On Davis–Januszkiewicz homotopy types I; formality and rationalisation

Abstract

For an arbitrary simplicial complex K , Davis and Januszkiewicz have defined a family of homotopy equivalent CW-complexes whose integral cohomology rings are isomorphic to the Stanley-Reisner algebra of K . Subsequently, Buchstaber and Panov gave an alternative construction (here called c(K)), which they showed to be homotopy equivalent to Davis and Januszkiewicz's examples. It is therefore natural to investigate the extent to which the homotopy type of a space is determined by having such a cohomology ring. We begin this study here, in the context of model category theory. In particular, we extend work of Franz by showing that the singular cochain algebra of c(K) is formal as a differential graded noncommutative algebra. We specialise to the rationals by proving the corresponding result for Sullivan's commutative cochain algebra, and deduce that the rationalisation of c(K) is unique for a special family of complexes K . In a sequel, we will consider the uniqueness of c(K) at each prime separately, and apply Sullivan's arithmetic square to produce global results for this family.

Moment-angle Complexes, Monomial Ideals and Massey Products

“…A crucial ingredient is provided by the fibration Z K → DJ (K) → BT n , where DJ (K) is the Davis-Januszkiewicz space associated to K. The formality of this space-a result of Franz [22] for smooth toric fans, and Notbohm and Ray [43] in general-leads to the collapse of the Eilenberg-Moore spectral sequence for the path-fibration of DJ (K), thereby allowing us to compute π * (Z K ) ⊗ k in terms of Tor S/I (k, k), the dual of the Yoneda algebra of S/I. The answer turns out to be particularly nice when K is a flag complex, in which case S/I is a Koszul algebra.…”

“…In other words, Z K (X, A) is the colimit of the diagram of spaces {(X, A) σ }, indexed by the category whose objects are the simplices of K (including the empty simplex), and whose morphisms are the inclusions between those simplices; see [46,43].…”

Subdivisions of Toric Complexes

“…Observe that the homomorphisms i ST * and i * ST are twins in the sense of Notbohm and Ray [16], that is i QT * i ST * = i RQ * i RS * and i ST * i QT * = i RS * i RQ * . The face ring of T and the face rings of Q and S are related as follows:…”

Generalized Davis-Januszkiewicz spaces, multicomplexes and monomial rings

“…In [7] Davis and Januszkiewicz showed that every StanleyReisner algebra R(K) is realized topologically by a space DJ(K). Subsequent work of Buchstaber and Panov ([3]) has shown the existence of a homotopy equivalent model for DJ(K) as a cellular subcomplex of BT m = (CP ∞ ) m , and Notbohm and Ray ( [13]) expressed it as a homotopy colimit of a certain diagram over the poset category of K.…”