Representing homology classes of closed orientable surfaces

We prove that the primitive elements of the first homology group of a genus g torus can be represented by a simple closed curve. Let T be a closed orientable surface with g handles (g > 0). The first homology group, HI (T), is the free abelian group on 2g generators. In general, if an element of H1 (T) is represented by a closed curve, the curve must cross itself. However, we prove that if the element is primitive, then it can be represented by a simple closed curve. Primitive means not divisible by an integer greater than one. Our proof is accomplished through use of the "twist" homeomorphism described by Lickorish [1]. Let ak and 8k be the standard oriented simple closed curves around the kth handle of T for 1 < k < g. So each ai or Pi represents a basis element of HI (T). For 1 < k < g, let Yk be a simple closed curve looping the kth and (k + l)st handles of T. The Lickorish "twist" homeomorphism about a simple closed curve J is defined by cutting T along J, giving a full twist to one edge, and then gluing T back together along J. Twists about ak, fk, and Yk induce automorphisms on HI (T) carrying ( * * *, ak, bk, ak+I, bk+,. * *) to ... * ak ?bk,bk,ak+1,bk+1, ... ), (... ,ak,bk ? ak,ak+l,bk+l ...), and (... ,ak ? bk ? bk+l,bk,ak+l ? bk ? bk+1, bk+1, ...) respectively. The "+" signs appear since there is always a choice of two ways of twisting. Now let (al, b1, a2, b2, ... ) be a fixed element of HI (T). By several appropriate iterations of twists about ak's and J8k'sq and using the Euclidean algorithm, we get a homeomorphism which induces a map carrying (a, , b, a2, b2, .. .) to (d1, 0, d2, 0, .. .), where di = g.c.d.{ai,bi}. By following this by further twists about ak'S, J8k'sq and yk's, we can produce a self-homeomorphism of T which induces a map carrying (a, , b, a2, b2, ... ) to (d, 0, 0, 0,...), where d = g.c.d.{dl, d2, ... ) = g.c.d.{al, bl, a2, b2 * I I }. For d = ? 1, (d, 0,0 ,0 , ... ) is represented by the simple closed curve ?al, and by applying the inverse of the homeomorphism we have constructed to a, we get the simple closed curve we desire. The converse is true, namely simple closed curves represent primitive (or trivial) elements. The following argument is due to Hans Samelson. A Received by the editors March 25, 1975. AMS (MOS) subject classifications (1970). Primary 57A05. Copyright e 1977, American Mathematical Society