets, then the torus subgroup associated to the interior of F is the product of those coordinate tori coming from each of the facets whose intersection is the face F . More succinctly, the following map is an isomorphism: im

 1≤j≤m

Tλ (Fj ) −−−−→ T (F )

Finally, it is required that the Ker of λ partitions P by the interior of the faces. By a Quasitoric manifold M (λ) one means the derived space D(λ) where λ is subject to the conditions described above. 2n

Remark 2.1. [16] refer to the subgroup GF of the lattice Zm “determined by T n and λ” cf., p. 423, 1.5. In addition, λ determines a map between integer lattices Zm → Zn . Note that this map is also referred to as “lambda” in many research articles. Recall, to each facet Fj of P , one associates a certain circle subgroup,

J. La Luz, D. Allen / Topology and its Applications 209 (2016) 347–366

353

say Tλ (Fj ). In [11] these subtori were made explicit, see p. 64, (5.3). Each such torus subgroup gives rise to a facet vector λij ∈ Zn , for 1 ≤ i ≤ n, 1 ≤ j ≤ m. These vectors are indexed by the facets and from this and the discussion above, a function can be defined from the set of facets of P , , to the integer lattice whose dimension depends on the dimension of P . This is how the characteristic function → Zn is obtained and by a simple identification, the function Zm → Zn can be derived. By condition (*) in [16], one requires that the free Z-module spanned by those facet vectors coming from the subtori that are the images of those facets whose intersection is the appropriate face of P , be a direct summand of Zn . [16] refer to such modules as unimodular. Notational convention: Throughout the paper the dimension of a quasitoric manifold may be dropped as well as any reference to λ when the context is clear. For example, we may write M or M (λ) instead of M 2n (λ). In addition, if M (or M (λ)) is a Quasitoric manifold with orbit P , then it is common to say that M is a Quasitoric manifold over P or M sits over P (this parlance is also adopted in [14]). Examples of quasitoric manifolds would include the following: CP n with orbit Δn . Buchstaber and Ray [13] show that the 2n-dimensional manifold – Bn of all bounded flags in Cn+1 is a Quasitoric manifold over I n . They also show that CP n CP n is a Quasitoric manifold over Δ1 × Δn−1 by defining a connect sum operation on the level of the polytopes [13]. [20] classified four dimensional Quasitoric manifolds that sit over polygons and showed that they are connect sum of the Hirzebruch surface with connect sums of CP 2 . Given a set X endowed with an action of a group G, the Borel Construction can be used to replace the orbit space X/G, by a space EG ×G X which is homotopy equivalent to the orbit if the action is free, the latter fitting into a fibration. In the case of a Quasitoric manifold M , we have the following, writing T n as T : Definition 2.2. Let M be a Quasitoric manifold. The Borel space BT M is the identification space ET × M/ ∼= ET ×T M where the equivalence relation is defined by: (e, x) ∼ (eg, g −1 x) for any e ∈ ET and x ∈ M , g ∈ T . Notational convention: Sometimes the notation BT P appears in the literature instead of BT M . The orbit of the T -action on a Quasitoric manifold M can be identified with P , this should give some insight into the notational convention. When the context is clear, the notation BT P will be used instead of BT M . The following fibration is well known [11,16]: M (λ) −−−−→ BT P −−−−→ BT The face ring is an invariant that will be useful in the sections that follow. Recall, Definition 2.3. Let F1 , .., Fm be the facets of P . For a fixed commutative ring R with unit we have R(P ) = R[v1 , .., vm ]/(vi1 · · · vik |Fi1 ∩ · · · ∩ Fik = ∅) where |vi | = 2 are indexed by the facets and the ideal I is generated by square free monomials coming from trivial intersection of facets. For example, if P = ∂Δ2 , then Z(P ) = Z[v1 , v2 , v3 ]/(v1 v2 v3 ). Sometimes we refer to R(P ) as the Stanley– Reisner algebra and I the Stanley–Reisner ideal. It is shown in [16] that H ∗ (BT P ) ∼ = Z(P ) and that the Borel Construction is a contravariant functor from the category of simplicial complexes to the category of homotopy types of spaces, see pages 436 and 437.

354

J. La Luz, D. Allen / Topology and its Applications 209 (2016) 347–366

3. Construction of Ri P (−), Li Q(−) and their relationship We recall the construction of the higher right derived functors of the primitive element functor Ri P (−) [12]. We dualize the construction to obtain the higher left derived functors of the indecomposables functor Li Q(−). Notational convention: We assume A is a commutative ring with unit. All A-coalgebras are free graded A-modules equipped with a coproduct map ∇ and counit  : C → A such that the restriction |C0 : C0 → A  is an isomorphism. Let C = ker(C → A). When reference to a coalgebra is made, it is assumed that it has the structure elucidated above. Finally, when reference is made to a module it is assumed to be non-negatively graded. Definition 3.1. If C is a coalgebra, then the module of primitive is defined and denoted as: ∇

P (C) = ker(C −→ C ⊗A C) This defines a non-additive functor from the category of A-coalgebras C to the category of A-modules, M. Let S be the cofree, cocommutative, coalgebra functor with counit over A as defined in [12]. That is, S : M → C where we restrict to the subcategory of free A-modules. It comes equipped with a natural transformation s : S → 1 satisfying the typical universal property. Namely, if M is a free A-module, C is an A-coalgebra with an A-module map f : C → M then there is a unique A-coalgebra map, f¯ : C → S(M ) such that the following diagram commutes: S(M )

s

M

f¯ f

C Let M = C, f = id, s−1 = s and d0 = id, then we obtain an augmented cosimplicial object over the category of A-coalgebras: S• (C) s0

s0 d0 d0

0

C

d

S(C)

S 2 (C)

d1

d1 d2

S 3 (C) · · ·

s1

where di = S n (d0S n−i (C) ) : S n (C) → S n+1 (C) for 0 ≤ i ≤ n and for 0 ≤ i ≤ n we have si = S n (s−1 S n−i (C) ) : n+1 n (C) → S (C). Applying the primitive element functor P we obtain an un-augmented chain complex, S n chu (P S• (C)) with δ n = i=0 (−1)i P (di ). Definition 3.2. Ri P (C; A) := H i (chu (P S• (C))) The functors Ri P have the following properties as proven in [12,10]: Theorem 3.3. For any coalgebras C and D we have: i) R0 P (C; A) = P C;

J. La Luz, D. Allen / Topology and its Applications 209 (2016) 347–366

355

ii) If C is cofree, then Ri P (C; A) = 0 for i > 0; iii) Ri P (C ⊗A D; A) ∼ = Ri P (C; A) ⊕ Ri P (D; A). Definition 3.4. An injective extension sequence is a sequence of coalgebras f

g

C1 → C2 → C3 such that i) g is an epimorphism of A-modules; ii) C2 is injective as a C1 -comodule; iii) f is the inclusion C2 C3 A → C2 . We have the following result from [12,10]: Theorem 3.5. If C1 → C2 → C3 is an injective extension sequence then there is an exact sequence · · · → Ri P (C1 ; A) → Ri P (C2 ; A) → Ri P (C3 ; A) → Ri+1 P (C1 ; A) → · · · Remark 3.6. The long exact sequence above is one of the few tools used to make specific computations. In what follows, it is desirable to work with algebras. As such, we wish to dualize the construction of Ri P (−; A). Notational convention: We assume that all A-algebras are graded, free A-modules endowed with a product map m and a unit. Let B be such an algebra with unit η : A → B. Furthermore, we require the restriction η η|A0 : A0 → B to be an isomorphism and we let B = Coker(A → B). Definition 3.7. The module of indecomposables of B is defined and denoted by Q(B) = B/B

2

Q defines a non-additive functor from the category of A-algebras to the category of A-modules. Let F be the free, commutative algebra functor with unit over A. It is a functor from the category of free A-modules to the category of A-algebras and comes equipped with a natural transformation s : 1 → F . There is a diagram that describes the universal property: if M is a free A-module and B is an A-algebra with an A-module map f : M → B, then there is a unique A-algebra map f¯ : F (M ) → B such that the following diagram commutes: s

M

(1)

F (M ) f¯

f

B Let M = B, f = id, s−1 = s and d0 = id, then we obtain an augmented simplicial object over the category of A-algebras: F• (B) s0

s0 d0

· · · F 3 (B) s1

d1 d2

d0

F 2 (B)

d1

F (B)

d0

B

356

J. La Luz, D. Allen / Topology and its Applications 209 (2016) 347–366

The di = F n ((d0 )F n−i (B) ) : F n (B) → F n−1 (B) for 0 ≤ i ≤ n and for 0 ≤ i ≤ n we have si = F ((s−1 )F n−i (B) ) : F n (B) → F n (B). Applying the functor Q we obtain an un-augmented chain complex, n chu (QF• (B)) with δn = i=0 (−1)i Q(di ). n

Definition 3.8. Li Q(B; A) = Hi (chu (QF• (B))) Dualizing the results from Theorem 3.3 we have Theorem 3.9. For any algebras B, D and A i) L0 Q(B; A) = QB; ii) If B is a free algebra, then Li Q(B; A) = 0 for i > 0; iii) Li Q(B ⊗A D; A) ∼ = Li Q(B; A) ⊕ Li Q(D; A). Remark 3.10. If we assume that the algebra B is of finite type and C is a coalgebra of finite type, then B ∗ is a coalgebra with coproduct m∗ and C ∗ is an algebra with product ∇∗ . We refer the reader to [19] for additional details. Also, by the universality of S and F , we have F (B)∗ = S(B ∗ ) and S(C)∗ = F (C ∗ ). From this it follows that F n (B)∗ = S n (B ∗ ) and S n (C)∗ = F n (C ∗ ). Remark 3.11. In [19], the authors define, for an augmented algebra B, the module of indecomposables as Q(B) = A ⊗B B and for an augmented coalgebra C, the module of primitives as P (C) = AC C. Because our algebras and coalgebras are augmented, it is easy to see that the definitions given by the authors for these modules coincide with the definition in [19]. The interested reader can refer to Proposition 3.2 (2) of [19] and 3.7 (p. 224). From this, Q(B)∗ = P (B ∗ ) follows. For the relation between these two functors we need a general result. Theorem 3.12. If B is an A-algebra of finite type then (F• (B))∗ = S• (B ∗ ). Proof. Applying HomA (−, A) to the simplicial object F• (B) we obtain s0

s0 d0

B∗

d0

F (B)∗

d0

d1

F 2 (B)∗

d2

d1

F 3 (B)∗ · · ·

s1

where di = d∗i . By Remark 3.10 we obtain the desired result. 2 Lemma 3.13. If B is an A-algebra of finite type which is free as an A-module, then Q(F (B))∗ = P (S(B ∗ )) Proof. We have (QF• (B))∗ = P (F• (B)∗ ) = P S• (B ∗ )

by Remark 3.11 by Remark 3.10

2

Remark 3.14. If B is Z-algebra, then it is easy to see that B ⊗A is an A-algebra. If B is also a free Z-module, then B ∗ ⊗ A ∼ = Hom(B, Z) ⊗ A ∼ = Hom(B, A).

J. La Luz, D. Allen / Topology and its Applications 209 (2016) 347–366

357

Corollary 3.15. If B is a Z-algebra of finite type and a free Z-module, then we have a short exact sequence 0 → ExtA (Ln−1 Q(B; Z), A) → Rn P (B ∗ ⊗ A; A) → Hom(Ln Q(B, Z), A) → 0 Proof. By the Universal coefficients theorem for cohomology there is a short exact sequence [17] 0 → Ext(Ln−1 Q(B; Z), A) → H n (HomA (chu QF• (B), A)) → HomA (Ln Q(B, Z), A) → 0 We have H n (HomA (chu QF• (B), A)) = H n (chu (QF• (B))∗ ⊗ A) by Remark 3.14 ∼ = H n (chu P (F• (B)∗ ⊗ A)) by Remark 3.11 ∼ = H n (chu P S• (B ∗ ) ⊗ A)) by Remark 3.10 ∼ = Rn P (B ∗ ⊗ A; A) The last isomorphism follows because of Lemma 3.5 [4]. 2 4. The bracket notation Parts of this section can be found in [5]. In particular the bracket notation. We recall it for the convenience of the reader. Notational convention: Let B = Z[v1 , . . . , vm ] with |v1 | = · · · = |vm | = 2 and let r1 , . . . , rn ∈ B be monomials in the vj with C = B/(r1 , . . . , rn ). There is a natural map ρ : B → C. We write Ln Q(B) for Ln Q(B; Z). Recall, by [12] there exists a forgetful functor J from the category of algebras to the category of Z-modules. Let F (B) be the free algebra generated by B (and by this we mean, F (JB)) and its elements can be described as [ai ] and products [a1 ] · · · [an ] where ai ∈ B. More specifically, the bracket of an element of B is an element of the free algebra, then one forms all possible products of such elements. From this it follows that QF (B), is the Z-span of the set {[a]|a ∈ B}. This is conveniently encoded in the diagram displaying the universal property of the free algebra (see diagram (1)). The map s−1 : B → F (B) takes a to [a] and d0 ([a1 ] · · · [an ]) = a1 · · · an . If D is an algebra and g : B → D is an algebra map then the induced map is: F (g)([m]) = [g(m)]. Given a collection of brackets: 

   · · · [aι1 ] · · · [aιp ] · · ·

A bracket of depth i is the ith bracket from the outermost bracket. When referring to the depth of a bracket it is understood to mean a pair of brackets where the left and right sides are simultaneously at depth i. For example, a bracket of depth zero is the outermost bracket. That is, the outermost pair of brackets. A bracket of depth one is the bracket that is one place over from the bracket of depth zero. Put another way, it is the second outermost bracket. This mechanism allows for an explicit formulation of the face maps. The map di : QF n (B) → QF n−1 (B) removes the bracket in depth i. The map si : QF n (B) → QF n+1 (B) will add a bracket in depth i. It is useful to observe that elements in QF n (B) will have n brackets. For illustrative purposes we list a few elements using this bracket notation from [5]:

358

J. La Luz, D. Allen / Topology and its Applications 209 (2016) 347–366

Example 4.1. Consider the following element in QF 3 (B). 

  a1 ][a2 ] [a3 ]

[   depth 2

      d0 [a1 ][a2 ] [a3 ] = [a1 ][a2 ] [a3 ] Observe that the im d0 ∈ QF 2 (B) and as such it is equal to zero in QF 2 (B).  d1

    [a1 ][a2 ] [a3 ] = [a1 ][a2 ][a3 ]

     = [a1 a2 ][a3 ] d2 [a1 ][a2 ] [a3 ]       s0 [a1 ][a2 ] [a3 ] = [a1 ][a2 ] [a3 ]       s1 [a1 ][a2 ] [a3 ] = [a1 ][a2 ] [a3 ]   Notational convention: We define [x](k) ∈ F k (B) inductively. Let [x](1) = [x] and [x](k+1) = [x](k) . Definition 4.2. Suppose B is a free A-algebra containing the non-zero product x = a1 · · · an . The core  n of x, c(x), is {a1 , · · · , an } and |x| = |c(x)| = i |ai |. For xn ∈ QF (B), let xn = [xι1 · · · xιj ] where  j n−1 xιk ∈ F (B). The core of xn , c(xn ) := i=1 c(xιi ) and the degree is the sum of the degrees of the elements contained in the union. Simply put, the core is nothing more than the set of elements contained within the innermost bracket. Example 4.3. Let xn ∈ QF n (B) and suppose it is of the form: 

   xn = · · · [xι1 ]· · ·[xιp ] · · · c(xn ) = {xι1 , . . . , xιp } and its degree is |xι1 | + · · · + |xιp |. We give a more explicit example Example 4.4. Given the element in QF 3 (B)  x3 =

  [a1 ][a2 ] [a3 ]

its core is: {a1 , a2 , a3 } and |x3 | = |a1 | + |a2 | + |a3 |. Remark 4.5. As the filtration increases the degree of the core remains constant. For computational purposes, it is useful to recall that the di are degree preserving maps. Recall, ρ : Z[v1 , . . . , vm ] → Z[v1 , . . . , vm ]/(r1 , . . . , rn ).

J. La Luz, D. Allen / Topology and its Applications 209 (2016) 347–366

359

Definition 4.6. The ring homomorphism ρ : B → C induces a surjective map ρ : QF• (B) → QF• (C). Let ρ∗ K• (C) = Ker(QF• (B) → QF• (C)). The complex K• (C) will simplify certain computations as will become evident after the proof of the next Lemma. Lemma 4.7. For i > 0 we have Li Q(C) ∼ = Hi−1 ((K)• (C)) Proof. The short exact sequence of chain complexes 0 → K• (C) → QF• (B) → QF• (C) → 0 induces a long exact sequence in homology · · · → Hi (K• (C)) → Li Q(B) → Li Q(C) → Hi−1 (K• (C)) → · · · Since B is a free algebra, Li Q(B) = 0 for i > 0 and L0 Q(B) ∼ = Q(B) = Q(C) ∼ = L0 Q(C) the result follows. 2 There is an explicit formulation of elements in the complex K• (C) in terms of divisibility. Proposition 4.8. K n (C) is generated by the elements of QF n (B) such that one of the elements of the core is divisible by one of the relations of C. Proof. Recall, for x ∈ F i (C), c(x) is the core. Let r be any relation in C, then the following is a basis for QF n (B): {[x : x ∈ F n−1 (C)] | r  c(x)} Assume x ∈ QF n (B) and ρ(x) = 0, then a relation r must divide x. That is, the core.

2

Corollary 4.9. L1 Q(C) is a free Z-module and is isomorphic to span{r1 , . . . rn }. Proof. By Lemma 4.7, it is enough to find the generators of H0 (K• (C)). K 1 (C) is generated by [rj ] and [rj x] where 1 ≤ j ≤ n and x ∈ B and nothing more. Then for δ1 = d0 − d1 we have    δ1 − [r j ][x] = [rj x] Thus the non-zero generators in homology are [rj ]. 2 5. The second higher derived functor In what follows we set the notation to address the second derived functor L2 Q(C). Recall, there are monomials r1 , . . . , rn ∈ B such that C = B/(r1 , . . . , rn ). For i = j let ri = vι1 · · · vιp and rj = vτ1 · · · vτk .

J. La Luz, D. Allen / Topology and its Applications 209 (2016) 347–366

360

There are two indexing sets coming from these relations and they are: Ii = {ι1 , . . . , ιp } and Ij = {τ1 , . . . , τk }. The intersection of the relations, ri ∩ rj = {vs |s ∈ Ii ∩ Ij }. A relation among relations is a polynomial Rij = ri xi − rj xj = 0 where xi , xj ∈ B. Continuing with the bookkeeping, a relation among relations can be   made explicit in this setting. Rij = ri va −rj vb where the products of v  s are indexed by a ∈ (Ii ∪Ij ) −Ii   and b ∈ (Ii ∪ Ij ) − Ij . In this case, xi = va and xj = vb . Rij is of minimal degree precisely when the cardinality of Ii ∩ Ij is greatest. In other words, when xi and xj are of minimal degree. Example 5.1. Consider BT P = ET ×T M and E∗ [v1 , . . . , vm ]/(r1 , . . . , rk ), its E∗ -face ring. Each relation ri is a square free monomial consisting of a product of v  s. For i = j assume that ri = vι1 · · · vιp and rj = vτ1 · · · vτk as above. In terms of the bracket notation we have the following representation:  Rij =



[v ι1 ] · · · [v ιp ]





(2)

[v a ]

 −

 [vτ1 ] · · · [vτk ]

a∈(Ii ∪Ij )−Ii



 (2)

[vb ]

b∈(Ii ∪Ij )−Ij

In what follows we will compute the second higher derived functor. We now show that multiplying elements in the core, say by elements in B produces homologous elements relating back to the requirement that the relation among relations must be of minimal degree. We note that elements commute within a bracket of depth i. Recall, [[x]] = [x](2) Lemma 5.2. Let ri be a relation and y1 , y2 ∈ B. Then       [ri y1 ][y2 ] ∼ [ri ][y1 ][y2 ] ∼ [ri ][y1 y2 ] Proof. The required elements in K 3 (C) are: 

 (2)  [ri ][y1 ] y2

and  (2)

[r i ]

  [y 1 ][y 2 ]

Note that   (2)      δ2 [ri ][y1 ] y2 = − [ri y1 ][y2 ] + [ri ][y1 ][y2 ] and  δ2

(2)

[r i ]

      [y 1 ][y 2 ] = − [ri ][y1 ][y2 ] + [ri ][y1 y2 ]

2

Before reading the next theorem, the reader should review the first part of §5. Theorem 5.3. L2 Q(C) is a free Z-module generated by the Z-span of the set of relations among relations in r1 , . . . , rn of minimal degree with non-empty intersection.

J. La Luz, D. Allen / Topology and its Applications 209 (2016) 347–366

361

Proof. First, consider the following elements in K 2 (C):     []  Rij = [ri y][xi ] − [rj y][xj ] [] is in the kernel of δ2 = d0 − d1 and it is not a relation among relations of where y ∈ B. Observe that R ij  [] is in the image; meaning that it does not contribute minimal degree. In what follows we will show that R ij

[] = 0. Since ri xi = rj xj we have: to homology. Moreover, if ri ∩ rj = ∅ then xi = rj and xj = ri implying R ij     []  Rij = [ri y][xi ] − [rj y][xj ]     ∼ [ri ][y][xi ] − [rj y][xj ]

by Lemma 5.2

    = [ri ][xi ][y] − [rj y][xj ]     = [ri xi ][y] − [rj ][y][xj ]

by Lemma 5.2

    ∼ [ri xi ][y] − [rj xj ][y]     ∼ [ri xi ][y] − [ri xi ][y] = 0 [] is a boundary. We will now show that the cycle below is not hit by a class in filtration three. Hence, R ij     [] Rij = [ri ][xi ] − [rj ][xj ] Recall, the di are degree preserving maps and multiplication by elements in varying filtration is not permitted. That is, we can not multiply elements with different brackets. As such, we focus on those classes in [] K 3 (B) that have core with the same degree as core Rij . We must add brackets—one pair, in all possible []

ways to Rij . This process preserves the degree of the core, producing a relatively small group of possible classes that have to be checked. The only possibilities are:     [ri ](2) [xi ](2) − [rj ](2) [xj ](2) The next possible class is:  (2)  (2) [ri ][xi ] − [rj ][xj ] Assuming xk xk = xk , the other class to consider is: 

 [ri ](2)

    [xi ][xi ] − [rj ](2) [xj ][xj ] []

Computing δ2 = d0 − d1 + d2 on each of the classes above shows that none hit Rij .

2

J. La Luz, D. Allen / Topology and its Applications 209 (2016) 347–366

362

Remark 5.4. One has to be careful when using Theorem 5.3 to find the generators of L2 Q(C) because the set of relations among relations may be linearly dependent. For example, the ring Z[v1 , v2 , v3 ]/(v1 v2 , v2 v3 , v1 v3 ) has relations r1 = v1 v2 , r2 = v2 v3 and r3 = v1 v3 . There are relations among relations: R12 = r1 v3 − r2 v1 , R23 = r2 v1 − r3 v2 and R13 = r1 v3 − r3 v2 . Since R12 + R23 = R13 , L2 Q(C) is generated by two [] [] elements, R12 and R23 . Corollary 5.5. Let T = (A[v1 , . . . , vm ]/(r1 , . . . , rn ))∗ then: i) R0 P (T ; A) = spanA {v1 , . . . , vm }; ii) R1 P (T ; A) = spanA {r1 , . . . , rm }; iii) R2 P (T ; A) = spanA {Rij } where Rij are relations among relations of minimal degree with non-empty intersection. Proof. The result follows from Corollary 3.15. 2 We have the following result concerning the first higher left derived functor which relates the torus action to the computations highlighted above more explicitly. First some notation; for a λ satisfying condition (∗) [16], there is a Quasitoric manifold M (λ) and a fibration M (λ) → BT P → BT n where the map H ∗ (BT n ) → H ∗ (BT P ) maps ti to the linear forms λi . We note that the dimension of the torus is important since it parameterizes the indecomposables. Recall, BT n is a free algebra and its E-cohomology is E ∗ [t1 , . . . , tn ] where E∗ is the coefficient ring for the complex orientable theory E. Proposition 5.6. Li QE ∗ (M (λ)) ∼ = Li QE ∗ (BT P ) for i ≥ 1. Proof. For a complex orientable theory E, there is a Projective Extension Sequence E ∗ [t1 , . . . , tn ] → E ∗ (BT P ) → E ∗ (M (λ)) [4] Theorem 3.22. This induces a long exact sequence [12] L1 QE ∗ (BT P )

0

L1 E ∗ (M (λ))

QE ∗ [t1 , . . . , tn ]

QE ∗ (BT P )

QE ∗ (M (λ))

0

We observe that the following is a short exact sequence 0

QE ∗ [t1 , . . . , tn ]

QE ∗ (BT P )

QE ∗ (M (λ))

0

The result follows. 2 We can strengthen Theorem 3.22 and Corollary 3.23 that appears in [4] concerning cohomological rigidity of polytopes. From [15] we recall the notion of cohomologically rigid polytope – a simple polytope P for which if a Quasitoric manifold M exists over P and whenever there exists a Quasitoric manifold N over a simple polytope Q with a graded ring isomorphism H ∗ (M ) ∼ = H ∗ (N ), then Q is combinatorially equivalent to P . For a given simple polytope P , let |IP | denote the cardinality of the generating set of the Stanley–Reisner ideal. Proposition 5.6 gives the following: Corollary 5.7. If M and N are Quasitoric manifolds over n-dimensional simple polytopes P and Q such that P → Q induces H ∗ (M ) ∼ = H ∗ (N ), then the neighborliness of P must be equal to the neighborliness of Q and |IP | = |IQ |.

J. La Luz, D. Allen / Topology and its Applications 209 (2016) 347–366

363

Proof. Suppose there is a map of polytopes P → Q that induces the isomorphism of cohomology rings H ∗ (M ) ∼ = H ∗ (N ). This isomorphism induces an isomorphism L1 QE ∗ (M ) ∼ = L1 QE ∗ (N ). The map of polyhedra induces a map of spaces BT P → BT Q [16] which by Lemma 5.6 induces an isomorphism L1 QE ∗ (BT Q) → L1 QE ∗ (BT P ). By Corollary 4.9 the first left higher derived functor is the span of the set of relations. Therefore, the relations must be the same degree and this can only happen if the neighborliness agrees. The statement concerning the cardinality of the generating sets of IP and IQ follows. 2 The conditions in Corollary 5.7 are necessary if a simple polytope is to be rigid as described in [15]. 6. Application to torus actions The computations made in the previous section have two main applications. The first is an answer to a question posed by Martin Bendersky during the 2005–2006 academic year concerning the existence of torus actions with certain properties. The second is a refinement of Equivariantly Formal Torus actions [18]. From a spectral sequence perspective, it was of interest to the second author and Bendersky to find a lambda that would ensure that the higher right derived functors of the primitive element functor of the homology coalgebra E∗ (M (λ)) would vanish. In other words, the search was on to find “nice torus actions”. The motivating example for this line of research was the four dimensional Quasitoric manifold with orbit the square. We recall a few details to motivate this line of inquiry. When P = Δ1 × Δ1 , the E∗ -face ring is E∗ [v1 , v2 , v3 , v4 ]/(v1 v3 , v2 v4 ). The cohomology ring of the Quasitoric manifold M (λ) is the E∗ -face ring modulo the ideal generated by the λi . In particular, (λ1 , λ2 ) where λ1 =λ11 v1 + λ12 v2 + λ13 v3 + λ14 v4 λ2 =λ21 v1 + λ22 v2 + λ23 v3 + λ24 v4 Following [16] there is a matrix 

λ11 λ21

λ12 λ22

λ13 λ23

λ14 λ24



This matrix must satisfy condition (∗) as described in [16]. This of course, provides ample flexibility in choosing λij that might be computationally useful from a spectral sequence perspective. In this motivating example, we can see that the following matrix, coming from certain choices of λij produce a nice homology coalgebra in the sense of [12]. That is, a coalgebra dual to a free algebra.   1 0 −1 0 0 1 0 −1 This matrix gives rise to λ1 = v1 − v3 and λ2 = v2 − v4 , from which it follows that H ∗ (M ) ∼ = H ∗ (S 2 × S 2 ). The latter ring being dual to a nice homology coalgebra. Of course, if the λij where chosen differently, say the following matrix [16]   0 1 −1 1 1 0 1 −2 then it is not obvious that the dual of the corresponding cohomology ring is nice as a homology coalgebra since the corresponding Quasitoric manifold is CP 2 CP 2 [20]. The results in [4] show that for a suitable theory, Ri P E∗ (M ) are independent of the torus action (for i > 1) which explains why λ is not present in the term. In terms of the example above, this means that for i > 1 there is an abstract isomorphism Ri P E∗ (S 2 × S 2 ) ∼ = Ri P E∗ (CP 2 CP 2 ).

364

J. La Luz, D. Allen / Topology and its Applications 209 (2016) 347–366

The Toric Topology along with the isomorphism in [2] shows that E∗ (CP 2 CP 2 ) is nice as a homology coalgebra. However, using the methods in §3 this can be seen directly. The cohomology ring is given by [7] page 1194. ⎧ ⎪ Z ;i = 0 ⎪ ⎪ ⎪ ⎨Z ⊕ Z ; i = 2 H i (CP 2 CP 2 ) = ⎪ Z ;i = 4 ⎪ ⎪ ⎪ ⎩ 0 ;i > 4 We note, from that paper, the cohomology group is generated by v2 and v4 subject to a variety of relations. For instance, the second cohomology group is generated by v2 and v4 , whereas the fourth cohomology group has generators subject to the relation: v22 = v42 and finally, the ith cohomology group is zero by vι1 vι2 vι3 = 0 for ιj ∈ {2, 4}. For additional details, the reader should refer to [7]. To illustrate the “niceness” we show for illustrative purposes that one possible relation among relations does not map to zero in the simplicial resolution. From the cohomology ring above, we have a relation among relations: (v22 v4 )v4 − (v2 v42 )v2 . The name of the corresponding class in the chain complex QF • , in particular, QF 3 is: 





[v 22 ][v 4 ] [v 4 ](2) −







[v 2 ][v 24 ] [v 2 ](2)

This class is not zero under δ = d0 − d1 + d2 . In particular, the d1 term does not cancel. A similar argument can be used to show that the other possible classes that come from the relations among relations (of minimal degree, of course) are not in the kernel of this map either. This information is crucial in determining to what extent the combinatorics plays in obstructing the existence of a vanishing line in the Composite Functor Spectral Sequence. The results in [1] held through a range specifically because these higher derived functors were not computable using injective extension sequences. A few preliminary definitions will be needed in the sequel. A space X is called a T-space if it comes equipped with a torus action T × X → X. Since we are primarily interested in the coalgebra structure, we restrict attention to reduced complex orientable theories for which E∗ (X) is a free module over the coefficient ring E∗ . This ensures there is a Kunneth isomorphism. Definition 6.1. Let X be a T -space and E a complex orientable theory. A torus action on X is nice if E∗ (X)/T is nice as a homology coalgebra. A case of particular interest is that of a torus action on a Quasitoric manifold M (λ). It is nice if E∗ (M (λ)) is nice as a homology coalgebra. That is, the E face ring modulo the T -action, E∗ (BT P )/T where the quotient by the torus is interpreted to be modding out by the ideal generated by the λ. By the results of [1], the E-Face ring controls, in a certain sense, the types of actions that are possible. By the results in §3 we write either Li Q(−) or Ri P (−) depending on the context and application. Since nice homology coalgebras were studied in [12] and was a key notion motivating this line of inquiry, we will often refer to Ri P (−) but when convenient we will also refer to Li Q(−). A torus action on the Quasitoric manifold M (λ) is nice if Ri P E∗ (M (λ)) = 0 for i > 1. By a nice Quasitoric manifold we mean a Quasitoric manifold endowed with a nice torus action. That is, there is a λ satisfying condition (∗) as described in [16] such that the manifold M (λ) is nice. We highlight a question of Bendersky and provide an answer below.

J. La Luz, D. Allen / Topology and its Applications 209 (2016) 347–366

365

Bendersky’s Question: “Is it possible to find a λ satisfying condition (∗) of [16] such that E∗ (M (λ)) is nice as a homology coalgebra?” This question has homotopy implications since if it were true, then certain unstable spectral sequences [9] would be somewhat tractable, in particular the Composite Functor Spectral Sequence that was analyzed in [1,9,10]. In the setting of simplicial resolutions (cosimplicial resolutions resp.), it is related to the vanishing of the cotangent complex, which lends to its difficulty. This is only complicated by the dearth of specific calculations of these higher derived functors. We are able to answer Bendersky’s question: briefly, if the E∗ -face ring has relations that intersect, then it is not possible to find a torus action that will produce a vanishing of the (left resp.) right higher derived functors referenced above. We have the following: Theorem 6.2. A Quasitoric manifold M (λ) is nice if and only if for all i = j, ri ∩ rj = ∅ for all relations ri , rj in the E-face ring. Proof. :⇒ By Theorem 3.22 [4] and §3, there is an isomorphism Li QE ∗ (BT P ) ∼ = Li QE ∗ (M (λ)) for i > 1. Since L2 QE ∗ (M (λ)) = 0 this implies that either E ∗ (BT P ) is a free algebra or any potential generator of a second derived functor is a relation among relations that is generated by two relations ri and rj such that ri ∩ rj = ∅. The result follows by appeal to Theorem 5.3. :⇐ If the relations ri and rj have empty intersection, then any possible relation among relations is in  [] where ri ∩ rj = ∅ then xi = rj and the image by Theorem 5.3. Specifically, in the proof see the class R ij xj = ri ; combine this with the isomorphism of the higher derived functors proven in [2–4]. 2 We note that if the intersection of the relations is trivial, then the face ring is the tensor product of the face rings of the boundary complexes of simplicial complexes. We obtain the following corollaries. Corollary 6.3. Let P be a fixed simple convex polytope. If for some indices i and j, there exist relations ri and rj such that ri ∩ rj = ∅ in E ∗ (BT P ), then there does not exist a nice torus action on M (λ). Given a Quasitoric manifold M (λ), an equivariantly formal T-action on M (λ) is a torus action on the manifold such that the Serre Spectral Sequence of the associated fibration M (λ) → BT P → BT collapses [18]. We recall that in analyzing this spectral sequence there are multiplicative extensions to be considered. It was proven in [16] that such a collapse occurs and as a result, H ∗ (BT P )—the equivariant cohomology—HT∗ (M (λ)) is a free H ∗ (BT )-module and this cohomology ring is indeed independent of λ. These results imply that torus actions on Quasitoric manifolds M (λ) are equivariantly formal. We define nice equivariantly formal torus actions to be those equivariantly formal torus actions that are nice. More explicitly, nice equivariantly formal torus actions are those torus actions for which Ri P (E∗ (M (λ)) vanish for i > 1 (or dually, for which Li QE ∗ (M (λ)) vanish in the same degrees). The following is evident: Corollary 6.4. An equivariantly formal torus action is nice if and only if the Quasitoric manifold M (λ) is nice. Acknowledgements The authors would like to thank Pete Bousfield for providing details and insights into his thoughts concerning the relations between the higher left derived functors and the work of Avramov and Iyengar. We are grateful for his encouragement in our attempt to compute these derived functors. We would also like to thank Paul Goerss for his comments concerning this work and related avenues of research. We also thank Haynes Miller for taking a look at a very early draft of the paper and for providing comments and pointing us toward Avramov’s work too. The second author is grateful to Fred Cohen for his encouragement (over

366

J. La Luz, D. Allen / Topology and its Applications 209 (2016) 347–366

many years) and for pointing out that this work is related to certain group actions on the cohomology of various toric spaces – a line of inquiry that is very interesting and potentially very fruitful. Finally, both authors would like to thank the referee for carefully reading the paper and making suggestions that improved the exposition. References [1] D. Allen, On the homotopy groups of toric spaces, Homology, Homotopy, and Applications 10 (1) (2008) 437–479. [2] D. Allen, J. La Luz, The higher derived functors of the primitive element functor of quasitoric manifolds, Topol. Appl. 158 (16) (2011) 2103–2110. [3] D. Allen, J. La Luz, Methods in unstable homotopy theory, Surv. Math., Math. Sci. 1 (1) (2012) 1–34. [4] D. Allen, J. La Luz, Certain generalized higher derived functors associated to quasitoric manifolds, Pac. J. Math., submitted for publication. [5] D. Allen, J. La Luz, The Determination of Certain Higher Derived Functors of Moment angle complexes, Topol. Proc., vol. 49, 2016, in press. [6] L. Avramov, S. Iyengar, Andre–Quillen homology of algebra retracts, Ann. Sci. Ec. Norm. Super. 36 (3) (May–June 2003) 431–462. [7] A. Bahri, M. Bendersky, The KO-theory of toric manifolds, Trans. Am. Math. Soc. 352 (3) (1999) 1191–1202. [8] A. Bahri, M. Bendersky, F. Cohen, S. Gitler, The polyhedral product functor: a method of computation for moment-angle complexes, arrangements and related spaces, Adv. Math. 225 (3) (2010) 1634–1668. [9] M. Bendersky, E.B. Curtis, H.R. Miller, The unstable Adams spectral sequence for generalized homology, Topology 17 (1978) 229–248. [10] M. Bendersky, E.B. Curtis, D.C. Ravenel, EHP sequences in BP theory, Topology 21 (1982) 373–391. [11] V.M. Buchstaber, T.E. Panov, Torus Actions and their Applications in Topology and Combinatorics, Univ. Lect. Ser., AMS, 2002. [12] A.K. Bousfield, Nice homology coalgebras, Trans. Am. Math. Soc. 148 (1970) 473–489. [13] V.M. Buchstaber, N. Ray, Tangential structures on toric manifolds, and connected sums of polytopes, Int. Math. Res. Not. 4 (2001) 193–219. [14] V.M. Buchstaber, N. Ray, An invitation to toric topology: vertex four of a remarkable tetrahedron, in: M. Harada, et al. (Eds.), Toric Topology, in: Contemp. Math., vol. 460, Amer. Math. Soc., Providence, RI, 1970, pp. 1–27. [15] S. Choi, M. Masuda, D.Y. Suh, Rigidity problems in toric topology: a survey, arXiv:1102.1359v2, 2011. [16] M. Davis, T. Januszkiewicz, Convex polytopes, coxeter orbifolds and torus actions, Duke Math. J. 62 (2) (1991) 417–451. [17] P.J. Hilton, U. Stammbach, A Course in Homological Algebra, Graduate Text in Math., Springer, 1997. [18] Oliver Goertshes, Dirk Toen, Torus actions whose equivariant cohomology is Cohen–Macaulay, arXiv:0912.0637v2. [19] J.W. Milnor, J. Moore, On the structure of Hopf algebras, Ann. Math. 81 (2) (1965) 211–264. [20] P. Orlik, F. Raymond, Actions of the torus on 4-manifolds I, Trans. Am. Math. Soc. 152 (1970) 531–559. [21] C. Weibel, An Introduction to Homological Algebra, Cambridge University Press, Cambridge, 1994.