Borromean surgery formula for the Casson invariant

Abstract

It is known that every oriented integral homology $3$–sphere can be obtained from $S^3$ by a finite sequence of Borromean surgeries. We give an explicit formula for the variation of the Casson invariant under such a surgery move. The formula involves simple classical invariants, namely the framing, linking number and Milnor’s triple linking number. A more general statement, for $n$ independent Borromean surgeries, is also provided.