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ANALYSIS SITUS.

149

form which generalizes to n dimensions) to P. Heegaard in his
dissertation, Forstudier til en topologisk teori for de algebraiske
fladers sammenhaeng, Copenhagen, 1898 (republished in the
Bulletin de la Soc. Math. de France, Vol. 44 (1916), p. 161).
It is also described very clearly by M. Dehn, Math. Ann., Vol.
69, p. 165. Dehn draws from it the corollary that any M3 can
be defined by a non-singular complex having four 3-cells.
42. The curves c1, c2,.*, c, are the boundaries of a set of
2-cells which reduce N3 to a single 3-cell. In like manner there
is a set of a curves, d1, d2, '* *, da no two of which have a point
in common and which bound a set of a 2-cells which reduce
L3 to a single 3-cell. Moreover M2 and the two sets of curves
fully determine M3.
In fact, suppose we have a manifold M2 of genus a and two
sets of curves c1, c2, *.., ca and d1, d2, ' ', da, each set being
such that no two of its curves have a point in common and such,
moreover, that by introducing a 2-cells each bounded by one
of the curves, M2 is converted into a complex which can bound
a 3-cell if each of the a 2-cells is counted twice. Then if we
introduce a set of 2-cells of this sort for the curves c1, c2, * *, c,
and a 3-cell bounded by the resulting complex we obtain an open
manifold L3 bounded by M2. If now we introduce another set
of a 2-cells bounded by dl, d2, d, da and having no points in
common with each other or with L3, or M2, we can introduce
another 3-cell bounded by M2 and these 2-cells. The resulting
three-dimensional complex is clearly a manifold which is homeomorphic with Ms if M2 and the curves were determined from M3
in the manner described in the paragraph above.
43. The problem of three-dimensional manifolds is thus reduced to one regarding systems of curves upon a two-dimensional
manifold. The modifications which can be made in the systems
of curves of a Heegaard diagram without changing the manifold
M3 represented by the diagram have been studied (though not
completely) by Heegaard in his dissertation. The most important results thus far obtained on systems of curves are those
of Poincare in his fifth complement, in which he was evidently
considering the problem of three-dimensional manifolds from