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SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

268 isolated singular

268 isolated singular points (where it is locally a cone on a projective space), and it can be triangulated so that each singular point is at a vertex (simplex of dimension 0). Now llft this triangulation up to L, and let Ck be the Q-vector space generated by the simpllces of dimension k. Note that r induces an action on each Ck by permutation matrices (there are no -1 enties by construction). Thus, the trace of r on Ck, which we denote by Tr(r; Ck), is equM to the number of simplexes of dimension k whidl are left fixed by r. Since all N fixed points are isolated, Tr(r; Ck) f 0 if k > 0, t N ifk = 0. Thus, ~=o(-1)kTr(r; Ck) = N. Now, since r is a chain map on the chain complex, 0--* C, --+ C,_1 --+ ...-, C0~ 0, an easy argument shows that ~=o(-1)kTr(r;Ck) = ~.=o(-1)kTr(r;Hk), where/irk = Hk(C.) =//k(L, Q). It remains therefore, to calculate Tr(r; EL.). Denote the map on Hr, induced by r by rk. Since r is an involution, so are the rl,. Moreover, since/5 is a torus, //k(L,Q) is the k-th exterior power of HI(L,Q) and rk is the k-th exterior power of rl. Since 7"1 is an involution, there is a basis of HI(L,Q) with respect to which it is diagonal with +l's down the diagonal. Let rl =/, E) -/8. It is not hard to show that Multiplying this by t k and summing over k gives, k=0 l=o k - l " Tr(rk)t k = (-1)"(1 + t)r(1 - t)'. To obtain the alternating sum put t = -1 and multiply by (-1)", so L(r) f 0 if r > 0, t 2" if r = 0. Thus N = 0 or 2" for any involution with isolated fixed points, t2 3 Lagrangian maps and generating families Let L C T*Q be a Lagrangian submamifold. Since ~r : T*Q -~, Q is a Lagraxtgian fibration its restriction rlL to L is by definition a Lagrangian map. (From now on we will denote rlL simpl# by ~r.) To study the local geometry of such maps we use generating families. Here we give a very brief outline of the theory of generating families ms developed by V.I. Arnold and V.M. Zakalyukin. The details can be found in [3]. RecM1 that the caus$ic of a Lagrangian map is the set of its singular values. Let f : (Rn,0) -, R be a function germ, and let F : (R n x R",0) --* R be a deformation of f. Denote F(x, u) by fu(X), so fo = f. Let C(F) = I d(/,,)(x) = 0}

269 be the set of critical points in the family F. We assume from now on that C(F) is a submanifold (germ) of R" x R a, in which case the projection ~f : C(F) --} R a is a Lagrangian map (germ). The family F is said to be a generating family for rE, and we will call f the organizing centre of F (and of rE). The set of singldax points of the map rrF axe precisely the points (x,u) for which f,, has a degenerate critical point at x, and thus the caustic of the Lagrangian map rF is precisely the discriminant of the generating family F. Given any Lagrangian map germ r : (L,0) -4 (Q,0) there is a family 2' as above with rE ..~ r (where ,.~ is Lagrangian equivalence). Of course, a = dim L = dim Q. Furthermore one can take n = dimkerdr0 (necessarily, n > dimkerdr0, and one can reduce to n = dim kerdr0 by a splitting lemma argument). Two Lagrangian map germs are Lagrangian equivalent if and only if their generating families are ~+-equivalent: there axe diffeomorphism germs ¢ : (R",0) --* (R",0), and !l~ : (R" x R", 0) --* (R n X R a, 0), related by ~(x, u) = (¢(x, ,,), ¢(n)), for some map germ ¢, and a function germ a on (R", 0) such that Fo~(z, u) = G(x, u) + a(u). Note that the map x ~ ¢(x, 0) is a diffeomorphism germ, so the organizing centres of T~+-equivalent families are themselves T&equivalent. Finally, a Lagrangian map germ is Lagrangian stable if and only if any associated generating family is an 7~+-versal defor- mation of its organizing centre. In our application, the Lagrangian map germs in question are invariant under a Z2 action: r(z) = r(-z). The only difference this makes to the discussion above is that the generating family F(z,u) is odd in x, i.e. F(-x,u) = -F(x,u). Furthermore, the Lagrangian equivalence respects the Z2 action if and only if the 7~+-equivalence between generating families does, i.e. ¢(-x, u) = -¢(z, u). We will call this Tl+2-equivalence, even though the '+' is redundant as a must be 0. Denote by En the ring of smooth function germs (R", 0) --* R, by E + the subring of those invariant under the action of Z2 (acting by x ~ -x), and denote by E~" the £+- module of odd function germs (i.e. f(x) = -f(-x)). Denote by m, and m + the ideals in E, and C + respectively of germs vanishing at 0. For f E ~" let J+(f) be the ideal in E+ generated by the partial derivatives of f. Define The 7~/~-codimension is defined to be J_(f) = J+(f).£~. cod(f) = dim.(£n/J- (f))" Applying the usual arguments of singularity theory, adapted to the world of odd functions, one obtains, Proposition 3 (i) Let f E C; and let k be an odd integer. If n n cZ c m+..l_(f)

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