In questo articolo studiamo i gruppi ${\mathrm{\Theta }}_h^{\mathcal{F}}$ di una sfera $S^n$ e proviamo che il gruppo ${\mathrm{\Theta }}_n^{\mathcal{F}}\left(S^n,x_0\right)$ è isomorfo all'ennesimo gruppo di omotopia di $\left(S^n,x_0\right)$, nell'ipotesi che $\mathcal{F}$ sia una classe coconnessa di links ammissibili.

In this note we give examples in every dimension $m\ge 9$ of piecewise linearly homeomorphic, closed, connected, smooth $m$-manifolds which admit two smoothness structures with differing spans, stable spans, and immersion co-dimensions. In dimension $15$ the examples include the total spaces of certain $7$-sphere bundles over $S^8$. The construction of such manifolds is based on the topological variance of the second Pontrjagin class: a fact which goes back to Milnor and which was used by Roitberg to give examples...

We classify the genus one compact (PL) 5-manifolds and prove some results about closed 5-manifolds with free fundamental group. In particular, let $M$ be a closed connected orientable smooth $5$-manifold with free fundamental group. Then we prove that the number of distinct smooth $5$-manifolds homotopy equivalent to $M$ equals the $2$-nd Betti number (mod $2$) of $M$.