Julius L. Shaneson

EMBEDDINGS AND IMMERSIONS OF FOUR DIMENSIONAL MANIFOLDS IN R6

This chapter discusses the embeddings and immersions of four dimensional manifolds in. It presents a theorem that states that the following conditions are equivalent, such as M is stably parallelizable, that is, the tangent bundle of M plus a trivial line bundle is trivial; ω 2 (M)=0 and p 1 (M)=0; M is almost parallelizable and the index of M is zero ; M embeds smoothly in R 6 ; and M has a locally flat P.L. embedding in R 6 . In this, ω 2 (M) and p 1 (M) denotes the second Stiefel–Whitney class. Almost parallelizable means that M- {a point} is parallelizable. According to another theorem statesd in the chapter, the following conditions are equivalent, such as, M is almost parallelizable; ω 2 (M)=0; M has a P.L. but not necessarily locally flat embedding in R 6 ; and M has a P.L. embedding in R 6 which is locally flat except possibly at one point of M .

Construction of some new four-dimensional manifolds

This note announces a new construction in the theory of 4-manifolds. Let \p\ T —» T, T = S x S x S, the torus of dimension three, be a diffeomorphism, with ip(x) = x, some x E T 3 , Let A be a matrix for the map y induces on nx T 3 = Z ® Z ® Z. Assume that det A = 1 and det(7 A) = ±1, ƒ = identity matrix. It is easy to see that such a map \p exists. Let the manifold M be obtained from T x [0, 1] by identifying (y, 0) with (<p(y), 1). Let M0 be the complement of the interior of a tubular neighborhood of the image of {x} x [0, 1] in the quotients. Clearly bM0 can be identified with the boundary S(p) of the nontrivial disk bundle D(p) over S with group 0(4). There is a (canonical) map h: M0 —> D(p) restricting to the identity on S(p). Let N be any connected nonorientable 4-manifold, and let N0 be the complement of the interior of a tubular neighborhood of a circle in N representing an element a. E nxN that reverses orientation. Then bN0 = 5(p). Let