Fixed Point Free Involution Research Articles

Squareness for the Monopole-Dimer Model

The monopole-dimer model introduced recently is an exactly-solvable signed generalisation of the dimer model. We show that the partition function of the monopole-dimer model on a graph invariant under a fixed-point free involution is a perfect square. We give a combinatorial interpretation of the square-root of the partition function for such graphs in terms of a monopole-dimer model on a new kind of graph with two types of edges which we call a dicot. The partition function of the latter can be written as a determinant, this time of a complex adjacency matrix. This formulation generalises T. T. Wu's assignment of imaginary orientation for the grid graph to planar dicots. As an application, we compute the partition function for a family of non-planar dicots with positive weights.

Special divisors on curves on K3 surfaces carrying an Enriques involution

We study the pencils of minimal degree on the smooth curves lying on a K3 surface X which carries a fixed-point free involution. Generically, the gonality of these curves is totally governed by the genus 1 fibrations of X

Plane Permutations and Applications to a Result of Zagier--Stanley and Distances of Permutations

In this paper, we introduce plane permutations, i.e., pairs $\mathfrak{p}=(s,\pi)$, where $s$ is an $n$-cycle and $\pi$ is an arbitrary permutation, represented as a two-row array. Accordingly, a plane permutation gives rise to three distinct permutations: the permutation induced by the upper horizontal ($s$), the vertical ($\pi$), and the diagonal ($D_{\mathfrak{p}}$) of the array. The latter can also be viewed as the three permutations of a hypermap. In particular, a map corresponds to a plane permutation, in which the diagonal is a fixed-point free involution. We study the transposition action on plane permutations obtained by permuting their diagonal blocks. We establish basic properties of plane permutations and analyze transpositions and exceedances, deriving various enumerative results. In particular we prove a recurrence for the number of plane permutations having a fixed diagonal and $k$ cycles in the vertical, generalizing a recursion of Chapuy for maps filtered by the genus. As applications of ...

The moduli space of maps with crosscaps: Fredholm theory and orientability

Just as a symmetric surface with separating fixed locus halves into two oriented bordered surfaces, an arbitrary symmetric surface halves into two oriented symmetric halfsurfaces, i.e. surfaces with crosscaps. Motivated in part by the string theory view of real Gromov-Witten invariants, we introduce moduli spaces of maps from surfaces with crosscaps, develop the relevant Fredholm theory, and resolve the orientability problem in this setting. In particular, we give an explicit formula for the holonomy of the orientation bundle of a family of real Cauchy-Riemann operators over Riemann surfaces with crosscaps. Special cases of our formulas are closely related to the orientability question for the space of real maps from symmetric Riemann surfaces to an almost complex manifold with an anti-complex involution and in fact resolve this question in genus 0. In particular, we show that the moduli space of real J-holomorphic maps from the sphere with a fixed-point free involution to a simply connected almost complex manifold with an even canonical class is orientable. In a sequel, we use the results of this paper to obtain a similar orientability statement for genus 1 real maps.

Automorphisms of Order $2p$ in Binary Self-Dual Extremal Codes of Length a Multiple of 24

Let C be a binary self-dual code with an automorphism g of order 2p, where p is an odd prime, such that gp is a fixed point free involution. If C is extremal of length a multiple of 24, all the involutions are fixed point free, except the Golay Code and eventually putative codes of length 120. Connecting module theoretical properties of a self-dual code C with coding theoretical ones of the subcode C(gp) which consists of the set of fixed points of gp, we prove that C is a projective F2〈g 〉-module if and only if a natural projection of C(gp) is a self-dual code. We then discuss easy-to-handle criteria to decide if C is projective or not. As an application, we consider in the last part extremal self-dual codes of length 120, proving that their automorphism group does not contain elements of order 38 and 58.

COURBES DE GENRE 5 MUNIES D'UNE INVOLUTION SANS POINT FIXE

We study genus 5 curves C with a fixed point free involution. We give a geometrical (embedded) characterisation of these curves among all genus 5 curves : the points of the Prym variety associated to the involution give embeddings of the curve C in P3 so that C has infinitely many quadrisecant lines. Conversely, any genus 5 curve having such an embedding is endowed with a fixed point free involution and the embedding is given by a point of the Prym variety.

A geometric parametrization for the virtual Euler characteristics of the moduli spaces of real and complex algebraic curves

We determine an expression 9 (1X) for the virtual Euler characteristics of the moduli spaces of s-pointed real (-y = 1/2) and complex (-y = 1) algebraic curves. In particular, for the space of real curves of genus g with a fixed point free involution, we find that the Euler characteristic is (-2)s-1(1-29-1)(g?s-2)!Bg/g! where Bg is the gth Bernoulli number. This complements the result of Harer and Zagier that the Euler characteristic of the moduli space of complex algebraic curves is (-1) S (g+s-2)!Bg+ /(g+1) (g -1) ! The proof uses Strebel differentials to triangulate the moduli spaces and some recent techniques for map enumeration to counit cells. The approach involves a parameter y that permits specialization of the formula to the real and complex cases. This suggests that 4 (-y) itself may describe the Euler characteristics of some related moduli spaces, although we do not yet know what these spaces might be.

Type II codes over F/sub 2/+uF/sub 2/

The alphabet F/sub 2/+uF/sub 2/ is viewed here as a quotient of the Gaussian integers by the ideal (2). Self-dual F/sub 2/+uF/sub 2/ codes with Lee weights a multiple of 4 are called Type II. They give even unimodular Gaussian lattices by Construction A, while Type I codes yield unimodular Gaussian lattices. Construction B makes it possible to realize the Leech lattice as a Gaussian lattice. There is a Gray map which maps Type II codes into Type II binary codes with a fixed point free involution in their automorphism group. Combinatorial constructions use weighing matrices and strongly regular graphs. Gleason-type theorems for the symmetrized weight enumerators of Type II codes are derived. All self-dual codes are classified for length up to 8. The shadow of the Type I codes yields bounds on the highest minimum Hamming and Lee weights.

An Upper and a Lower Bound Theorem for Combinatorial 4-Manifolds

In this paper we prove two inequalities. The first one gives a lower bound for the Euler characteristic of a tight combinatorial 4-manifold M under the additional assumptions that |M| is 1-connected, that M is a subcomplex of H(M) , and that H(M) is a centrally symmetric and simplicial d -polytope. The second inequality relates the Euler characteristic with the number of vertices of a combinatorial 4-manifold admitting a fixed-point free involution. Furthermore, we construct a new and highly symmetric 12-vertex triangulation of S2 x S2 realizing equality in each of these inequalities.

Variations on a theorem of lusternik and schnirelmann

A fixed-point free map f: X → X is said to be colorable with k colors if there exists a closed cover β of X consisting of k elements such that C∩ f( C) = φ for every C in β. It is shown that each fixed-point free involution of a paracompact Hausdorff space X with dim X ≤ n can be colored with n + 2 colors. Each fixed-point free homeomorphism of a metrizable space X with dim X ≤ n is colorable with n + 3 colors. Every fixed-point free continuous selfmap of a compact metrizable space X with dim X ≤ n can be colored with n + 3 colors

On Schur's Q -functions and the Primitive Idempotents of a Commutative Hecke Algebra

Let Bn denote the centralizer of a fixed-point free involution in the symmetric group S2n. Each of the four one-dimensional representations of Bn induces a multiplicity-free representation of S2n, and thus the corresponding Hecke algebra is commutative in each case. We prove that in two of the cases, the primitive idempotents can be obtained from the power-sum expansion of Schur's Q-functions, from which follows the surprising corollary that the character tables of these two Hecke algebras are, aside from scalar multiples, the same as the nontrivial part of the character table of the spin representations of Sn.

Riemann surfaces, Clifford algebras and infinite dimensional groups

We introduce a class of Riemann surfaces which possess a fixed point free involution and line bundles over these surfaces with which we can associate an infinite dimensional Clifford algebra. Acting by automorphisms of this algebra is a “gauge” group of meromorphic functions on the Riemann surface. There is a natural Fock representation of the Clifford algebra and an associated projective representation of this group of meromorphic functions in close analogy with the construction of the basic representation of Kac-Moody algebras via a Fock representation of the Fermion algebra. In the genus one case we find a form of vertex operator construction which allows us to prove a version of the Boson-Fermion correspondence. These results are motivated by the analysis of soliton solutions of the Landau-Lifshitz equation and are rather distinct from recent developments in quantum field theory on Riemann surfaces.

Riemann surfaces, Clifford algebras and infinite dimensional groups

We introduce a class of Riemann surfaces which possess a fixed point free involution and line bundles over these surfaces with which we can associate an infinite dimensional Clifford algebra. Acting by automorphisms of this algebra is a “gauge” group of meromorphic functions on the Riemann surface. There is a natural Fock representation of the Clifford algebra and an associated projective representation of this group of meromorphic functions in close analogy with the construction of the basic representation of Kac-Moody algebras via a Fock representation of the Fermion algebra. In the genus one case we find a form of vertex operator construction which allows us to prove a version of the Boson-Fermion correspondence. These results are motivated by the analysis of soliton solutions of the Landau-Lifshitz equation and are rather distinct from recent developments in quantum field theory on Riemann surfaces.

Towards a classification of transitive group actions on finite metric spaces

Based on J. R. Isbell's “completion” of metric spaces (cf. Comment. Math. Helv. 39 (1964–1965), 65–74) the combinatorial dimension dim comb X of a metric space X has been defined in [A. W. M. Dress, Adv. in Math. 53(3)(1984), 321–402] and shown to satisfy dim comb X ⩽ ⌊ #X 2 ⌋ . In this paper finite metric spaces X with a transitive group G of isometries which satisfy in addition dim comb X = ⌊ #X 2 ⌋ are investigated. It is shown that—for almost trivial reasons—in case # X ≡ 0(2) there exists a fixed point free involution on X which commutes with the action of G and that vice versa any finite transitive G-space admitting a fixed point free G-automorphism of order 2 carries a G-invariant metric satisfying the above equality, while for not so trivial reasons in case # X ≡ 1(2) the group G must be either cyclic of order # X or dihedral of order 2 · # X. As a corollary the Thompson-Feit Theorem turns out to be actually equivalent to its simple consequence that any finite simple group acts transitively as a group of isometries on some finite metric space X satisfying dim comb X = ⌊ #X 2 ⌋ .

Counting semiregular permutations which are products of a full cycle and an involution

Character theoretic methods and the group algebra of the symmetric group are used to derive properties of the number of permutations, with only p p -cycles, for an arbitrary but fixed p p , which are expressible as the product of a full cycle and a fixed point free involution. This problem has application to single face embeddings of p p -regular graphs on surfaces of given genus.

A transfer spectral sequence for fixed point free involutions with an application to stunted real projective spaces

We show that if X is a finite CW-complex admitting a fixed point free involution then there is a singly graded spectral sequence with E ∗ 1 ≅ H ∗(X; Z 2) and E ∗∞ = 0 . As an application we prove that for any n > 0 there is a natural number k( n) such that if n > k( n) and X is a homotopy RP n+k RP n , then X will not admit a fixed point free involution.

Framed manifolds with a fixed point free involution.

Framed manifolds with a fixed point free involution.

Groups of free involutions of homotopy $S\sp{[n/2]}\times S\sp{[(n+1)/2]}$’s

Let $M$ be an oriented $n$-dimensional manifold which is homotopy equivalent to ${S^l} \times {S^{n - l}}$, where $l$ is the greatest integer in $n/2$. Let $Q$ be the quotient manifold of $M$ by a fixed point free involution. Associated to each such $Q$ are a unique integer $k\bmod {2^{\varphi (l)}}$, called the type of $Q$, and a cohomology class $\omega$ in ${H^1}(Q;{Z_2})$ which is the image of the generator of the first cohomology group of the classifying space for the double cover of $Q$ by $M$. Let ${I_n}(k)$ be the set of equivalence classes of such manifolds $Q$ of type $k$ for which ${\omega ^{l + 1}} = 0$, where two such manifolds are equivalent if there is a diffeomorphism, orientation preserving if $k$ is even, between them. It is shown in this paper that if $n \geq 6$, then ${I_n}(k)$ can be given the structure of an abelian group. The groups ${I_8}(k)$ are partially calculated for $k$ even.

A note on fixed point free involutions and equivariant maps

The space P(Sn) of all paths a) in Sn with given initial point x and endpoint -x admits an involution (To)(t) = -co(1 t). With the standard antipodal involution on Sn-1 an equivariant map P(Sn) -* Sn-' is constructed for n = 2, 4, or 8. Afixedpointfree involution on a space Xis a map T:X T Xsatisfying T' = identity and Tx $ x for every x E X. Three examples of interest are (i) (Sn, T1) with T1x = -x; (ii) (V(Sn), T2) where V(Xn) = the unit tangent sphere bundle of Sn and T2 = the antipodal action on each fibre; and (iii) (P(Sn), T3) where P(Sn) = the space of paths with given initial point x E Sn and endpoint -x, and (T3w))(t) = -w(1 t). For (X, T) a fixed point free involution the co-index of X is the least integer n for which there exists an equivariant map (X, T) --(Sn, T1). A classical result of Borsuk [1] asserts that the co-index of (Sn, T1) equals n. In [2] Conner and Floyd determined the co-index of (V(Sn), T2) (for all n) and the co-index of (P(Sn), T3) for all n except n = 2, 4, and 8. Their results assert (i) co-index(V(Sn), T2) = n or n 1 according as n 0 {1, 3, 7} or n E {1, 3, 7} and (ii) co-index(P(Sn), T3) = n if n 0 {1, 2, 4, 8} and = n 1 if n = 1. The remaining cases of (ii) are resolved by PROPOSITION. For n = 2, 4, or 8 there exists an equivariant map (P(Sn), T3) -_ (Sn-i, T1). PROOF. For n = 2, 4, and 8 there are the Hopf fibrations Sn-1 F S2n-1 P+ Sn. Here S2n-1 is the unit sphere in F2 (F = complexes for n = 2, the quaternions for n = 4 and the Cayley numbers for n = 8) and the map p assigns to each unit vector the 1-dimensional (over F) subspace it spans. Fix a point x E Sn and a point y E S7n= the fibre of p over x. S2n-1 iS the join of Sxnand Snx71, where Sf7xl is both the fibre over -x and the unit sphere in the 1-dimensional (over F) subspace orthogonal to the subspace spanned by Sn-1. Moreover p maps the great circle arc Received by the editors December 8, 1970. AMS 1969 subject classifications. Primary 5536.

Coinciding directions for imbeddings of spaces

1. Equivariant cohomology. Let A be a space with a continuous fixed point free involution a: A --A. Consider the two possible actions of a on the coefficient group Z of integers: the trivial action and the nontrivial one. Let I9+(A) (resp. H7+(A)) denote the equivariant (resp. residual) ith cohomology group of A for the trivial action; and let HL (A) (resp. m (A)) denote the corresponding equivariant (resp. residual) cohomology groups for the nontrivial action of a on Z (the singular homology is considered throughout; see [1]). The groups 0 i ~~~0 H+ (A) and Hi (A) can also be described as the corresponding cohomology groups of the orbit space A/a, with the constant and twisted coefficients { Z}, respectively. There are exact sequences: