Almost framed bordism (Ex)

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A closed almost framed $<wikitex>; 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 stable bundle isomorphism $TM|_{M-x} \cong (M - x) \times \Rr^n$. 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 \cite{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 $\Omega_n^{\text{alm}}$ and there is an obvious forgetful homomorphism $$ f \colon \Omega_n^{\text{fr}} \to \Omega_n^{\text{alm}} $$ which lies in a sequence $$ \dots \stackrel{\partial}{\rightarrow}\pi_n(SO) \stackrel{\bar J}{\rightarrow} \Omega_n^{\text{fr}} \stackrel{f}{\rightarrow} \Omega_n^{\text{alm}} \stackrel{\partial}{\rightarrow} \pi_{n-1}(SO) \stackrel{\bar J}{\rightarrow} \Omega^{\text{fr}}_{n-1} \stackrel{f}{\rightarrow} \dots ~~.$$ The task of this exercise is to prove that this sequence is exact: this is the statement of \cite{Lück2001|Lemma 6.16} and may also be found in \cite{Levine1983|Appendix (ii)}. The homomorphisms $\bar J$ and $\partial$ are described in \cite{Lück2001|6.14 & 6.15}. '''Hint''': Use the [[J-homomorphism]] and [[Pontrjagin-Thom]] construction for framed bordism as explained after the \cite{Lück2001|Lemma 6.16} see also \cite{Lück2001|Definition 6.23 and Lemma 6.24}. </wikitex> == References == {{#RefList:}} [[Category:Exercises]]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 stable bundle isomorphism $TM|_{M-x} \cong (M - x) \times \Rr^n$. 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 $\Omega_n^{\text{alm}}$ and 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 Pontrjagin-Thom construction for framed bordism as explained after the [Lück2001, Lemma 6.16] see also [Lück2001, Definition 6.23 and Lemma 6.24].

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