A note on the modular representation on the $\mathbb Z/2$-homology groups of the fourth power of real projective space and its application

We write $BV_h$ for the classifying space of the elementary Abelian 2-group $V_h$ of rank $h,$ which is homotopy equivalent to the cartesian product of $h$ copies of $\mathbb RP^{\infty}.$ Its cohomology with $\mathbb Z/2$-coefficients can be identified with the graded unstable algebra $P^{\otimes h} = \mathbb Z/2[t_1, \ldots, t_h]= \bigoplus_{n\geq 0}P^{\otimes h}_n$ over the Steenrod ring $\mathcal A$, where grading is by the degree of the homogeneous terms $P^{\otimes h}_n$ of degree $n$ in $h$ generators with the degree of each $t_i$ being one. Let $GL_h$ be the usual general linear group of rank $h$ over $\mathbb Z/2.$ The algebra $P^{\otimes h}$ admits a left action of $\mathcal A$ as well as a right action of $GL_h.$ A central problem of homotopy theory is to determine the structure of the space of $GL_h$-coinvariants, $\mathbb Z/2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}H_n(BV_h; \mathbb Z/2) ,$ where ${\rm Ann}_{\overline{\mathcal A}}H_n(BV_h; \mathbb Z/2) ={\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}_n]^{*}$ denotes the space of primitive homology classes, considered as a representation of $GL_h$ for all $n.$ Solving this problem is very difficult and still unresolved for $h\geq 4.$ The aim of this Note is of studying the dimension of $\mathbb Z/2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}_n]^{*}$ for the case $h = 4$ and the "generic" degrees $n$ of the form $k(2^{s} - 1) + r.2^{s},$ where $k,\, r,\, s$ are positive integers. Applying the results, we investigate the behaviour of the Singer cohomological "transfer" of rank $4$, which is a homomorphism from a certain subquotient of the divided power algebra $\Gamma(a_1^{(1)}, \ldots, a_4^{(1)})$ to mod-2 cohomology groups ${\rm Ext}_{\mathcal A}^{4, 4+n}(\mathbb Z/2, \mathbb Z/2)$ of the algebra $\mathcal A.$ Singer's algebraic transfer is one of the relatively efficient tools in determining the cohomology of the Steenrod algebra.

Quadratic Differential Systems with a Finite Saddle-Node and an Infinite Saddle-Node (1,1)SN - (A)

Our goal is to make a global study of the class [Formula: see text] of all real quadratic polynomial differential systems which have a finite semi-elemental saddle-node and an infinite saddle-node formed by the coalescence of a finite and infinite singularities. This class can be divided into two different families, being (A) possessing the finite saddle-node as the only finite singularity and (B) possessing the finite saddle-node and also a finite simple elemental singularity. In this paper we provide the complete study of the geometry of family (A). The family (A) modulo the action of the affine group and time homotheties are four-dimensional and we give the bifurcation diagram of its closure with respect to a specific normal form, in the four-dimensional real projective space [Formula: see text]. As far as we know, this is the first time that a complete family is studied in the four-dimensional real projective space. The respective bifurcation diagram yields 36 topologically distinct phase portraits for systems in the closure [Formula: see text] within the representatives of [Formula: see text] given by a specific normal form.

Cyclic p-group actions on ℝℙ3

In this paper, we classify the smooth orientation preserving cyclic [Formula: see text]-group actions on the real projective space [Formula: see text] up equivalence, where two actions are equivalent if their images are conjugate in the group of self-diffeomorphisms. We view [Formula: see text] as the lens space [Formula: see text]. We show that any such action on [Formula: see text] is conjugate to a standard action explicitly defined, and we identify the quotient spaces of these actions. In addition, we enumerate the equivalence classes.

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.

Balanced Triangulations on few Vertices and an Implementation of Cross-Flips

A $d$-dimensional simplicial complex is balanced if the underlying graph is $(d+1)$-colorable. We present an implementation of cross-flips, a set of local moves introduced by Izmestiev, Klee and Novik which connect any two PL-homeomorphic balanced combinatorial manifolds. As a result we exhibit a vertex minimal balanced triangulation of the real projective plane, of the dunce hat and of the real projective space, as well as several balanced triangulations of surfaces and 3-manifolds on few vertices. In particular we construct small balanced triangulations of the 3-sphere that are non-shellable and shellable but not vertex decomposable.

Rigidity of complex convex divisible sets

An open convex set in real projective space is called divisible if there exists a discrete group of projective automorphisms which acts cocompactly. There are many examples of such sets and a theorem of Benoist implies that many of these examples are strictly convex, have [Formula: see text] boundary, and have word hyperbolic dividing group. In this paper we study a notion of convexity in complex projective space and show that the only divisible complex convex sets with [Formula: see text] boundary are the projective balls.

Dynamics of real projective transformations

<div data-canvas-width="358.1721365118214">The dynamics of a projective transformation on a real projective space are studied in this paper. The two main aspects of these transformations that are studied here are the topological entropy and the zeta function. Topological entropy is an inherent property of a dynamical system whereas the zeta function is a useful tool for the study of periodic points. We find the zeta function for a general projective transformation but entropy only for certain transformations on the real projective line.</div>