 
Summary: PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 100, Number 2, June 1987
THE THURSTON N O R M A N D 2HANDLE ADDITION
MARTIN SCHARLEMANN
ABSTRACT. Suppose a 2handle is attached to a compact orientable 3mani
fold M along an annulus A contained in a subsurface N of a M . If N is
compressible in M, but N  A is not, then the Thurston norm is unaffected.
This generalizes a series of results due to Przytycki, Jaco, and Johannson.
Following Thurston [Th],define the complexity X (S) of an oriented surface S
to be x(C), where C is the union of all nonsimply connected components of S
and x(C) is its Euler characteristic. Hence X(S) 2 0. For M a compact oriented
3manifold and N a (possibly empty) surface in dM, assign to any homology class a
in H2(M,N; Z) the minimum complexity v of all oriented imbedded surfaces whose
fundamental class represents a, Thurston shows that there is a unique continuous
extension of v to H2(M,N; R). It is a pseudonorm and, if the fundamental class
of every embedded sphere, torus, disk, and annulus is trivial in H2(M,N), it is a
norm.
Any continuous map p : (X,Y) + (M,N) induces a pseudonorm p*(v) on
H2(X,Y) definedby p*(v)(a) = v(p*(a)). If p: (M', N') + (M, N) is an inclusion
