Free Involutions Research Articles

The Eulerian Distribution on the Fixed-Point Free Involutions of the Hyperoctahedral Group Under the Natural Order

The Eulerian Distribution on the Fixed-Point Free Involutions of the Hyperoctahedral Group Under the Natural Order

Orbit spaces of free involutions on the product of three spheres

In this paper, we have determined the orbit spaces of free involutions on a finitistic space having mod 2 cohomology of the product of three spheres . This paper generalizes the results proved by Dotzel et al. [7] for free involutions on the product of two sphere . As an application, we have also derived the Borsuk-Ulam type results.

Cohomology algebra of orbit spaces of free involutions on the product of projective space and 4-sphere

Let X be a finitistic space with the mod 2 cohomology of the product space of a projective space and a 4 -sphere. Assume that X admits a free involution. In this paper we study the mod 2 cohomology algebra of the quotient of X by the action of the free involution and derive some consequences regarding the existence of \mathbb{Z}_2 -equivariant maps between such X and an n -sphere.

The Eulerian distribution on the fixed-point free involutions of the hyperoctahedral group

The Eulerian distribution on the fixed-point free involutions of the hyperoctahedral group

Free involutions on odd dimensional Wall manifolds and cohomology of orbit spaces

The main purpose of this paper is to compute all possible mod 2 cohomology algebra of orbit spaces of free involutions on odd dimensional Wall manifolds of type Q(m,n), which are the closed differentiable manifolds constructed as mapping torus of specific involutions on Dold manifolds P(m,n), for m even and n odd.

Fixed point free actions of spheres and equivariant maps

The concept of index and co-index of a paracompact Hausdorff space X equipped with free involutions relative to the antipodal action on spheres were introduced by Conner and Floyd [2]. In this paper, we extend the notion of index and co-index for free G-spaces X, where X is a finitistic space and G=S1 (with complex multiplication) and G=S3 (with quaternionic multiplication). We prove that the index of X is less than or equal to the mod 2 cohomology index of X. We also compute the ring cohomology of the orbit space X/G, where G=S1 or G=S3 and X is a finitistic space whose cohomology ring is the same as the product of spheres Sn×Sm,1≤n≤m. Using these cohomological calculations, we obtain some Borsuk-Ulam type results.

Cohomology Algebra of Orbit Spaces of Free Involutions on Some Wall Manifolds

In this paper, we investigate the existence of free involutions on some Wall manifolds and we compute the mod 2 cohomology algebra of the correspondent orbit space.

Central limit theorem for descents in conjugacy classes of Sn

The distribution of descents in fixed conjugacy classes of Sn has been studied, and it is shown that its moments have interesting properties. Fulman proved that the descent numbers of permutations in conjugacy classes with large cycles are asymptotically normal, and Kim proved that the descent numbers of fixed point free involutions are also asymptotically normal. In this paper, we generalize these results to prove a central limit theorem for descent numbers of permutations in any conjugacy class of Sn.

Kazhdan–Lusztig R-polynomials for pircons

The purpose of this work is to provide a common combinatorial framework for some of the analogues and generalizations of Kazhdan–Lusztig R-polynomials that have appeared since the introduction of these remarkable polynomials (e.g., parabolic Kazhdan–Lusztig R-polynomials, Kazhdan–Lusztig R-polynomials of zircons, and Kazhdan–Lusztig–Vogan polynomials for fixed point free involutions).

The Borsuk–Ulam property for homotopy classes of self-maps of surfaces of Euler characteristic zero

Let M and N be topological spaces such that M admits a free involution $$\tau $$ . A homotopy class $$\beta \in [ M , N ] $$ is said to have the Borsuk–Ulam property with respect to $$\tau $$ if for every representative map $$f:\,M \rightarrow N$$ of $$\beta $$ , there exists a point $$x \in M$$ such that $$f ( \tau ( x) ) = f(x)$$ . In the case where M is a compact, connected manifold without boundary and N is a compact, connected surface without boundary different from the 2-sphere and the real projective plane, we formulate this property in terms of the pure and full 2-string braid groups of N, and of the fundamental groups of M and the orbit space of M with respect to the action of $$\tau $$ . If $$M=N$$ is either the 2-torus $$\mathbb {T}^2$$ or the Klein bottle $$\mathbb {K}^2$$ , we then solve the problem of deciding which homotopy classes of [M, M] have the Borsuk–Ulam property. First, if $$\tau :\,\mathbb {T}^2\rightarrow \mathbb {T}^2$$ is a free involution that preserves orientation, we show that no homotopy class of $$[ \mathbb {T}^2, \mathbb {T}^2]$$ has the Borsuk–Ulam property with respect to $$\tau $$ . Second, we prove that up to a certain equivalence relation, there is only one class of free involutions $$\tau :\,\mathbb {T}^2\rightarrow \mathbb {T}^2$$ that reverse orientation, and for such involutions, we classify the homotopy classes in $$[\mathbb {T}^2, \mathbb {T}^2]$$ that have the Borsuk–Ulam property with respect to $$\tau $$ in terms of the induced homomorphism on the fundamental group. Finally, we show that if $$\tau :\,\mathbb {K}^2\rightarrow \mathbb {K}^2$$ is a free involution, then a homotopy class of $$[\mathbb {K}^2, \mathbb {K}^2]$$ has the Borsuk–Ulam property with respect to $$\tau $$ if and only if the given homotopy class lifts to the torus.

The Eulerian distribution on involutions is indeed γ-positive

Let In and Jn denote the set of involutions and fixed-point free involutions of {1,…,n}, respectively, and let des(π) denote the number of descents of the permutation π. We prove a conjecture of Guo and Zeng which states that In(t):=∑π∈Intdes(π) is γ-positive for n≥1 and J2n(t):=∑π∈J2ntdes(π) is γ-positive for n≥9. We also prove that the number of (3412,3421)-avoiding permutations with m double descents and k descents is equal to the number of separable permutations with m double descents and k descents.

Distribution of Descents in Matchings

The distribution of descents in certain conjugacy classes of $$S_n$$ has been previously studied, and it is shown that its moments have interesting properties. This paper provides a bijective proof of the symmetry of the descents and major indices of matchings (also known as fixed point free involutions) and uses a generating function approach to prove an asymptotic normality theorem for the number of descents in matchings.

Hidden Symmetries of Complex Analysis

We are dealing with domains of the complex plane which are not symmetric in common sense, but support fixed point free antianalytic involutions. They are fundamental domains of different classes of analytic functions and the respective involutions are obtained by composing their canonical projections onto the complex plane with the simplest antianalytic involution of the Riemann sphere. What we obtain are hidden symmetries of the complex plane. The list given here of these domains is far from exhaustive.

Diagonal Involutions and the Borsuk–Ulam Property for Product of Surfaces

In this work we study a generalization of the Borsuk–Ulam Theorem. Namely, we replace the sphere $${\mathbb {S}}^n$$ by a product of two closed surfaces $$M^2 \times N^2$$ equipped with the diagonal involution $$T \times S$$ where T and S are free involutions on $$M^2$$ and $$N^2$$ , respectively, and the indexes $$i(M^2, T)=i(N^2, S)=2$$ . Then we compute the index of the pair $$(M^2 \times N^2,T \times S)$$ and we obtain a Borsuk-Ulam Theorem for $$M^2 \times N^2$$ .

THE COHOMOLOGY RING OF ORBIT SPACES OF FREE -ACTIONS ON SOME DOLD MANIFOLDS

We determine the possible $\mathbb{Z}_{2}$-cohomology rings of orbit spaces of free actions of $\mathbb{Z}_{2}$ (or fixed point free involutions) on the Dold manifold $P(1,n)$, where $n$ is an odd natural number.

Smooth structures on nonorientable four-manifolds and free involutions

In this paper, we investigate existence of inequivalent smooth structures on closed smooth nonorientable 4-manifolds building upon results of Akbulut, Cappell–Shaneson, Fintushel–Stern, Gompf and Stolz. We add to the number of known constructions and provide new examples of exotic manifolds that are obtained as an application of Gluck twists to the standard smooth structure. Inspection of the smooth structure on the orientation 2-covers yields existence results of orientation-reversing exotic free involutions.

Combinatorial invariance of Kazhdan\u2013Lusztig\u2013Vogan polynomials for fixed point free involutions

When mathrm {Sp}(2n,mathbb {C}) acts on the flag variety of mathrm {SL}(2n,mathbb {C}), the orbits are in bijection with fixed point free involutions in the symmetric group S_{2n}. In this case, the associated Kazhdan–Lusztig–Vogan polynomials P_{v,u} can be indexed by pairs of fixed point free involutions vge u, where ge denotes the Bruhat order on S_{2n}. We prove that these polynomials are combinatorial invariants in the sense that if f:[u,w_0]rightarrow [u',w_0] is a poset isomorphism of upper intervals in the Bruhat order on fixed point free involutions, then P_{v,u} = P_{f(v),u'} for all vge u.

Free involutions on torus semi-bundles and the Borsuk-Ulam Theorem for maps into $\mathbf{R}^n$

In this article we classify the free involutions of every torus semi-bundle Sol 3-manifold. Moreover, we classify all the triples $M, \tau, \mathbf{R}^n$, where $M$ is as above, $\tau$ is a free involution on $M$, and $n$ is a positive integer, for which the Borsuk-Ulam Property holds.

Equivariant Maps from Stiefel Bundles to Vector Bundles

AbstractLet E → B be a fibre bundle and let Eʹ → B be a vector bundle. Let G be a compact Lie group acting fibre preservingly and freely on both E and Eʹ – 0, where 0 is the zero section of Eʹ → B. Let f : E → Eʹ be a fibre-preserving G-equivariant map and let Zf = {x ∈ E | f(x) = 0} be the zero set of f. In this paper we give a lower bound for the cohomological dimension of the zero set Zf when a fibre of E → B is a real Stiefel manifold with a free ℤ/2-action or a complex Stiefel manifold with a free 𝕊1-action. This generalizes a well-known result of Dold for sphere bundles equipped with free involutions.

Avoiding patterns in irreducible permutations

We explore the classical pattern avoidance question in the case of irreducible permutations, <i>i.e.</i>, those in which there is no index $i$ such that $\sigma (i+1) - \sigma (i)=1$. The problem is addressed completely in the case of avoiding one or two patterns of length three, and several well known sequences are encountered in the process, such as Catalan, Motzkin, Fibonacci, Tribonacci, Padovan and Binary numbers. Also, we present constructive bijections between the set of Motzkin paths of length $n-1$ and the sets of irreducible permutations of length $n$ (respectively fixed point free irreducible involutions of length $2n$) avoiding a pattern $\alpha$ for $\alpha \in \{132,213,321\}$. This induces two new bijections between the set of Dyck paths and some restricted sets of permutations.