On Davis-Januszkiewicz homotopy types I; Formality and rationalisation

By D. Notbohm, Nigel Ray. 2003

72 citations recorded by the source corpus.

For an arbitrary simplicial complex K, Davis and Januszkiewicz have defined a family of homotopy equivalent CW-complexes whose inte- gral 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 co- homology 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 rationali- sation 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. AMS Classification 55P62, 55U05; 05E99

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cc92aa59922c8f1e708f221c3fb4ce69216325d1

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In Discrete Mathematics and Combinatorics

1. T. Panov
2. V. Buchstaber
3. Nigel Ray
4. S. Theriault
5. J. Grbić
6. M. Franz
7. M. Masuda
8. Y. Félix

In Geometry and Topology

1. T. Panov
2. V. Buchstaber
3. S. Theriault
4. J. Grbić
5. M. Franz
6. Nigel Ray
7. M. Masuda
8. Y. Félix

In Mathematical Physics

1. T. Panov
2. V. Buchstaber
3. M. Franz
4. S. Theriault
5. J. Grbić
6. Nigel Ray
7. Y. Félix
8. M. Masuda

Citations in the corpus

• Categorical aspects of toric topology (2007, background): “As functors top → cdga, both rational cohomology and APL map homotopy limits to homotopy colimits on diagrams PK , and map homotopy colimits to homotopy limits on diagrams T V/K.”
• Surveys in Contemporary Mathematics: Cohomology of face rings, and torus actions (2005, background): “homotopy type of X? In other words, for given K, does there exist a ‘fake’ Davis–Januszkiewicz space, whose cohomology is isomorphic to Z[K], but which is not homotopy equivalent to DJ (K)? This question is addressed in [21].”
• The integral cohomology of toric manifolds (2006, methodology): “1 is another formality result, which was independently obtained by Notbohm and Ray [19]: Theorem 1.”
• Algebraic models in geometry (2008, background): “25 ([212]) The space DJ(K) is rationally formal.”
• The homotopy type of the complement coordinate subspace arrangement (2006, background): “Notbohm and Ray [10] showed that this is true rationally, that is, the rational homotopy type of D J (K ) is determined by H(D J (K ); Q).”
• Toric Topology (2012, background): “) Second, formality of the Davis–Januszkiewicz space DJ (K) (or, equivalently, the polyhedral product (CP)) was established by Notbohm and Ray in [243].”