# Almost framed bordism (Ex)

A closed almost framed -manifold is a closed smooth -manifold together with a stable framing of its tangent bundle away from a point : that is, a bundle isomorphism

for some . A bordism of closed almost framed manifolds and is a smooth bordism , a nicely embedded arc from and and a stable framing of away from : see [Lück2001, Defintions 6.8 & 6.9] for more details.

The set of bordism classes of closed almost framed -manifolds forms a group under connected sum (at the unframed point) and this group is denoted . There is an obvious forgetful homomorphism

which lies in a sequence

The task of this exercise is to prove that this sequence is exact: this is the statement of [Lück2001, Lemma 6.16] and may also be found in [Levine1983, Appendix (ii)]. The homomorphisms and are described in [Lück2001, 6.14 & 6.15].

**Hint**: Use the J-homomorphism and the Pontrjagin-Thom construction for framed bordism as explained after the [Lück2001, Lemma 6.16] see also [Lück2001, Definition 6.23 and Lemma 6.24].

## [edit] References

- [Levine1983] J. P. Levine,
*Lectures on groups of homotopy spheres*, Algebraic and geometric topology (New Brunswick, N.J., 1983), Lecture Notes in Math.,**1126**(1983), 62–95. MR802786 (87i:57031) Zbl 0576.57028 - [Lück2001] W. Lück,
*A basic introduction to surgery theory*,**9**(2001), 1–224. Available from the author's homepage. MR1937016 (2004a:57041) Zbl 1045.57020