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The purpose of this paper is to give a decorated version of the Eynard–Orantin topological recursion using a 2D Topological Quantum Field Theory. We define a kernel for a 2D TQFT and use an algebraic reformulation of a topological recursion to define how to decorate a standard topological recursion by a 2D TQFT. The A-model side enumerative problem consists of counting cell graphs where in addition vertices are decorated by elements in a Frobenius algebra, and which are a decorated version of the generalized Catalan numbers. We show that the function that counts these decorated graphs, which is a decoration of the counting function of the generalized Catalan numbers by a Frobenius algebra, satisfies a topological recursion with respect to the edge-contraction axioms. The path we follow to pass from the A-model side to the remodeled B-model side is to use a discrete Laplace transform as a mirror symmetry map. We show that a decorated version by a 2D TQFT of the Eynard–Orantin differentials satisfies a decorated version of the Eynard–Orantin recursion formula. We illustrate these results using a toy model for the theory arising from the orbifold cohomology of the classifying space of a finite group. In this example, the graphs are orbifold cell graphs (graphs drawn on an orbifold punctured Riemann surface) defined out of the moduli space M¯¯¯¯¯¯g,n(BG) of stable morphisms from twisted curves to the classifying space of a finite group G. In particular we show that the cotangent class intersection numbers on the moduli space M¯¯¯¯¯¯g,n(BG) satisfy a decorated Eynard–Orantin topological recursion and we derive an orbifold DVV equation as a consequence of it. This proves from a different perspective the known result which states that the ψ-class intersection numbers on M¯¯¯¯¯¯g,n(BG) satisfy the Virasoro constraint condition.

We consider the maps between classifying spaces of /7-compact groups of the form BX x B Y -* BZ. The main theorem shows that if the restriction map on B Y is a weak epimorphism, then the restriction on BX should factor through the classifying spaces of the center of the /7-compact group Z. Introduction In [8], the author investigated certain pairing problems for classifying spaces of compact Lie groups. The main work in this paper can be regarded as a /?-compact group version. Dwyer-Wilkerson [3] defined a /7-compact group and studied its properties. The purely homotopy theoretic object appears to be a good generalization of a compact Lie group at the prime p. A ^-compact group has rich structure, such as a maximal torus, a Weyl group, etc. A note of Moller [13] summarizes their work. Further development on the homotopy theory of /7-compact groups can be seen, for example, in [4], [14], [15], [2] and [18]. We first recall some basic things about the/7-compact groups and pairing problems, and then state our main results. A /7-compact group, [3], is a loop space X such that X is Fp-finite and that its classifying space BX is F^-complete. The /7-completion of a compact Lie group G is a /7-compact group if n0(G) is a /7-group. For an odd dimensional Communicated by K. Saito, February 3, 1998. 1991 Mathematics Subject Classification(s): 55R35, 55P15, 55P60. * Fukuoka University, Fukuoka 814-0180, Japan.

In this paper we calculate the squaring operations in the 4connective fibre spaces over the classifying spaces of the exceptional Lie groups. Let G be a compact, 1 -connected and simple Lie group. As is well known, its classifying space BG is 3-connected and H*(BG', Z) ~Z. Choose a generator y^H^(BG',Z). Then the 4-connective fibre space BG over BG is, by definition, the homotopy fibre of y^:BG— > K(Z,ty. Note that BG is a classifying space of G, the 3-connective fibre space over G. Here we quote the results in [2] and [4]. Define the sets Jl (/ = 2, 4, 6, 7, 8) as follows

Let p be a prime number and Z/p be a cyclic group of order p '. We choose BZ/p, the classifying space of Z/p, as the colimit of the lens spaces L(p), It is well known that K(BZ/p) =0Z^ by Atiyah [1], Atiyah-Segal [2], and K^-^BZ/p*} =®Z/p°° by Vick [9] (and the groups of other degrees are trivial) . We consider the connective X-theory k. Using the Atiyah-Hirzebruch spectral sequence, connective K(co) homology group of BZ/p is a subgroup of periodic K(co) homology group of BZ/p. And the Atiyah Hirzebruch spectral sequence also shows that k2n-i (BZ/p ) is a finite group for any n. Moreover, the Atiyah-Hirzebruch spectral sequence determines its order. The purpose of this paper is to calculate the additive structure of

Let CP°° be the infinite dimensional complex projective space and BU the classifying space of stable complex vector bundles. Then there is the natural inclusion j: CP°°—>BU and the structure map of the infinite loop space structure defined by the Bott periodicity ^:d(BU) >BU where Q,( ) =ColimllO 2"( ). Let Z: Q(CP~) > BU be the composition £°Q(j). The results of Segal [8] and Becker [2] show us that there exists a map s: BU >Q^(CP°°) such that Aos = id. The main result of this paper is to show that one can take s satisfying that SQJ~inclusion : CP°° >Q(CP°°}. To show this, we will use the results of Brumfiel-Madsen [4] for the evaluation of the transfer map.