Normal maps and submanifolds (Ex)

Let $(f, b) \colon (M, \nu_M) \to (X, \xi)$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}(f, b) \colon (M, \nu_M) \to (X, \xi)$ be a degree one normal map. For simplicity, assume that $M$$M$ and $X$$X$ are closed oriented $\text{Cat}$$\text{Cat}$-manifolds of dimension $n$$n$. Suppose that $i \colon Y \subset X$$i \colon Y \subset X$ is the inclusion of a codimension $k$$k$ oriented submanifold $X$$X$ with normal bundle $\nu_{Y \subset X}$$\nu_{Y \subset X}$ and that that $f$$f$ is transverse to $Y$$Y$.

Exercise 0.1. Prove the following:

1. There is a canonical degree one normal map $(f|_N, b') \colon (N, \nu) \to (Y, \xi|_Y \oplus \nu_{Y \subset X})$$(f|_N, b') \colon (N, \nu) \to (Y, \xi|_Y \oplus \nu_{Y \subset X})$.
2. This defines well-defined maps
   $\displaystyle \pitchfork_{Y, \xi} \colon \mathcal{N}(X,\xi)\rightarrow \mathcal{N}(Y,\xi|_Y \oplus \nu_{Y \subset X}) \quad \text{and} \quad \pitchfork_Y \colon \mathcal{N}(X)\rightarrow \mathcal{N}(Y).$ $$\displaystyle \pitchfork_{Y, \xi} \colon \mathcal{N}(X,\xi)\rightarrow \mathcal{N}(Y,\xi|_Y \oplus \nu_{Y \subset X}) \quad \text{and} \quad \pitchfork_Y \colon \mathcal{N}(X)\rightarrow \mathcal{N}(Y).$$
3. If we use

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   ${\rm Id}_X$ and

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   ${\rm Id}_Y$ as base-points to identify $\mathcal{N}(X) \equiv [X, G/Cat]$$\mathcal{N}(X) \equiv [X, G/Cat]$ and $\mathcal{N}(Y) \equiv [Y, G/Cat]$$\mathcal{N}(Y) \equiv [Y, G/Cat]$, show there is a commutative diagram:
   $\displaystyle \xymatrix{ \mathcal{N}(X) \ar[d]^{\pitchfork_Y} \ar[r] & [X, G/Cat] \ar[d]^{i^*} \\ \mathcal{N}(Y) \ar[r] & [Y, G/Cat]. }$
4. Of course we have the surgery obstruction map

$\displaystyle \sigma \colon \mathcal{N}(Y, \xi_y \oplus \nu_{Y \subset X}) \to L_{n-k}(\pi_1(Y))$ $$\displaystyle \sigma \colon \mathcal{N}(Y, \xi_y \oplus \nu_{Y \subset X}) \to L_{n-k}(\pi_1(Y))$$

and the composite map

$\displaystyle \sigma \circ \pitchfork_Y \colon \mathcal{N}(X, \xi) \to L_{n-k}(\pi_1(Y))$ $$\displaystyle \sigma \circ \pitchfork_Y \colon \mathcal{N}(X, \xi) \to L_{n-k}(\pi_1(Y))$$

which is called the splitting obstruction map along $Y$$Y$. Prove the following:

If $(f, b)$$(f, b)$ is normally bordant to a homeomorphism then the splitting obstruction along $Y$$Y$ vanishes.