Almost framed bordism (Ex)

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

\displaystyle  TM|_{M-x}\times \Rr^{a} \cong (M - x) \times \Rr^{n+a}

for some a > 0. A bordism of closed almost framed manifolds M_0 and M_1 is a smooth bordism (W; M_0, M_1), a nicely embedded arc I \subset W from x_0 and x_1 and a stable framing of TW away from I: see [Lück2001, Defintions 6.8 & 6.9] for more details.

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

\displaystyle  f \colon \Omega_n^{\text{fr}} \to \Omega_n^{\text{alm}}

which lies in a sequence

\displaystyle  \dots \stackrel{\partial}{\rightarrow}\pi_n(SO) \stackrel{\bar  J}{\rightarrow} \Omega_n^{\text{fr}} \stackrel{f}{\rightarrow}  \Omega_n^{\text{alm}} \stackrel{\partial}{\rightarrow} \pi_{n-1}(SO)  \stackrel{\bar J}{\rightarrow} \Omega^{\text{fr}}_{n-1}  \stackrel{f}{\rightarrow} \dots ~~.

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 \bar J and \partial 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].

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