In this note we show that $1$-skeletons and $2$-skeletons of $n$-pseudomanifolds with full boundary are $(n+1)$-connected graphs and $n$-connected $2$-complexes, respectively. This generalizes previous results due to Barnette and Woon.

For $K=K_1\bigsqcup ...\bigsqcup K_s$ and a link map $f:K\to {ℝ}^m$ let $K^{\sim }={\bigsqcup }_{i<j}{K}_i\times K_j$, define a map $f^{\sim }:K^{\sim }\to S^{m-1}$ by $f^{\sim }(x,y)=(fx-fy)/|fx-fy|$ and a (generalized) Massey-Rolfsen invariant $\alpha \left(f\right)\in {\pi }^{m-1}\left(K\right)$ to be the homotopy class of $f^{\sim }$. We prove that for a polyhedron K of dimension ≤ m - 2 under certain (weakened metastable) dimension restrictions, α is an onto or a 1 - 1 map from the set of link maps $f:K\to {ℝ}^m$ up to link concordance to ${\pi }^{m-1}\left(K^{\sim }\right)$. If $K_1,...,K_s$ are closed highly homologically connected manifolds of dimension $p_1,...,p_s$ (in particular, homology spheres), then ${\pi }^{m-1}\left(K^{\sim }\right)\cong {\oplus }_{i<j}{\pi }_{p_i+p_j-m+1}^S$.

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...