Talk:Normal bundles in products of spheres (Ex)

1. The normal bundle of the diagonal embedding $M\to M\times M$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}M\to M\times M$ is isomorphic to the tangent bundle of $M$$M$. The tangent bundle of $S^k$$S^k$ is trivial if and only if $k=0,1,3,7$$k=0,1,3,7$, by Adam's theorem.

2. We present two solutions.

Low-tech solution: Given immersions $M\to W$$M\to W$ and $N\to W$$N\to W$ of closed manifolds, one can produce an immersion $M \# N \to W$$M \# N \to W$ by "connecting the immersions by a tube". (See for instance the definition of the sum in $I_k(M)$$I_k(M)$ in Lück's book.) If both immersions happen to have a trivial normal bundle, then the immersion $M\# N\to W$$M\# N\to W$ also has a trivial normal bundle.

We use this for $M=N=S^k$$M=N=S^k$ and $W=S^k\times S^k$$W=S^k\times S^k$, where the first immersion is the inclusion $i_1\colon S^k\to S^k\times \{*\}\to S^k\times S^k$$i_1\colon S^k\to S^k\times \{*\}\to S^k\times S^k$ and the second one is the corresponding inclusion $i_2$$i_2$ into the second factor. Both these embeddings apparently have trivial normal bundles. Hence so does their "sum"

$\displaystyle S^k \cong S^k \# S^k \to S^k\times S^k.$ $$\displaystyle S^k \cong S^k \# S^k \to S^k\times S^k.$$

The homotopy class of this map is given by $S^k\to S^k \vee S^k \subset S^k\times S^k$$S^k\to S^k \vee S^k \subset S^k\times S^k$ where the first map is the pinch map. By definition of the sum in $\pi_k(S^k\times S^k)$$\pi_k(S^k\times S^k)$, it is thus the sum $i_1 + i_2$$i_1 + i_2$. Its Hurewicz image is given by the diagonal class $(1,1)$$(1,1)$.

High-tech solution: The Hirsch-Smale classification of immersions implies that

$\displaystyle \pi_0 \mathrm{Imm}(S^k, S^k\times S^k) \cong \mathrm{colim}_n \; \pi_0 \mathrm{Mono}(TS^k\oplus \varepsilon^n, TS^k \times TS^k\oplus \varepsilon^n)$ $$\displaystyle \pi_0 \mathrm{Imm}(S^k, S^k\times S^k) \cong \mathrm{colim}_n \; \pi_0 \mathrm{Mono}(TS^k\oplus \varepsilon^n, TS^k \times TS^k\oplus \varepsilon^n)$$

where $\varepsilon^n$$\varepsilon^n$ denotes the $n$$n$-dimensional trivial bundle.

We will use surjectivity. Both the bundles $TS^k\oplus \varepsilon$$TS^k\oplus \varepsilon$ and $TS^k\oplus TS^k\oplus\varepsilon$$TS^k\oplus TS^k\oplus\varepsilon$ over $S^k$$S^k$ are trivial, thus isomorphic to $\varepsilon ^{k+1}$$\varepsilon ^{k+1}$ and $\varepsilon ^{2k+1}$$\varepsilon ^{2k+1}$ respectively. However the inclusion $\varepsilon^{k+1}\to \varepsilon^{2k+1}$$\varepsilon^{k+1}\to \varepsilon^{2k+1}$ of the first $k+1$$k+1$ summands is a bundle monomorphism with trivial complement.

By Smale-Hirsch, the homotopy class of the composite

$\displaystyle \xymatrix{ TS^k \oplus \varepsilon \ar[r] \ar[d] & TS^k \oplus TS^k \oplus \varepsilon \ar[r]\ar[d] & (TS^k \times TS^k) \oplus \varepsilon \ar[d]\\ S^k \ar[r]^{\mathrm{id}} & S^k \ar[r]^{\Delta} & S^k\times S^k }$

is the differential of an immersion.

Its normal bundle is still trivial. Moreover, by construction, this immersion is (non-regularly) homotopic to the diagonal. So its Hurewicz image is the diagonal class $(1,1)$$(1,1)$.