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].
- [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