RESEARCH STATEMENT RUIAN CHEN 1. Overview Chen is currently working on a large-scale program that aims to unify the theories of generalized cohomology and of perverse sheaves. This program is a major development in topology, as it not only sets up theoretical foundations and opens up a vast amount of possible future research directions, but also provides many intriguing applications to geometry, especially to the study of singularities. As the first step of the program, Chen improves, in the recent joint work with Kriz and Pultr [4], the foundations of Kan s approach to stable homotopy theory via combinatorial spectra. Using the Cartan Eilenberg formalism of homotopical algebra, Chen [3] defines a series of right derived functors on sheaves of Kan spectra and establishes a crucial special case of Verdier duality. This, in turn, grants the sixfunctor formalism as well as the constructible derived category of Kan spectral sheaves. These suggest the strong connections with perverse sheaves, and are leading to a theory of perverse Postnikov towers. Chen proposes to investigate this, with the objective to develop a common generalization of both generalized cohomology and perverse sheaves. The last theoretical piece of Chen s program is the spectral algebra structure on Kan spectral sheaves, that is, the construction of a symmetric monoidal smash product on Kan spectra and their sheaves. Ultimately, this program could (1) in terms of theory, connect to and potentially analogize the geometric Langlands program, and (2) in terms of applications, provide a powerful new tool to study singularities and possibly address questions like Arnold s conjecture in symplectic geometry. 2. Results of recent projects 2.1. Foundations of Kan spectral sheaves. The starting point of Chen s program is the foundational improvement on Kan spectral sheaves. Kan s combinatorial spectra give the first construction of the point-set category underlying the stable homotopy category. The 1
2 RUIAN CHEN notion of Kan spectrum is the direct stabilization of simplicial sets, just as the notion of May spectrum is to topological spaces. Since its discovery by Kan [7] in the 1960s, it had not received much attention or follow-up. One reason is that alternate approaches to the stable homotopy category, most notably May spectra, prove very effective, especially in equivariant homotopy theory. In sheaf theory, however, Kan spectra display a unique advantage. K.S. Brown [2] defines generalized sheaf cohomology by developing a theory of sheaves of Kan spectra. On the other hand, a fully functional category of May spectral sheaves has not been constructed. Recently, Chen, Kriz and Pultr [4] substantially improved the foundations of Kan spectral sheaves by constructing, in the language of Guillén et al. [6], a right Cartan Eilenberg structure on the category of Kan spectral sheaves on sites, building on a previous result by Rodríguez González and Roig [8]. Consequently, many right derived functors can be defined and computed via the canonical cosimplicial Godement resolution, including the derived direct image functor Rf as defined by K.S. Brown [2] and hence generalized sheaf cohomology, as well as the derived proper direct image functor Rf! and the derived exceptional inverse image Rj! recently defined by Chen [3]. 2.2. Verdier duality on constructible Kan spectral sheaves. The construction of the functor Rf! naturally leads to the question of Verdier duality for Kan spectral sheaves, analogously to the classical case of abelian sheaves [9], which is itself a refinement of Poincaré duality on manifolds. Specifically, one asks whether there exists a derived functor f! right adjoint to Rf!. In his recent work [3], Chen gives an affirmative answer in the case of a locally closed embedding j, by constructing a strict right adjoint j! to j! and proving that Rj! exists and is right adjoint to Rj!. In parallel with the locally closed Verdier duality, Chen develops, in the same recent work [3], a theory of constructible Kan spectral sheaves on stratified spaces, which is compatible with the six-functor formalism. This suggests the notion of constructible derived category of Kan spectral sheaves, which inherits the triangulated structure from the stable homotopy category, as well as the six-functor formalism from the Kan spectral sheaves. Moreover, the six-functor formalism on the constructible Kan spectral sheaves exhibits a recollement [1]. This allows the gluing of t-structures on the strata, which is provided by the theory of Postnikov towers in this setting, into a perverse t-structure of the entire space. The heart
RESEARCH STATEMENT 3 of this t-structure are again perverse sheaves, hence giving rise to a theory of perverse Postnikov towers. With the foundations set up, Chen investigates the possibility of generalized intersection cohomology, in particular intersection K- theory. Although it has been found out that such a naive, direct generalization does not exist, an incredibly rich structure is discovered in the course of the investigation. This indicates that a more sophisticated common generalization of both generalized cohomology and intersection cohomology is necessary, which Chen proposes to pursue. 3. Ongoing projects and future plans 3.1. Spectral algebra on constructible Kan spectral sheaves. One disadvantage of Kan spectra compared to other approaches to stable homotopy category, particularly May spectra, is the difficulty to define a smash product that is associative, commutative and unital on the point-set level. In the work of EKMM [5], this foundational issue is addressed for May spectra by structuring with the operadic actions. Such studies are now referred to as spectral algebra by the Harvard school of algebraic topology. Chen proposes explore an analogous construction of the symmetric monoidal smash product of Kan spectra, eventually putting a spectral algebra structure on constructible Kan spectral sheaves. The difficulty of defining a smash product already arises even when smashing a Kan spectrum with a based simplicial set. The reason is that, unlike for based topological spaces, the canonical suspension of based simplicial sets does not commute with the smash product. Furthermore, although there exists a comparison map (ΣK) T Σ(K T ), its direction is opposite to what one desires. In [4], this issue is overcome by keeping track of the degrees of the simplicial cells of the Kan spectrum, and to smash the based simplicial set with cells before attaching them to the appropriate degrees. Jointly with Kriz, Chen will extend this technique to smash two Kan spectra. More specifically, one again keeps track of degrees cells in both spectra, and attaches the smashed cells. Different ways of getting a Kan spectrum from such Kan bi-spectrum after smashing are encoded by infinite sequences of 0 s and 1 s, both of which appearing infinitely many times. These are called Adams data. The key of the construction is to organize the Adams data into a simplicial operad, so
4 RUIAN CHEN that the method in [5] can be applied to construct a symmetric monoidal smash product for Kan spectra. Following that, extending the smash product to Kan spectral sheaves is formal. The construction of such a smash product would be a huge breakthrough in algebraic topology. Generally speaking, one obtains this way a model of spectral algebra with every object Quillen-cofibrant, dual to that of May spectra. It would open up a whole new area of research. Specifically for this program, one would not only be able to obtain the full-fledged Verdier duality and six-functor formalisms, but also make possible the consideration of sheaves of Kan module spectra. The latter, in particular, is crucial to applications, as one will see below. 3.2. Applications to singularity theory: classification of extensions of twisted K-theories across singularities. In light of the foundational researches above, Chen pursues applications of the method of generalized sheaf cohomology to the study of singularities. Assuming a spectral algebra structure on constructible Kan spectral sheaves is granted, then the K-theory Kan spectrum K is an E -ring spectrum, and one may consider the K-module spectra and their sheaves. One problem within reach is the classification of extensions of locally constant sheaves of K-modules (corresponding to twisted K- theory) in singular spaces, especially complex varieties. As a preview of such study, a basic example of computations is given below. Example. Consider X = A 2 C stratified by a one-point stratum i : {0} X (the singular locus ) and its complement j : U = A 2 C \{0} X (the regular part ). Then a constructible sheaf F on X restricts to locally constant sheaves on U, which are just twisted K-theories K τ corresponding to twistings τ H 3 (U, Z) = Z. From the stratification, there is an open-closed distinguished triangle i! M F Rj K τ Σ in the constructible derived category, where M is the K-module i! F. To calculate Rj K τ, it suffices to consider its sections supported at {0}. The Postnikov tower of Rj K τ has constant sheaves Z in even dimensions and skyscraper sheaves i Z in odd dimensions. Taking sections supported at {0}, the Postnikov tower has two copies of Z in each odd dimension, connected by the k-invariant of K τ, i.e. the d 3 in the Atiyah Hirzebruch spectral sequence, which is multiplication by n Z associated to τ. Therefore, Γ 0 (X, Rj K τ ) = ΣK/n.
RESEARCH STATEMENT 5 This shows that the category of K-module extensions of K τ from U to X is equivalent to the category of K-modules maps K/n M. Note that this category is, by taking homotopy fibers, equivalent to the category of K-module maps N K/n. Immediate generalizations of this example are the extensions of locally constant sheaves of finitely generated free K-modules in higher dimensions and with other simple singularities. The goal of this program is to develop a method to study singularities on complex varieties using generalized sheaf cohomology theories, which is expected to be a powerful new tool, as normal singularities have complex codimension at least two. Ultimately, the program could potentially lead to an exciting analogue of the geometric Langlands program valued in K-theory, in complex codimension two, and could perhaps also address questions like Arnold s conjecture. References [1] A. A. Beĭ linson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981), Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5 171. [2] Kenneth S. Brown, Abstract homotopy theory and generalized sheaf cohomology, Trans. Amer. Math. Soc. 186 (1973), 419 458. [3] Ruian Chen, Locally closed Verdier duality on constructible Kan spectral sheaves, in preparation, 2017. [4] Ruian Chen, Igor Kriz, and Aleš Pultr, Kan s combinatorial spectra and their sheaves revisited, Theory Appl. Categ. (2017), to appear, preprint available on webpage at https://arxiv.org/abs/1710.00429. [5] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997, With an appendix by M. Cole. MR 1417719 [6] F. Guillén, V. Navarro, P. Pascual, and Agustí Roig, A Cartan-Eilenberg approach to homotopical algebra, J. Pure Appl. Algebra 214 (2010), no. 2, 140 164. [7] Daniel M. Kan, Semisimplicial spectra, Illinois J. Math. 7 (1963), 463 478. [8] Beatriz Rodrí guez González and Agustí Roig, Godement resolutions and sheaf homotopy theory, Collect. Math. 66 (2015), no. 3, 423 452. [9] Jean-Louis Verdier, Dualité dans la cohomologie des espaces localement compacts, Séminaire Bourbaki, Vol. 9, Soc. Math. France, Paris, 1995, pp. Exp. No. 300, 337 349. MR 1610971