arXiv subjects
Publications and source records attributed to Panagiotis Konstantis.
We investigate a combinatorial notion of orientability for abstract GKM graphs and its connections to graph cohomology in the sense of Guillemin--Zara. In particular, we prove that orientability of the GKM graph is equivalent to Poincar\'e duality of the rational (non-equivariant) graph cohomology algebra. As an application, we prove that orientable GKM graphs remain connected after removing any single vertex.
In this paper, we establish three finiteness and boundedness theorems for compact positive monotone symplectic manifolds endowed with special actions, called GKM$_3$, which generalize smooth toric varieties. Specifically, we prove that, for fixed dimension and Euler characteristic, there are only finitely many complex cobordism classes of such spaces. Moreover, modulo lattice transformations, the moment map image can be embedded into a box of explicitly bounded size, and all Chern numbers satisfy quantitative bounds. In particular, this yields a bound on the volume of the underlying symplectic manifold, analogous to the one obtained by Koll\'{a}r-Miyaoka-Mori for Fano varieties.
We prove that the only CROSSes that admit a (not necessarily homogeneous) almost quaternionic structure are ${\mathbb{H}} {\mathbb{P}}^n$ and ${\mathbb{C}} {\mathbb{P}}^2$.
We classify $T^2$-GKM fibrations in which both fiber and base are the GKM graph of $S^4$, with standard weights in the base. For each case in which the total space is orientable, we construct, by explicit clutching, a realization as a $T^2$-equivariant linear $S^4$-bundle over $S^4$. We determine which of the total spaces of these examples are non-equivariantly homotopy equivalent, homeomorphic or diffeomorphic, thereby finding many examples of a) pairs of homotopy equivalent, non-homeomorphic GKM manifolds with different first Pontryagin class, and b) pairs of GKM actions on the same smooth manifold whose GKM graphs do not agree as unlabeled graphs.
We investigate under which conditions an equivariant fiber bundle whose base, total space and fiber are GKM manifolds induces a fibration or fiber bundle of the corresponding GKM graphs. In particular, we give several counterexamples. Concerning the converse direction, i.e., the realization problem for fiber bundles of GKM graphs, we restrict to the setting of fiberwise signed GKM fiber bundles over $n$-gons whose fiber is the GKM graph of a full flag manifold. While it was known that any such bundle is realizable for a $\mathbb{CP}^1$-fiber, we observe that new phenomena occur in higher dimensions where realizability depends on the twist automorphism of the GKM fiber bundle. We classify possible twist isomorphsims and show that realizability can be decided in terms of our classification.
We show that under standard assumptions on the isotropy groups of an integer GKM manifold, the equivariant Stiefel-Whitney classes of the action are determined by the GKM graph. This is achieved via a GKM-style description of the equivariant cohomology with coefficients in a finite field $\mathbb Z_{p}$ even though in this setting the restriction map to the fixed point set is not necessarily injective. This closes a gap in our argument why the GKM graph of a $6$-dimensional integer GKM manifold determines its nonequivariant diffeomorphism type. We introduce combinatorial Stiefel-Whitney classes of GKM graphs and use them to derive a nontrivial obstruction to realizability of GKM graphs in dimension $8$ and higher.
Jang has proven a remarkable classification of $6$-dimensional manifolds having an almost complex circle action with $4$ fixed points. Jang classifies the weights and associated multigraph into six cases, leaving the existence of connected manifolds fitting into three of the cases unknown. We show that one of the unknown cases may be constructed by a surgery construction of Kustarev, and the underlying manifold is diffeomorphic to $S^4 \times S^2$. We show that the action is not equivariantly diffeomorphic to a linear one, thus giving a new exotic $S^1$-action of on a product of spheres that preserves an almost complex structure. We also prove a uniqueness statement for the almost complex structures produced by Kustarev's construction and prove some topological applications of Jang's classification.
It follows from the GKM description of equivariant cohomology that the GKM graph of a GKM manifold has free equivariant graph cohomology, and satisfies a Poincaré duality condition. We prove that these conditions are sufficient for an abstract $3$-valent $T^2$-GKM graph to be realizable by a simply-connected $6$-dimensional GKM manifold. Our realization has the property that any closed stratum of a finite isotropy group contains a fixed point. Furthermore, we argue that in case there exists a fixed point in whose vicinity there occur at most two distinct finite nontrivial isotropy groups such a realization is unique up to equivariant homeomorphism, thus establishing a complexity one GKM correspondence in dimension $6$. We show that the statement on equivariant uniqueness is false without the two conditions on the finite isotropies by providing counterexamples in presence of a fixed point with three distinct neighbouring finite isotropy groups, as well as an example of a simply-connected integer GKM manifold with a closed stratum of a finite isotropy group which does not contain any fixed point.
GKM theory is a powerful tool in equivariant topology and geometry that can be used to generalize classical ideas from (quasi)toric manifolds to more general torus actions. After an introduction to the topic this survey focuses on recent results in low dimensions, where the interaction between geometry and combinatorics turns out to be particularly fruitful.
We construct effective GKM $T^3$-actions with connected stabilizers on the total spaces of the two $S^2$-bundles over $S^6$ with identical GKM graphs. This shows that the GKM graph of a simply-connected integer GKM manifold with connected stabilizers does not determine its homotopy type. We complement this by a discussion of the minimality of this example: the homotopy type of integer GKM manifolds with connected stabilizers is indeed encoded in the GKM graph for smaller dimensions, lower complexity, or lower number of fixed points. Regarding geometric structures on the new example, we find an almost complex structure which is invariant under the action of a subtorus. In addition to the minimal example, we provide an analogous example where the torus actions are Hamiltonian, which disproves symplectic cohomological rigidity for Hamiltonian integer GKM manifolds.
Using the classification of $6$-dimensional manifolds by Wall, Jupp and Žubr, we observe that the diffeomorphism type of simply-connected, compact $6$-dimensional integer GKM $T^2$-manifolds is encoded in their GKM graph. As an application, we show that the $6$-dimensional manifolds on which Tolman and Woodward constructed Hamiltonian, non-Kähler $T^2$-actions with finite fixed point set are both diffeomorphic to Eschenburg's twisted flag manifold $SU(3)//T^2$. In particular, they admit a noninvariant Kähler structure.
We classify fibrations of abstract $3$-regular GKM graphs over $2$-regular ones, and show that all fiberwise signed fibrations of this type are realized as the projectivization of equivariant complex rank $2$ vector bundles over quasitoric $4$-folds or $S^4$. We investigate the existence of invariant (stable) almost complex, symplectic, and Kähler structures on the total space. In this way we obtain infinitely many Kähler manifolds with Hamiltonian non-Kähler actions in dimension $6$ with prescribed one-skeleton, in particular with prescribed number of isolated fixed points.
The set of unrestricted homotopy classes $[M,S^n]$ where $M$ is a closed and connected spin $(n+1)$-manifold is called the $n$-th cohomotopy group $π^n(M)$ of $M$. Moreover it is known that $π^n(M) = H^n(M;\mathbb Z) \oplus \mathbb Z_2$ by methods from homotopy theory. We will provide a geometrical description of the $\mathbb Z_2$ part in $π^n(M)$ analogous to Pontryagin's computation of the stable homotopy group $π_{n+1}(S^n)$. This $\mathbb Z_2$ number can be computed by counting embedded circles in $M$ with a certain framing of their normal bundle. This is a analogous result to the mod $2$ degree theorem for maps $M \to S^{n+1}$. Finally we will observe that the zero locus of a section in an oriented rank $n$ vector bundle $E \to M$ defines an element in $π^n(M)$ and it turns out that the $\mathbb Z_2$ part is an invariant of the isomorphism class of $E$. At the end we show, that if the Euler class of $E$ vanishes this $\mathbb Z_2$ invariant is the final obstruction to the existence of a nowhere vanishing section.
Let $M$ be a symplectic manifold and $G$ a connected, compact Lie group acting on $M$ in a Hamiltonian way. In this paper, we study the equivariant cohomology of $M$ represented by basic differential forms, and relate it to the cohomology of the Marsden-Weinstein reduced space via certain residue formulae using resolution of singularities and the stationary phase principle. In case that $ M $ is a compact, symplectic manifold or the co-tangent bundle of a $G$-manifold, similar residue formulae were derived by Jeffrey, Kirwan et al. for general equivariantly closed forms and by Ramacher for basic differential forms, respectively.
We construct symplectic structures on roughly half of all equal rank biquotients of the form $G//T$, where $G$ is a compact simple Lie group and $T$ a torus, and investigate Hamiltonian Lie group actions on them. For the Eschenburg flag, this action has similar properties as Tolman's and Woodward's examples of Hamiltonian non-Kähler actions. In addition to the previously known Kähler structure on the Eschenburg flag, we find another Kähler structure on a biquotient $\mathrm{SU}(4)//T^3$.
The purpose of this paper is two-fold: On the one side we would like to close a gap on the classification of vector bundles over $5$-manifolds. Therefore it will be necessary to study quaternionic line bundles over $5$-manifolds which are in $1-1$ correspondence to elements in the first cohomotopy group $π^4(M)=[M,S^4]$ of $M$. From previous results this group fits into a short exact sequence, which splits into $H^4(M;\mathbb Z)\oplus\mathbb Z_2$ if $M$ is spin. The second intent is to provide a bordism theoretic splitting map for this short exact sequence, which will lead to a $\mathbb Z_2$-invariant for quaternionic line bundles. This invariant is related to the generalized Kervaire semi-characteristic.
We show that the m-fold connected sum $m\#\mathbb{C}\mathbb{P}^{2n}$ admits an almost complex structure if and only if m is odd.
We give necessary and sufficient topological conditions for the existence of an irreducible ${\rm SO}(3)$-structure on a $5$-manifold. Using these conditions we provide some new examples of $5$-manifolds with an irreducible ${\rm SO}(3)$-structure.