In Ayzenberg, Masuda and myself'23 it was claimed (without giving a correct argument) that the 2-skeleton Q_2 of the orbit space Q for a GKM_4 manifold is simply connected. Such a claim was further used in the proof of extension for the respective GKM graph to a torus graph (in complexity 1 case, which was later proved rigorously by Goertsches and myself'25).

In this paper, a rich source of examples is provided, each disproving this claim. Namely, for every topological or smooth manifold W^4 bounding a Brieskorn homology 3-sphere there exists a topological or smooth, respectively, (equivariantly formal) torus manifold with the orbit space Q=W^{4}, and the face structure on Q_{3} is regular. This implies, that the fundamental group of the respective Q_2 is that of the Brieskorn sphere. As an example, in the topological case one can take the Poincaré sphere S(2,3,5); in the smooth case one can take S(2,5,7), for instance. Certainly, the respective Q_2 is not shellable (as a regular CW complex), which was an open question of myself (and studied in the PhD dissertation of N. Wardenski) since at least 2018.

The above topological result is fairly short and self-contained, relying on the Davis-Januszkiewicz construction for a certain triangulation of the boundary for W^4. As it is well known, this construction is not smooth, in general. Instead of the DJ construction, the smooth case relies on the recent realization result for locally standard smooth torus manifolds by Karshon and Kuroki. The required smooth manifold with corners structure on the collar of a Brieskorn sphere is obtained by real oriented blow-ups (following Gaiffi) for a certain smooth conical stratification. This stratification is explicitly constructed on the above homology spheres using the S^1-bundle structure on Seifert 3-manifolds. The regularity of the resulting CW complex follows directly from Poincaré conjecture solution.

A by-product positive result is extension to a torus graph for the GKM graph of any GKM_4 manifold in complexity 2, and 3 provided that the fundamental group of Q_2 is finite. (As the above examples show, this group is not necessarily trivial.) The proof follows that from [GS'25], with additional facts on perfect or finite subgroups in SL_c(Z) for c=2,3.

In this paper, we prove that the only CROSSes that admit an almost quaternionic structure are HP^{n} and CP^{2}. Arguably, the most interesting case is that of CP^{4n+2}, n>0. My co-authors found an alternative argument using divisibility properties for Chern numbers of certain complex vector bundles given by the classical Kummer and Lucas theorems. However, this argument did not work in the complex dimensions 2^k-2 for all k>1.

My contribution was in finding an alternative proof (using an algebraic topology approach, by a contradiction) that covered all dimensions for CP^{4n+2}, n>0. The first observation is that the complexified tangent bundle has a rank 2 complex vector bundle tensor factor. The second observation is that the respective Chern roots amount to roots of a certain quadratic equation, and the respective expression (2-adic series) in complex K-theory ring tensor 2-adic numbers converges. This leads to a contradiction by parity after evaluating the respective series at -2, by comparison with the Chern class of the tangent bundle for the complex projective space.. 

Remarkably, a very similar argument to the above forbids existence of pluricomplex/paraconformal structures (about which I learned from D. Harland) on CP^{4n+2}, n>0.

Eilenberg-MacLane spaces are well-known to be rationally formal. In this paper, we prove that abelian EML are naturally formal over Q, i.e. the respective zigzag of quasi-isomorphisms is compatible with the respective continuous maps. (The choice of a representative for such a map is not important by a simple rectification result, and therefore can be chosen to be induced by the respective abelian group homomorphism.) In the particular case of tori this claim was proved by Franz in his dissertation. We generalize it to arbitrary abelian EML spaces of any degree. The proof uses the iterated bar and W-construction of simplicial abelian groups, following Gugenheim and May.

Formality of diagrams for (abelian) EML spaces over a fixed small category is defined in a similar way. As an application, for any diagram of EML spaces of a fixed degree we deduce a cohomological spectral sequence collapse over Q at its second page (converging to the cohomology of the respective homotopy colimit). This class of spaces includes toric varieties, partial quotients of moment-angle complexes and other interesting spaces equipped with a compact torus action, appearing in toric topology (as well their Borel spaces).  

By contrast, we show that formality for EML diagrams over k does not hold for diagrams over the category generated by an abelian group C such that: C\otimes k is nonzero and k is not a Q-algebra. The proof relies on a functor calculus argument, together with the classical results on cohomology operations for EML spaces, following H. Cartan.

My contributions to this project are the proofs of rational formality (restated in a more elegant form by my co-author) and of the spectral sequence collapse. 

In this paper, we study in greater detail when extensions of GKM graphs induce epmorphisms of the respective graph equivariant cohomology. The claim appeared unproved in Ayzenberg, Masuda and myself'2023.  We find a counterexample to this claim, and prove it (alongside with Cohen-Macaulayness) in the narrower class of GKM_3 manifolds (satisfying standard assumption of connected spabilizers for integral coefficients case) using commutative algebra. The main ingrediend of the proof is a new version of Atiyah-Bredon-Franz-Puppe sequence (up to degree 2) for GKM_3 graphs, which is nothing other than the cellular homology of the respective GKM sheaf over the CW poset of i-faces, i<3, in the corresponding GKM graph. As in [AMS23], this implies that the graph equivariant cohomology algebra of any GKM_3 manifold is the quotient of the face ring by the ideal generated by certain degree 2 linear forms.

The result is formulated for unsigned GKM manifolds (not requiring existence of an invariant almost complex structure). In order to have equivariant Thom classes and face ring isomorphism in this case, we prove that every torus graph has a lift to a T-graph, for which both such notions hold. This by-product result is useful on its own, because it explains how various (at least 3) definitions of GKM-type graphs are related to each other in complexity 0.

My contribution to this project was the ABFP-type sequence definition (in the sheaf as well as cellular form, of which only second part is recorded in the present text), the verification for non-CM property for the counterexample (reproved more elegantly by my co-author), and the proof of the lift existence.

This joint work continues the study of Masuda's problem in the family of homogeneous GKM manifolds. This time, we were able to prove it in the strong form (i.e. extension of the torus action on a GKM_4 manifold to a half-dimensional GKM action) by means of classification results in the particular case of homogeneous spaces. An important feature of homogeneous GKM manifolds is that the linear dependencies for weights of the T-representation at a fixed point (i.e. respective matroid of root subsystem complement) do not depend (up to vector space isomorphism) on the point choice. Borel-de Siebenthal classification reduces the study of subgroups to finitely many subgraphs of Dynkin diagrams, however it quickly becomes intractable outside maximal subgroup cases.

A crucial new observation: a homogeneous GKM_3 space is symmetric (class 1 of inner type). (The converse also holds modulo several exceptions, which we record in this work.) The beautiful and short proof of this fact uses Lie bracket structure. We proceed with the characterization of GKM_4 homogeneous manifolds (a stronger condition than GKM_3) by Cartan's classification. The case study of the respective subgroups splits into classical and exceptional cases. In the classical case, we describe all small linear dependencies (i.e. 3-circuits of matroids of root subsystem complement) using signed weighted graphs (a generalization of classical signed graphs of Harari and Zaslavsky). 

In the exceptional cases, the 3-circuits are easy to describe explicitly. If such linear dependence exists then the complement lies in an affine hyperplane with one coordinate being 1. Then, it is enough to study only maximal subgroups of exceptional group (given by Borel-de Siebenthal and carefully recorded e.g. by I. Yokota). The resulting case study is much more tractable than we initially expected. As a corollary, we provide new vanishing results for the degree 4 homology (with Z or Z[1/2] coefficients) of orbit spaces for homogeneous GKM manifolds using acyclicity results of Ayzenberg, Masuda and myself. (Equivariant formality, i.e. odd cohomology vanishing for the homogeneous space, holds either with coefficients are Z or Z[1/2].)

My contribution to this project was mainly in the study of exceptional group cases, and in the part of the acyclicity results. 

In this work, we realize an operad morphism from 2-HyCom to HyCom in DGA by those in topological spaces. The morphism is given object-wisely by the quotient for the Stanley-Reisner ring by a certain linear ideal. Davis-Januszkiewicz spaces and the wonderful compactification of the braid arrangement serve as the respective collections. However, the operad structure is more elusive. (The operads on DJ space constructed by T. Bahri et al., and by A. Ayzenberg, do not induce the correct map in cohomology, which is not hard to see.)

I contributed here by the following key idea: to take the colimit of a stabilization for the dC--P operad given by Neumann's c-plexification operation for arrangements in C^d. It clearly induces the correct map in cohomology after taking colimit, by virtue of Milnor's lim1 theorem. I showed that the respective colimit is the DJ space up to homotopy, by describing the homotopy type for the colimit of blow-ups along centers with normal bundle ranks tending to the infinity. Following a conjecture of E. Hoefel, I proved that the intermediate operad CGK_d is isomorphic to the operad of stable trees of P^d of Chen-Gibney-Krashen (having an obvious operad structure). It compactifies the quotient of configuration spaces by the group of affine transformations.

The intermediate-dimensional operads CGK_d serve as a natural generalization for Grav and HyCom operads studied by Getzler. We prove a number of relevant results. We give a generators-and-relations description for the homology of the open part and cohomology of the compactification. The first part refines a result of Westerland. I contributed to the proof that relies on the Getzler spectral sequence collapse and trivial abutment, by explaining splitting for the mixed Hodge structure of the open part cohomology. The remaining computation is conducted in both cases by Gröbner bases. We prove that the associated graded for the compactification cohomology is Koszul dual to homology of the open part. Surprisingly, the abutment problem for the compactification is nontrivial, and the expected Koszul duality lifts only on the level of oo-operads.

In this overview of applications for sheaf theory to modern Computer Science and Topological Data Analysis, my input was merely in writing certain parts of mathematical addendums (C, in particular). I explained to my co-authors (who know much more about the applied part of the project than me) Grothendieck's theorem on universal delta functor (a well known result for algebraic geometers), pointed out that any presheaf is a sheaf on a finite topological space, a reference to the original work of Roos for cochain complexes computing right derived limits, an earlier result of Everitt and Turner on isomorphism for cellular and sheaf homology on finite spaces, provided details on general sheaf theory, and discovered many interesting works of Husainov.

In this paper (a companion paper for [ST26]) rational (singular or equivariant) cohomology for homotopy colimits of toric diagrams. The focus is on skeletons toric varieties in this class, where a number of explicit formulas is written down. The proofs are based on the Mayer-Vietoris spectral sequence collapse from [ST26]. Caution: the respective "natural formality" argument holds only over Q (see [ST26]). 

I re-prove in a different way the orbit spectral sequence collapse previously established (for Q coefficients) in works of B. Totaro, and V. Danilov. This relies on the isomorphism (Everitt, Turner'15) between sheaf cohomology of posets (appearing at the second page of MVSS) and cellular homology for the respective sheaf (showing up at OSS second page). 

Another central result of this work is Cohen-Macaulayness of the torus action on the skeletons for smooth compact toric varieties, serving as the equivariant formality counterpart without fixed points.

I compare MVSS to Eilenberg-MacLane SS by a Grothendieck-type 3-graded spectral sequence. I computed all bigraded Betti numbers for skeletons of any smooth projective toric variety X. The resulting formulas (for ordinary Betti numbers) agree with those following from Xin Fu's homotopy decomposition in the particularly nice case of skeletons for a complex projective space. The comparison of our results involves a particular identity for the generalized hypergeometric function. 

This joint work with O. Goertsches follows the problem of torus acton extension on "intermediately" independent GKM manifolds (see [8] below), that was posed by M. Masuda. We resolved the combinatorial part of the problem in two interesting cases. Namely, we proved that any 4-independent (a.k.a. GKM_4) GKM-graph of a GKM manifold (of dimension 2n with acting torus having dimension k) is extendible to a complexity 0 GKM graph (a.k.a. torus graph), provided that the manifold is either Hamiltonian or has complexity n-k=1.

My input was first in providing with a coordinate-free (global sections of a certain sheaf on graph) description for S. Kuroki obstruction to GKM graph extensions (an axial function group), and second in reducing the computation for this group to that for subgraphs in the GKM graph. (The first part was presented in my talk on 2019 "Toric Topology in Okayama" conference; thanks to Oliver and his collaborators, I was able to apply these ideas successsfully much later.) 

An important data for GKM graph extension problem is the connection on the GKM graph. It provides with the notion of parallel transport on the graph, and with i-faces (i.e. i-valent connected subgraphs that are invariant with respect to the parallel transport). The reduction is possible if the fundamental group of the graph is generated by 2-faces. Establishing this property is open in general (and has something to do with shellings of the following regular CW complex: the 2-skeleton for the respective GKM manifold orbit space). We were able to show this property in the Hamiltonian case (using convexity of the moment polytope by Atiyah, Guillemin and Sternberg), and in complexity 1 case (using universal covers of GKM graphs and cohomology of covers as invariants for the deck transformation group). Our results seem to fill the gaps in several preceeding works. 

In this joint work with S. Kuroki, we obtained a graph-theoretical proof of the well-known formula for the equivariant cohomology of a projectivization for a complex toric vector bundle over a GKM manifold. Passage from topological space to combinatorial objects (graphs with labels) is given by GKM theory, provided that the projectivization satisfies pairwise independence of weights at fixed points (i.e. is a GKM manifold). The main tool is the notion of leg bundle corresponding to the above vector bundle (which is an invention of my co-author, following ideas from his earlier joint paper with V. Uma). 

My input in this project was in proving that any projective GKM bundle (in sense of Guillemin, Sabatini and Zara) is a projectivization of a leg bundle if one allows rational labels of the legs. This, together with the BH type formula, provided with graph equivariant cohomology of any projective GKM bundle. (Namely, with Q coefficients; with Z coefficients this seems to be an open problem for abstract GKM projective bundles). Notice that GSZ proved only free module structure for the respective ring (i.e. Leray-Hirsch type theorem), but did not describe the multiplicative relation. In general, a projective GKM bundle may not have a horizontal leaf isomorphic to the base, so it is totally unclear how to represent it as a projectivization. 

I also explained that projectives are projectivizations in the category of topological fiber bundles, which is based on an old result of J.C. Su. Caution: in the adopted definition for T-vector bundles, the respective action on fibers is not necessarily linear (e.g. see Petrie's vector bundles over P^3), which was pointed out to us by O. Goertsches.

This work takes homology vanishing of orbit spaces for GKM manifolds (by Ayzenberg and Masuda) further using homotopy colimits. Caution: the proofs of crucial Prop.5.18 (2-skeleton of the orbit space is not necessarily simply connected) and Prop.6.7 (extension of GKM graphs does not induce a surjection in graph equivariant cohomology, in general) are incorrect as given. Nevertheless, all these problems are resolved, and the main results are generalized in the joint works of myself [GS'25], [GS'26], [S'26].

For any GKM manifold with a sufficiently generic (j-independent, i.e. any i<=j weights at the same fixed point are linearly independent) fixed point data the respective orbit space has vanishing (reduced with Z coefficients) homology up to degree j+1, by Ayzenberg and Masuda. An interesting phenomenon is that all known GKM manifolds (of dimension 2n, with compact k-dimensional torus action) that are j-independent (and not extendible to a higher dimensional effective torus action) satisfy either j<4 or j=n (i.e. torus manifold), which was observed by M. Masuda. I constructed GKM graphs that are k-independent (for any n>k>2). The first step of this construction is to take a periodic (with respect to a certain subgroup of translations) GKM graph in R^n (this was considered earlier by S. Kuroki, unpublished). The second step adds edges to this graph, preserving periodicity and increasing "complexity" n-k. The third step is to take quotient of this infinite GKM graph, resulting in a finite GKM graph embedded to a torus. I show that this GKM graph cannot be realized by a (equivariantly formal) GKM manifold, using the generalization of mentioned above homology vanishing for orbit space, see [9]. The argument is based on the computation for the Euler characteristic of certain face posets of GKM graph using P. Hall formula. (The possibly nontrivial homology is only in one degree, which is a condition on the sign of the Euler characteristic, which is not satisfied for my examples.) An interesting open problem is to find the graph equivariant cohomology ring for these examples.

Milnor hypersurfaces H_{i,j}, i<=j, are rational projective nonsingular varieties known (in algebraic topology) to be additive generators for the complex bordism ring by Hirzebruch.  The hypersurface is invariant with respect to a subtorus in the naturally acting torus on the ambient toric variety (product of complex projective spaces). It is well known that this subtorus does not extend to a dense open orbit on Milnor surfaces (except i=0 or the case studied in [5] below) by looking at the respective cohomology ring. I proved that this torus action cannot be extended to any higher dimensional torus algebraic action, by explicitly describing the respective automorphism group. (The result is probably known to experts in algebraic geometry by using very ample embeddings.) I linked Milnor hypersurfaces with the hypersurfaces of Ray, and Buchstaber and Ray by iterated blow-up constructions. The latter two hypersurfaces are degenerate, therefore the cohomology ring description is not known in general (but there is a formula in terms of annihilator). Then I showed that these hypersufraces do not have a structure of a toric variety in general. This justified my previous work [4], namely, there exist no short proof using their hypersurfaces. The proof involved a certain generalization of GKM graph notion to hypergraphs (because the isotropy representation for the torus action at fixed points had multiplicities >1, in general), and a study of monodromy with respect to a "connection" on these hypergraphs.

M. Franz showed that any toric variety has a homotopy decomposition into a homotopy colimit of a certain diagram (whose objects are compact tori, arrows are group homomorphisms, and the indexing category corresponds to the cone poset of the fan). In this paper, I proved a slight generalization of this result in the category of moment-angle complexes and their quotients. Another input of mine was the Eilenberg-MacLane spectral sequence collapse (on page 2 with Z coefficients) for the associated Borel fibration. (This fits well into a series of EMSS collapse theorems in toric topology.) The proof is based on "natural formality" argument following Notbohm and Ray. Caution: the argument holds only over Q, which was pointed out to me by M. Franz. (The diagram (26) is not commutative, in general; this can be fixed by choosing a different natural quasi-isomorphism.) The correct proof over Q is given in my joint work [ST26].

In the appendix to a paper by S..P. Novikov (written joint with A. Mischenko) it was observed that certain Minor hypersurfaces are complex bordant to Cartesian product of complex projective spaces. (This was used in their proof for the computation for the logarithm of the universal formal group law for complex cobordism theory.) The proof was implicit (by comparing the respective Chern numbers). Since 1967, a problem remained open to construct explicitly a complex bordism between these two complex manifolds (which gathered attention since 2000). I was first to solve this problem (in particular, in two different ways). The first construction relied on a "well known construction", a gluing procedure based on hexagon, from a paper by B. Totaro. The second construction was based on a general construction for bordisms in the category of stably complex orbifolds with quasitoric boundary by S. Sarkar. Notice that both constructions lead to a null-bordant connected component. The construction of bounding manifold for it remains an open problem.

A compact manifold with stably complex structure is equivalent to a sum of line bundles is called stably tangentially split manifold, or a TTS manifold. The definition of a TNS manifold is given in a similar way by replacing the stably tangential stably complex structure with the complementary vector bundle (i.e. such that the Whitney sum of both fiber bundles is a trivial complex vector bundle). These two classes appeared in the works of Arthan and Bullet, and Ochanine. N. Ray proved that every complex bordism class (of degree >2) has a manifold that is TNS and TTS at the same time. Buchstaber and Ray proved s similar result, but for quasitoric manifolds. I prove a unification for these two results, namely, in the class of quasitoric TNS manifolds. (Any such manifold is TTS automatically.) The proofs followed my previous work with Y. Ustinovsky (in particular, computations for Chern numbers), with an increment in number-theoretical difficulty related to divisibility properties of binomial coefficients (based on known generalizations of Lucas theorem in number theory).

The class of compact TNS manifolds of dimension 4 (see [4] above) was characterized by J. Lannes in terms of non-semidefiniteness for the intersection form. I took this further in the class of quasitoric manifolds of arbitrary dimension, by replacing intersection form with degree k polynomial forms (defined by pairing the cup-product of any fixed x and k-th power of the argument y, with the fundamental class of the manifold). My criterion of TNS property for quasitoric manifolds was in terms of non-semidefiniteness of these forms, or in terms of K-theory, or in terms of the volume polynomial (by virtue of Khovanskii and Pukhlikov description for the cohomology). The proof relied on the study of certain cones (over particular convex bodies) in the complex K-theory (as a vector space over Q) of the quasitoric manifold. (The passage from cohomology to K-theory tensor Q is by Chern character.) Algebraically, this corresponds to Hilbert's 17th problem, Pfister theory etc (e.g. see nice book "Sums of even powers of real linear forms" by Bruce Reznick). Even though the obtained criterion involves infinitely many forms, it is algorithmically verifiable by Tarski's theorem (which was suggested to me by A. Ayzenberg). In (complex) dimension 3 I characterized explicitly (nonsingular projective) TNS toric varieties in terms of the combinatorics for the fan. In higher dimension I proposed a conjecture, potentially related to Hodge-Riemann relations and Horrocks theorem in algebraic geometry.

This work completes a partial series of generators built by A. Wilfong (among nonsingular projective toric varieties) for the complex bordism ringin the remaining dimensions 2n (where n is even and n+1 is not a power of a prime). By a theorem of Milnor and Novikov, this problem reduces to divisibility propoerties of the s-number, a certain Chern number for toric varieties. I found a series of two stage blow-up operations B_k defined for any toric variety that changes the s-number in a way that depends only on its dimension (using Sage). Then I computed these changes precisely, which turned out to be a rather long task with cohomology of toric varieties and their Chern numbers. An additional task was to prove that the required divisibility property can be achived by sufficiently many compositions of these operations. This step relied on the classica; Lucas theorem, which was achieved by joint effort with Y. Ustinovsky. (He also explained a classical Frobenius result on conductors for semigroups to me, and suggested studying this problem using my operations, for which I am grateful to him.)

The Poincaré series of a multi-indexed filtration on the ring of germs for a germ of plane cruve coincides with the Alexander polynomial of the corresponding link, which was proved by Ebeling and Gusein-Zade. The obtained expression had an elegant A'Campo type formula (ratio of product for cyclotomic polynomials). It was conjectured (motivated by HOMFLY polynomial interpretations) that a similarly looking multi-index filtration on the same ring provides with another topological invariant of the singularity, potentially described by a similar rational function formula. The first part turned out not to be the case, which is my observation. I provided with a very simple example of two (nondegenerate with respect to the same Newton diagram) curves with different Poincaré series, by an explicit computation. (It required writing a rather long C++ code to find the first coefficients of the Poincaré series.) In general, computation of the Poincaré series for this filtration seems to be an open problem.