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We show vanishing results about the infimum of the topological entropy of thegeodesic flow of homogeneous smooth four manifolds. We prove that any closedoriented geometric four manifold has zero minimal entropy if and only if it haszero simplicial volume. We also show that if a four manifold M admits ageometric decomposition, in the sense of Thurston, and does not have geometricpieces modelled on hyperbolic four-space, the complex hyperbolic plane or theproduct of two hyperbolic planes, then M admits an F-structure. It follows thatM has zero minimal entropy and collapses with curvature bounded from below. Wethen analyse whether or not M admits a metric whose topological entropycoincides with the minimal entropy of M and provide new examples of manifoldsfor which the minimal entropy problem cannot be solved.Comment: 38 pages, corrected flat case; Theorem A is now in terms of  F-structures, improved exposition; new final section collects all results  needed for the proofs of the main theorem

Minimal entropy and geometric decompositions in dimension four