Recently, I was reading the paper "Finite Group Actions on Kervaire Manifold" by Crowley, Hambolton. But I am having problem understanding a definition. Here it is,
In page 15-16 they are defining a map $N^t_{\mathrm{diff}}(X) \rightarrow [X,SG]$ . X is closed manifold of dimension $m$ embedded in $R^{m+k}$, $k$ large. SG is space of stable orientation-preserving self-homotopy
equivalences of the sphere. In doing so, they are defining a map which is isomorphism of abelian groups (Lemma 6.3), -
$\hat{D} : \pi_{m+k}(T(\nu_{X})) \rightarrow [X,\Omega^{\infty}S^{\infty}]$ . Take $\rho : S^{m+k} \rightarrow T(\nu_{X})$. The S-Dual is a stable map $D(\rho) : X_+ \rightarrow S^0$. Which I think just means that, an element in $ \{X_+,S^0\} $. The adjoint of this is, $\hat{D}(\rho) : X \rightarrow \Omega^{\infty}S^{\infty}$. I think they are just taking adjoint for big $k$. Here are my questions,-
Why + is missing in the adjoint $\hat{D}(\rho)$? If we are just taking restrictions to X, then is this still isomorphism?
Is $[X , \Omega^{\infty}S^{\infty}]$ unpointed homotopy here? Because for pointed homotopy, I think any map from X goes into $(\Omega^{\infty}S^{\infty})_a$, only one component. Which does not satisfy the criteria in Lemma $6.3$ .
Are there some references where they computed explicit elements of $N^t_{\mathrm{diff}}(X)$ ?
Thank you.
