Structures in topological recursion relations

In this paper, we study the basic structures of degree- g g topological recursion relations on the moduli space of curves M ¯ g , n M̄g,n : (i) the coefficient of the bouquet class on M ¯ g , n M̄g,n , which gives the answer to a conjecture of T. Kimura and X. Liu [Comm. Math. Phys. 262 (2006), pp. 645–661]; (ii) linear relations among the coefficients of certain rational tails locus of M ¯ g , n M̄g,n . Three applications of topological recursion relations will be discussed: (i) coefficients of universal equations for Gromov–Witten invariants for any smooth projective variety; (ii) the coefficient of the bouquet class in the double ramification formula of the top Hodge class λ g λ _g ; (iii) a new recursive formula for computing the intersection numbers on the moduli space of stable curves.