arXiv subjects

Gustavo Granja

Publications and source records attributed to Gustavo Granja.

12 recordsLinked to original sources

arXiv2024

Intersections of complex structures

We study the sets of planes in an even dimensional real vector space $V$ which are simultaneously stabilised by a pair of complex structures on $V$. We completely describe these sets of planes for pairs of orthogonal complex structures. Generically, the number of such planes is finite. We compute this number for orthogonal complex structures and prove that it gives a lower bound for the number of planes simultaneously stabilised by a generic pair of complex structures on $V$.

math.RA↗

arXiv2022

Topology of Almost Complex Structures on Six-Manifolds

We study the space of (orthogonal) almost complex structures on closed six-dimensional manifolds as the space of sections of the twistor space for a given metric. For a connected six-manifold with vanishing first Betti number, we express the space of almost complex structures as a quotient of the space of sections of a seven-sphere bundle over the manifold by a circle action, and then use this description to compute the rational homotopy theoretic minimal model of the components that satisfy a certain Chern number condition. We further obtain a formula for the homological intersection number of two sections of the twistor space in terms of the Chern classes of the corresponding almost complex structures.

math.DG↗

arXiv2021

On the topology of the space of almost complex structures on the six sphere

The space of orientation-compatible almost complex structures on the six-dimensional sphere naturally contains a copy of seven-dimensional real projective space. We show that the inclusion induces an isomorphism on fundamental groups and rational homotopy groups. We also compute the homotopy fiber of the inclusion and the homotopy groups of the space of almost complex structures in terms of the homotopy groups of the seven-dimensional sphere. Our approach lends itself to generalization to components of almost complex structures with vanishing first Chern class on six-manifolds.

math.AT↗

arXiv2021

The cohomological index of free $\Z/p$-actions is not additive with respect to join

Let $p$ denote an odd prime. We show by example that the inequalities obtained in arXiv:1704.05827 for the behaviour of the cohomological index of a join of free $\Z/p$-actions are sharp. Namely, for all odd integers $k,l$ at least one of which is greater than one, we give examples of finite free $\Z/p$-CW complexes of cohomological indices $k$ and $l$ whose join has index $k+l+1$ and also examples where the join has index $k+l-1$.

math.AT↗

arXiv2020

Givental's non-linear Maslov index on lens spaces

Givental's non-linear Maslov index, constructed in 1990, is a quasimorphism on the universal cover of the identity component of the contactomorphism group of real projective space. This invariant was used by several authors to prove contact rigidity phenomena such as orderability, unboundedness of the discriminant and oscillation metrics, and a contact geometric version of the Arnold conjecture. In this article we give an analogue for lens spaces of Givental's construction and its applications.

math.SG↗

arXiv2016

Symplectic embeddings in infinite codimension

Let $X$ be a union of a sequence of symplectic manifolds of increasing dimension and let $M$ be a manifold with a closed $2$-form $ω$. We use Tischler's elementary method for constructing symplectic embeddings in complex projective space to show that the map from the space of embeddings of $M$ in $X$ to the cohomology class of $ω$ given by pulling back the limiting symplectic form on $X$ is a weak Serre fibration. Using the same technique we prove that, if $b_2(M)<\infty$, any compact family of closed $2$-forms on $M$ can be obtained by restricting a standard family of forms on a product of complex projective spaces along a family of embeddings.

math.SG↗

arXiv2014

Self maps of HP^n via the unstable Adams spectral sequence

We use obstruction theory based on the unstable Adams spectral sequence to construct self maps of finite quaternionic projective spaces. As a result, a conjecture of Feder and Gitler regarding the classification of self maps up to homology is proved in two new cases.

math.AT↗

arXiv2014

A continuous deRham antidifferential

Let M be a manifold, possibly with boundary. We show that the deRham differential from k-forms to exact (k+1)-forms has a continuous right inverse when both spaces are given the weak Whitney topology. This antidifferential operator is given a fairly explicit formula depending on the choice of a suitable good cover of M and local coordinates on the elements of the cover. If the antidifferential operator is applied to a smooth family of exact forms, it produces a smooth family of forms.

math.DG↗

arXiv2009

Compatible complex structures on symplectic rational ruled surfaces

In this paper we study the topology of the space $\I_ω$ of complex structures compatible with a fixed symplectic form $ω$, using the framework of Donaldson. By comparing our analysis of the space $\I_ω$ with results of McDuff on the space $\cat J_ω$ of compatible almost complex structures on rational ruled surfaces, we find that $\I_ω$ is contractible in this case. We then apply this result to study the topology of the symplectomorphism group of a rational ruled surface, extending results of Abreu and McDuff.

math.SG↗

arXiv2007

Realizing modules over the homology of a DGA

Let A be a DGA over a field and X a module over H_*(A). Fix an $A_\infty$-structure on H_*(A) making it quasi-isomorphic to A. We construct an equivalence of categories between A_{n+1}-module structures on X and length n Postnikov systems in the derived category of A-modules based on the bar resolution of X. This implies that quasi-isomorphism classes of A_n-structures on X are in bijective correspondence with weak equivalence classes of rigidifications of the first n terms of the bar resolution of X to a complex of A-modules. The above equivalences of categories are compatible for different values of n. This implies that two obstruction theories for realizing X as the homology of an A-module coincide.

math.AT↗

arXiv2005

Moment maps, symplectomorphism groups and compatible complex structures

In this paper we apply Donaldson's general moment map framework for the action of a symplectomorphism group on the corresponding space of compatible (almost) complex structures to the case of rational ruled surfaces. This gives a new approach to understanding the topology of their symplectomorphism groups, based on a result of independent interest: the space of compatible integrable complex structures on any symplectic rational ruled surface is (weakly) contractible. We also explain how in general, under this condition, there is a direct relationship between the topology of a symplectomorphism group, the deformation theory of compatible complex structures and the groups of complex automorphisms of these complex structures.

math.SG↗

arXiv2003

Homotopy decomposition of a group of symplectomorphisms of $S^2 \times S^2$

We continue the analysis, started by Abreu, McDuff and Anjos, of the topology of the group of symplectomorphisms of $S^2 \times S^2$ when the ratio of the areas of the two spheres lies in the interval (1,2]. We express the group, up to homotopy, as the amalgam of certain of its compact Lie subgroups. We use this to compute the homotopy type of the classifying space of the group of symplectomorphisms and the corresponding ring of characteristic classes for symplectic fibrations.

math.AT↗