Topology and its Applications 69 (1996) 251-264
Least weight injective surfaces are fundamental
To detect if there is an injective surface in a compact irreducible 3-manifold it suffices to
triangulate the manifold and check only the fundamental surfaces (Jaco and Oertel, 1984). Here
we show that this is true simply because an injective surface of least weight will be fundamental.
Keywords: Haken manifold; Injective surface; Normal surface; Fundamental surface;
AMS c./as.s(fificatim: Primary 57M50, Secondary 57Q I5
In  Jaco and Oertel show there is an algorithm to decide if an irreducible 3-manifold
is Haken. The critical step is to show that in any closed 3-manifold there is a finite
constructible set of surfaces in M so that M contains an injective surface (different from
S*) if and only if one of the members of this finite set is injective.
A central ingredient is Haken's theory of normal surfaces [ 11. Haken shows that each
normal surface can be constructed from a finite set of "fundamental" surfaces. The main