In the notes of Vector Bundles and K-theory by Prof Allen Hatcher, on page 12 he proved a Proposition that for each vector bundle $E\to B$ with $B$ compact hausdorff there exists a vector bundle $E'\to B$ such that $E\oplus E'$ is the trivial bundle.
Later in example 3.6 he showed that the compactness of $B$ is an important condition, otherwise canonical line bundle over $\mathbb{RP^\infty}$ would be a counter example.


My doubt is... Is this Proposition still valid for finite dimensional manifold??? Or can someone provide me a counter example of a finite manifold where it is not true.



Is every vector bundle over a noncompact finite-dimensional manifold a summand of a trivial bundle?
The proposition you refer to holds for any space homotopy equivalent to a finite dimensional CW complex. Here are the main points.


1. The property that any vector bundle $E$ over a space has a complementary bundle $E^\prime$ is preserved by homotopy equivalences.
2. Any finite dimensional CW complex is homotopy equivalent to a smooth manifold. (A classical result of Whitehead shows that any finite dimensional CW complex is homotopy equivalent to a locally finite finite dimensional CW complex. Then simplicial approximation gives a homotopy equivalent finite dimensional simplicial complex. Embed it into a Euclidean space, take a regular neighborhood, and smooth it.)
3. The total space of any vector bundle $E$ over a smooth manifold $M$ smoothly embeds into a Euclidean space. The normal bundle bundle of $E$ restricted to $M$ is the desired complementary bunlde $E^\prime$.
4. Note that any topological manifold is homotopy equivalent to a finite dimensional CW complex.


