Microbundle

Abstract: Recently, generalised homology theories have received much attention ([5], [6], [19] etc.). An interesting question which arises in these theories is the orientability of manifolds [20]; as Whitehead remarks, this is a delicate question. In this paper we shall show that if there exists a microbundle 4 over a closed manifold M such that 4 is A-orientable in the sense of Dold [6], and such that the top homology class of the Thom space M4 is spherical, then M is A-orientable in the sense of Whitehead [19]. Hence M satisfies Poincare duality with coefficient spectrum A. Let N be a compact smooth manifold whose boundary ON= Y is a homotopy sphere which bounds a 7r-manifold: form a manifold M by adjoining to N a cone on its boundary S. We shall call such manifold-s M almost-smooth: the manifolds of Kervaire [11], Smale [16], Eells and Kuiper [8] (see also [18]), which do not have the same homotopy type as any closed smooth manifold, are all almost smooth. We shall show that, over any such M, there is a vector bundle 4 such that the top homology class of its Thom complex MX is spherical. It follows that (provided M is orientable in the usual sense), M is orientable for K-theory and KO-theory [2] and for bordism theory [1]. It also follows from the existence of such bundles that the hypothesis on the signature in the theorem of Browder [4], which gives necessary and sufficient conditions for a finite, simply-connected CW-complex to have the homotopy type of a closed smooth manifold (of dimension : 3,4), cannot be dispensed with. Microbundles and Thom spaces. We consider pairs (i,j) of maps, with ji the identity map of some fixed space B. Two pairs (i1,jl) and (i2,j2) are equivalent if there is a third pair (i3,j3) and a commutative diagram,

Abstract: ON PIECEWISE LINEAR IMMERSIONS Reference 1. H. Rossi, Vector fields on analytic spaces, Ann. of Math. 78 (1963), 455-467. Princeton University ON PIECEWISE LINEAR IMMERSIONS MORRIS W. HIRSCH The purpose of this note is to prove an existence theorem for im- mersions of piecewise linear manifolds in Euclidean space. A more comprehensive theory of piecewise linear immersions has been worked out by Haefliger and Poenaru [l]. All maps, manifolds, microbundles, etc. are piecewise linear unless the contrary is explicitly indicated. Let M be a manifold without boundary, of dimension n. Denote the tangent microbundle of M by tm, and the trivial microbundle over M of (fibre) dimension k by e*. Let be a microbundle of dimension k such that £ is a manifold. mersion of M in Rn+k is a locally one-one map /: M—>Pn+*. An im- I say / has a normal bundle of type v if there is an immersion g: E—>Rn+k such that gi=f. (It is unknown whether / necessarily has a normal bundle, or whether all normal bundles of / are of the same type.) The converse of the following theorem is trivial. Theorem. Assume that if k = 0, then M has no compact component. There exists an immersion of M in Rn+k having a normal bundle of type v if there exists an isomorphism 4>:rm® v->e+k Proof. We may assume that i(M) is a deformation retract of the total space E of v. By Milnor [3], te \i(M) is isomorphic to tm®v; it follows from the existence of that te is trivial. According to [3] Received by the editors July 29, 1964. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Abstract: a normal microbundle [3]. Thus we are led to inquire into the extentto which the local flatness gives rise to some normal structure. Inanalogy with the development of normal structures for the differen-tiable case, we define several possible normal structures which arerelated to the normal microbundle and show that the Rourke-Sanderson example admits none of them. From this we conclude thatthe homotopy normal bundle [2] is the strongest normal structureavailable, in general. We also give an example of interest in the theory

Abstract: A differentiable manifold M is said to be parallelizable if the tangent vector bundle of M is trivial A topological manifold M is said to be topologically parallelizable if the tangent microbundle of M is trivial In [2] Milnor has shown that on some open set M in some Euclidean space Rn there exists a differentiable structure with respect to which the integral Pontrjagin class p(M) of M is different from 1 It follows that on a topologically parallelizable manifold it is possible to have a differentiable structure with respect to which the manifold is not parallelizable It is known that the only spheres (of dimension _1) which are differentiably parallelizable are SI, S3 and S7 [1] It is also known that the only spheres which have fibre homotopically trivial tangent sphere bundles are SI, SI and S7 [3] In this note we prove