Normal invariants I am having a hard time finding examples of computations of normal invariants of surgery theory (or more generally the set of homotopy classes of maps $[X,G/O]$). Does anybody have good references?
 A: Unfortunately I do not know any good references for detailed computations, but let me point out that the rational picture is rather simple: we have $$G/O \sim_{\mathbb Q} BO \sim_{\mathbb Q} = \prod_{i \geq 1} K(\mathbb Q,4i)$$ and thus
$$ [M,G/O] \sim_{\mathbb Q} \prod_{i \geq 1} H^{4i}(M;\mathbb Q)$$
In particular, $\mathcal N(S^n)$ is finite if $n \not\equiv 0 \ (\text{mod} \ 4)$, and infinite cyclic if $n \equiv 0 \ (\text{mod} \ 4)$, in line with the analogous statement for the $L$-groups of $\mathbb Z$.
A: Many examples of computations of $[M,G/O]$ appear in papers which apply surgery theory. Here are some examples:

*

*Brumfiel did the complex projective spaces $\mathbb{C}P^n$.

*Land did the complex projective space $\mathbb{C}P^2$.

*Kirby-Siebenmann did high-dimensional tori in Appendix V.B.

*Crowley did products of spheres $S^p \times S^q$ for $p,q \geq 2$ and $p+q \geq 5$.

A: If you want specifically low-dimensional calculations, the Kirby-Taylor article A survey of 4-manifolds through the eyes of surgery does this for 4-manifolds. The discussion highlights the difference between the topological and smooth (= PL in this dimension) cases.
