## Highlights - DEFINITION\. A real vector bundle £ over B consists of the following: 1\) a topological space E = E\(£\) called the total space, 2\) a \(continuous\) map 7: E » B called the projection map, and 3\) for each b ¢ B the structure of a vector space” over the real numbers in the set 7~1\(b\)\. (John Milnor & James D. Stasheff 17) - These must satisfy the following restriction: Condition of local triviality\. For each point b of B there should exist a neighborhood U C B, an integer n> 0, and a homeomorphism h:Ux RP > 71\) so that, for each b ¢ U, the correspondence x + h\(b, x\) defines an isomorphism between the vector space R™ and the vector space 7 1\(b\)\. (John Milnor & James D. Stasheff 17) - Such a pair \(U,h\) will be called a local coordinate system for & about b\. If it is possible to choose U equal to the entire base space, then & will be called a trivial bundle\. (John Milnor & James D. Stasheff 17) - In Steenrod’s terminology an R™-bundle is a fiber bundle with fiber R™ and with the full linear group GL\(R\) in n variables as structural group\. (John Milnor & James D. Stasheff 18) - Example 2\. The tangent bundle ry; of a smooth manifold M\. (John Milnor & James D. Stasheff 18) - If ry is a trivial bundle, then the manifold M is called parallelizable\. (John Milnor & James D. Stasheff 19) - THEOREM 2\.1\. The bundle Ve over P" is not trivial, for n> 1\. (John Milnor & James D. Stasheff 20) - \(A cross-section of the tangent bundle of a smooth manifold M is usually called a vector field on M\.\) (John Milnor & James D. Stasheff 20) - THEOREM 2\.2\. An R™-bundle ¢ is trivial if and only if & admits n cross-sections s,,\.\.\.,S, n which are nowhere dependent\. n (John Milnor & James D. Stasheff 22) - Hence S° is parallelizable\. (John Milnor & James D. Stasheff 24) - DEFINITION\. A Euclidean vector bundle is a real vector bundle ¢& together with a continuous function §: E\(\) > R such that the restriction of pu to each fiber of ¢ is positive definite and quadratic\. The function pu itself will be called a Euclidean metric on the vector bundle ¢£\. (John Milnor & James D. Stasheff 25) - In the case of the tangent bundle ry; of a smooth manifold, a Euclidean metric yp: DM -> R (John Milnor & James D. Stasheff 25) - is called a Riemannian metric, and M together with pu is called a Riemannian manifold\. (John Milnor & James D. Stasheff 26) - ote\. In Steenrod’s terminology a Euclidean metric on & gives-rise to a reduction of the structural group of £ from the full linear group to the orthogonal group\. (John Milnor & James D. Stasheff 26) - A priori there appear to be two different concepts of triviality for Euclidean vector bundles; (John Milnor & James D. Stasheff 26) - Show that the unit sphere S" admits a vector field which is nowhere zero, providing that n is odd\. Show that the normal bundle of S$ c R™! is trivial for all n\. (John Milnor & James D. Stasheff 27) - If S" admits a vector field which is nowhere zero, show that the identity map of S" is homotopic to the antipodal map\. (John Milnor & James D. Stasheff 27) - For n even show that the antipodal map of S™ is homotopic to the reflection {CSP Xn41\) = \(=X, Xy, 000, Xp\) (John Milnor & James D. Stasheff 27) - and therefore has degree —1\. (John Milnor & James D. Stasheff 27) - show that S? is not parallelizable for n even, n> 2\. (John Milnor & James D. Stasheff 27) - More generally a smooth map f:M -» N between smooth manifolds is called an immersion if the Jacobian Df, : DM, -> DN¢\(x\) maps the tangent space DM, injectively \(i\.e\., with kernel zero\) for each x ¢ M\. [It follows from the implicit function theorem that an immersion is locally an embedding of M in N, but in the large there may be selfintersections\. (John Milnor & James D. Stasheff 34) - if M C N with normal bundle v, where N is a smooth Riemannian manifold, then the ‘second fundamental form’’ can be defined as a smooth symmetric cross-section of the bundle Hom \(ry; ® ney\) (John Milnor & James D. Stasheff 39) - A module is projective if it is a direct summand of a free module\. (John Milnor & James D. Stasheff 40) - THEOREM 4\.9 [Pontrjagin]\. If B is a smooth compact \(n+1\)dimensional manifold with boundary equal to M \(compare §17\), then the Stiefel-Whitney numbers of M are all zero\. (John Milnor & James D. Stasheff 56) - THEOREM 4\.10 [Thom]\. If all of the Stiefel-Whitney numbers of M are zero, then M can be realized as the boundary of some smooth compact manifold\. (John Milnor & James D. Stasheff 57) - DEFINITION\. Two smooth closed n-manifolds M; and M, belong to the same unoriented cobordism class iff their disjoint union M; UM, is the boundary of a smooth compact \(n+l\)-dimensional manifold\. (John Milnor & James D. Stasheff 57) - DEFINITION\. The Grassmann manifold G\(Ro+K\) is the set of all n-dimensional planes through the origin of the coordinate space ROK, (John Milnor & James D. Stasheff 60) - An n-frame in R™K is an n-tuple of linearly independent vectors of R™K, (John Milnor & James D. Stasheff 60) - The collection of all n-frames in R™¥ forms an open subset of the n-fold Cartesian product ROHK X eee X RO+K called the Stiefel manifold v\(ROK\), (John Milnor & James D. Stasheff 60) - THEOREM 7\.1\. The cohomology ring H\*G,; 2/2\) is a polynomial algebra over 7/2 freely generated by the Stiefel-Whitney classes w, \(YD, ery w,\(y™\)\. (John Milnor & James D. Stasheff 85) - PROPERTY 9\.7\. If the oriented vector bundle ¢ possesses a nowhere zero cross-section, then the Euler class e\(£\) must be zero\. (John Milnor & James D. Stasheff 103) - LEMMA 14\.1\. If w is a complex vector bundle, then the underlying real vector bundle wg has a canonical preferred orientation\. (John Milnor & James D. Stasheff 154) - Applying this lemma to the special case of a tangent bundle, it follows that any complex manifold has a canonical preferred orientation\. (John Milnor & James D. Stasheff 154) - every orientation for the tangent bundle of a manifold gives rise to a unique orientation of the manifold\. (John Milnor & James D. Stasheff 154) - But we already know from Section 14 that any characteristic class for complex vector bundles can be expressed as a polynomial in the Chern classes\. (John Milnor & James D. Stasheff 294)