Orienting Transverse Fiber Products
Ramshaw, Lyle; Basch, Julien
Keyword(s): Boundary, convolution, fiber product, Minkowski sum, mixed associativity, offset, orientation, pullback, quiver
Abstract: Direct products (a.k.a. Cartesian products) are familiar: Given linear spaces A and B of dimensions a and b, their direct product A X B is the linear space of dimension a + b consisting of all ordered pairs (a,b), for a in A and b in B; and direct products of smooth manifolds are an analogous story. Fiber products (a.k.a. pullbacks) are a less familiar generalization. Given three linear spaces A, B, and S of dimensions a, b , and s and given linear maps f:A -> S and g:B -> S, their fiber product A Xs B is that subspace of the direct product A X B on which the maps f and g agree, that is, the set of ordered pairs (a,b) with f(a)=g(b) in S. When the image spaces f(A) and g(B) together span all of S, the maps f and g are said to be transverse, and the dimension of the fiber product is then a + b - s. Fiber products make sense also for manifolds: Given smooth manifolds A, B, and S of dimensions a, b, and s and given smooth maps f:A -> S and g:B -> S that are transverse in the appropriate sense, it is a standard result that the fiber product A Xs B is itself a smooth manifold of dimension a + b - s.
But what if the input manifolds A, B, and S are oriented? Is there then some natural rule for orienting the fiber-product manifold A Xs B? Such a rule is needed for an application of fiber products in computer-aided geometric design and robotics --- in particular, for computing the boundary of a Minkowski sum from the boundaries of its summands. We show that there is a unique rule for orienting transverse fiber products that satisfies the Axiom of Mixed Associativity: (A Xs B) XT C = A XS (B XT C).
External Posting Date: June 21, 2009 [Fulltext]. Approved for External Publication
Internal Posting Date: June 21, 2009 [Fulltext]