The following is a problem in Munkres "Algebraic Topology" that I do not know how to tackle. Let $A$ be the union of two once linked circles in $S^3$ and let $B$ be the union of two unlinked circles. Show that the (integral) cohomology groups of $S^3 - A$ and $S^3 - B$ are isomorphic but the cohomology rings are not.
At this stage in the book Munkres has just defined cohomology and the cup product and done some hands on calculations so I think that is what he has in mind for this problem. I am not sure if Munkres wants explicit triangulations and explicit cocyles generating the cohomology but this would be nice.