Talk:Inertia group II (Ex)

1. $\Sigma \in I_H(M)$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\Sigma \in I_H(M)$ iff $\exists f\colon \Sigma \# M \cong M$$\exists f\colon \Sigma \# M \cong M$ with $f\sim id_M$$f\sim id_M$, iff the following diagram commutes up to homotopy: $\xymatrix{ &\Sigma\# M \ar[r]^-{id_M} \ar[d]^-{f} & M \\ &M \ar[ru]_-{id_M}}$

2. $\Hh P^2$$\Hh P^2$ fits into the cofibration sequence $\xymatrix{S^4 \ar[r] &\Hh P^2 \ar[r]^-{c} &S^8 }$ where $c$$c$ is the collapse map of the $7$$7$-skeleton. Then the Puppe exact sequence of the cohomology theory represented by $G/O$$G/O$ gives an exact sequence

$\xymatrix{\pi_5(G/O) \ar[r] & \pi_8 (G/O) \ar[r]^-{c^*} &[\Hh P^2, G/O]\ar[r] & \pi_4(G/O).}$

It is known that $\pi_5(G/O)=0$$\pi_5(G/O)=0$, and so $c^*$$c^*$ is injective. From the surgery exact sequences of $\Hh P^2$$\Hh P^2$ and $S^8$$S^8$ we get the following commutative diagram (see for example [Crowley2010, Lemma 3.4]):

$\xymatrix{&0 \ar[r]\ar[rd] &\Theta_8 \ar[r]^-{\eta} \ar[d]^-{a} &\pi_8(G/O) \ar[r] \ar[d]^-{c^*} &\Zz \\ & & \mathcal{S}(\Hh P^2) \ar[r]^-{\tilde{\eta}} &[\Hh P^2, G/O] \ar[ru]}$

Here we are using the facts that $L_9(e)=0$$L_9(e)=0$, $L_8(e)\cong \Zz$$L_8(e)\cong \Zz$ and $\Hh P^2$$\Hh P^2$ is simply connected. Injectivity of $\eta$$\eta$ and $\tilde{\eta}$$\tilde{\eta}$ follow from the diagram, and combine with the injectivity of $c^*$$c^*$ to show that $a$$a$ is injective. But by part 1 $I_H(\Hh P^2)$$I_H(\Hh P^2)$ is exactly the kernel of $a$$a$, and so $I_H(\Hh P^2)=0$$I_H(\Hh P^2)=0$.

3. Let $\Sigma$$\Sigma$ be the non-zero element of $\Theta_8$$\Theta_8$, and let $f\colon \Sigma\# \Hh P^2 \cong \Hh P^2$$f\colon \Sigma\# \Hh P^2 \cong \Hh P^2$ be a diffeomorphism. Then $id_{\Hh P^2}\circ f^{-1}$$id_{\Hh P^2}\circ f^{-1}$ is a homotopy self-equivalence. Suppose this composition is homotopic to a diffeomorphism $\phi$$\phi$. Then the following diagram would commute up to homotopy:

$\xymatrix{ &\Sigma\# \Hh P^2 \ar[r]^-{id_{\Hh P^2}} \ar[d]^-{f} & \Hh P^2 \\ &\Hh P^2 \ar[ur]_-{\phi} }$

In other words, $\phi\circ f$$\phi\circ f$ is a diffeomorphism homotopic to $id_{\Hh P^2}$$id_{\Hh P^2}$. Therefore $\Sigma$$\Sigma$ would be an element of $I_H(\Hh P^2)$$I_H(\Hh P^2)$, which is a contradiction.

Note that this proof generalizes to show: If $M$$M$ is a manifold such that $I(M)\neq I_H(M)$$I(M)\neq I_H(M)$, then $M$$M$ admits a homotopy self-equivalence which is not homotopic to the identity.

References

• [Crowley2010] D. Crowley, The smooth structure set of $S^p\times S^q$$S^p\times S^q$, Geom. Dedicata 148 (2010), 15–33. MR2721618 (2012a:57041) Zbl 1207.57043