Math. Nachr. zyxwvuts 174 (1995) 265-272 zyxwvut The Topology of the Real Part of a Holomorphic Function By LAURENTIU PAUNESCU of Sydney (Received March 22, 1993) (Revised Version June 13, 1994) Abstract. We describe the connection between the Milnor fibre of a holomorphic function and the Milnor fibre of its real part and we derive some simple equations for manifolds appearing as real links of real parts of holomorphic functions. For a given analytic functionf: zyxwv (IR", 0) zyxwv + (IR, 0) we want to study its positive and negative Moreover, we want as well some information about the linkf - zyx '(6) n aB, orf - ( - 6) n z aB,. In the general case of an arbitrary analytic function few things are known. In that particular case whenfis the real part of a holomorphic function, we can obtain information about the real fibres and the links by using the extensive knowledge of the complex Milnor fibre. fibres, f zyxwvu -'(6),f-'( -6), 6 > 0 small enough, in a small disk Be centered at the origin. These results improve the results from [PI where we worked in the homotopy category. Also we shall use these results to derive some simple equations for manifolds appearing as the real link associated to the real part of a holomorphic function. The real links which will appear in our context are four-dimensional smooth manifolds, closed and oriented, embedded in zyxwvut S5. The links and the Milnor fibres of holomorphic functionsf: (C", 0) -, (C, 0) have been intensively studied, and therefore then one may expect, using the information obtained in this way, to give a fair picture of the links and the Milnor fibres associated to the real parts of holomorphic functions. The complex links determined by holomorphic functions are (2n - 1)-dimensional, smooth, closed, orientable manifolds embedded in S2"+ If we consider the link determined by the real part we will deal with 2n-dimensional smooth, closed, orientable manifolds, embedded in SZnf1. Our paper will investigate these real links and the corresponding real fibres in the particular case n = 2, and will give some simple equations in a number of cases. For example, the connected sum of k-copies of Sz x S2, denoted by #$ x Sz, k 2 1, is simply given by the link associated to the real part of the holomorphic function .t (c3,o) -+ zyxwvutsrqpo (c, 01, zyxwvu f(x, y, z) = xz + yz + Zk+l ,