Almost framed bordism (Ex)

A closed almost framed $n$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}n$-manifold is a closed smooth $n$$n$-manifold $M$$M$ together with a stable framing of its tangent bundle away from a point $x \in M$$x \in M$: that is, a bundle isomorphism

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

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

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

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

which lies in a sequence

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

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$$\bar J$ and $\partial$$\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].

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