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We prove that the space of mathematical instantons with second Chern class 5 over P 3 is smooth and irreducible. Unified and simple proofs for the same statements in case of second Chern class ≤ 4 are contained.

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We show that the idea used by Kempf (1990) in order to obtain a splitting criterion for vector bundles on projective spaces leads to an elementary proof of the Babylonian tower theorem for this class of bundles, a result due to Barth- Van de Ven (1974) in the rank 2 case and to and Tyurin (1976) in the case of arbitrary rank. As a byproduct we obtain a slight improvement of the numerical criterion of Flenner (1985) in the particular case under consideration.