A topologically induced 2-in/2-out operation on loop cohomology

We apply the Transfer Algorithm introduced in arXiv:1106.5090 to transfer an A_\infty-algebra structure that cannot be computed using the classical Basic Perturbation Lemma. We construct a space X whose (base pointed) loop cohomology H = H^*(\Omega X; Z_2) comes equipped with a nontrivial operation \omega : H x H --> H x H.Comment: 9 pages. The title has changed in version 9, and some indexing issues in the transfer algorithm have been correcte

The Deformation Complex For DG Hopf Algebras

Let H be a differential graded Hopf algebra over a field k. This paper gives an explicit construction of a triple cochain complex that defines the Hochschild-Cartier cohomology of H. A certain truncation of this complex is the appropriate setting for deforming H as an H(q)-structure. The direct limit of all such truncations is the appropriate setting for deforming H as a strongly homotopy associative structure. Sign complications are systematically controlled. The connection between rational perturbation theory and the deformation theory of certain free commutative differential graded algebras is clarified.Comment: 21 pages, 4 figure

A Diagonal on the Associahedra

Let C_*(K) denote the cellular chains on the Stasheff associahedra. We construct an explicit combinatorial diagonal \Delta : C_*(K) --> C_*(K) \otimes C_*(K); consequently, we obtain an explicit diagonal on the A_\infty-operad. We apply the diagonal \Delta to define the tensor product of A_\infty-(co)algebras in maximal generality

Diagonals on the Permutahedra, Multiplihedra and Associahedra

We construct an explicit diagonal \Delta_P on the permutahedra P. Related diagonals on the multiplihedra J and the associahedra K are induced by Tonks' projection P --> K and its factorization through J. We introduce the notion of a permutahedral set Z and lift \Delta_P to a diagonal on Z. We show that the double cobar construction \Omega^2(C_*(X)) is a permutahedral set; consequently \Delta_P lifts to a diagonal on \Omega^2(C_*(X)). Finally, we apply the diagonal on K to define the tensor product of A_\infty-(co)algebras in maximal generality.Comment: 45 pages, 13 figures. This (final) version is significantly more detailed than the previou

An A∞A_{\infty}-coalgebra Structure on a Closed Compact Surface

Let $P$ be an $n$-gon with $n\geq3.$ There is a formal combinatorial $A_\infty$-coalgebra structure on cellular chains $C_*(P)$ with non-vanishing higher order structure when $n\geq5$. If $X_g$ is a closed compact surface of genus $g\geq2$ and $P_g$ is a polygonal decomposition, the quotient map $q:P_g\to X_g$ projects the formal $A_\infty$-coalgebra structure on $C_*(P_g)$ to a quotient structure on $C_*(X_g)$, which persists to homology $H_{\ast}\left( X_g;\mathbb{Z}_{2}\right)$, whose operations are determined by the quotient map $q$, and whose higher order structure is non-trivial if and only if $X_g$ is orientable or unorientable with $g\geq3$. But whether or not the $A_{\infty}$-coalgebra structure on homology observed here is topologically invariant is an open question.Comment: 13 pages, 6 figure

Obstructions to Deformations of DG Modules

Let k be a field and n > 0. There exists a DG k-module (V,d) and various approximations d + t d_1 + t^2 d_2 + ... + t^n d_n to a differential on V[[t]], one of which is a non-trivial deformation, another is obstructed, and another is unobstructed at order n. The analogous problem in the category of k-algebras in characteristic zero remains a long-standing open question.Comment: 9 page

Minimal Paths on Some Simple Surfaces with Singularities

Given two points on a soup can or conical cup with lid, we find and classify all paths of minimal length connecting them. When the number of minimal paths is finite, there are at most four on a can and three on a cup. At worst, minimal paths are piece-wise smooth with three components, each of which is a classical geodesic. Minimal paths are geodesics in the sense of Banchoff.Comment: 12 pages, 7 figures (v6:typo fixes in Lemma 1) (v5:revised the Ascoli argument in the first paragraph) (v4:Editorial revisions pursuant to referee's report) (v3:Revision of Title from "Minimal Paths on Unicone and Bicylinder Boundaries"; Other editorial revisions

Periodic Orbits of Billiards on an Equilateral Triangle

We give a complete solution of the following problem: Find, classify and count the (classes of) periodic orbits on an equilateral triangle. We prove that Fagnano's period 3 orbit is the only periodic orbit with odd period. A periodic orbit is either prime or some d-fold iterate thereof. We count prime and iterate periodic orbits of period $2n$ via a bijection with a certain partition of $n$, then count only prime orbits using the prime factorization of $n$.Comment: Version 7: Minor edits made, some figures redrawn, Table 1 moved from appendix to body of text

A-infinity Bialgebras of Type (m,n)

An A-infinity bialgebra of type (m,n) is a Hopf algebra H equipped with a "compatible" operation \omega : H^{\otimes m} \to H^{\otimes n} of positive degree. We determine the structure relations for A-infinity bialgebras of type (m,n) and construct a purely algebraic example for each m \geq 2 and m+n \geq 4.Comment: 10 pages, 4 figure

Generalizations of the Brachistochrone Problem

Consider a frictionless surface S in a gravitational field that need not be uniform. Given two points A and B on S, what curve is traced out by a particle that starts at A and reaches B in the shortest time? This paper considers this problem on simple surfaces such as surfaces of revolution and solves the problem two ways: First, we use conservation of mechanical energy and the Euler-Lagange equation; second, we use geometrical optics and the eikonal equation. We conclude with a discussion of the relativistic effects at relativistic velocities.Comment: 12 pages, 6 figure