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The famous Kunneth formula expresses the homology of a product manifold as the tensor product of the two algebras. Now suppose we know that a manifold $X$ has a decomposition $H_*(X) \simeq A \otimes ...

Step 1: We glue two sets of complement space of $D^2\times T^2$ out of the 4-sphere $S^4$, through their $T^3$ boundary to obtain a new 4-manifold: $$(S^4 \smallsetminus D^2\times T^2) \cup (S^4 \...