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Consider a (real) projective plane which is topologically locally flatly embedded in S 4 . It is known that it always admits a 2-disk bundle neighborhood, whose boundary is homeomorphic to the quaternion space Q, the total space of the nonorientable S'-bundle over

The abstract link L d of the complex isolated singularity x 2 + y 2 + z 2 + v 2d = 0 in (C 4 , 0) is diffeomorphic to S 3 × S 2 . We classify the embedded links of these singularities up to regular homotopies precomposed with diffeomorphisms of S 3 × S 2 . Let us denote by i d the inclusion L d ⊂ S 7 . We show that for arbitrary diffeomorphisms ϕ d : S 3 × S 2 −→ L d the compositions i d • ϕ d are image regularly homotopic for two values d 1 and d 2 of d if and only if d 1 ≡ d 2 mod 2.

In this paper, we completely determine the diffeomorphism types of the 5-dimensional links of 3-dimensional log-canonical singularities defined by Brieskorn polynomials. Moreover, we show that if k is an integer with 1 ≤ k < 6 11 , then there is no link K defined by a Brieskorn polynomial in C 4 such that the order of H2(K) is 6k.