## Formal groups, functional equations and generalized cohomology theories.(Russian)Zbl 0694.55008

The theory of formal groups plays an important role in algebraic topology because the formal groups of geometrical cobordism $$F_{MU}(u,v)$$ can be identified with the universal formal groups of Lazard. There exists [V. M. Bukhshtaber, Mat. Sb., Nov. Ser. 83(125), 575-595 (1970; Zbl 0204.235)] the following formula: $(1)\quad F_{MU}(u,v)=\frac{u+v+\sum {i,j\geq 1}[H(i,j)]u^ iu^ j}{CP(u)CP(v)}$ where $$CP(u)=\sum_{i\geq 0}[CP(i)]u^ i$$ and [CP(i)], [H(i,j)] are, respectively, the cobordism classes of a projective complex space and of a Milnor manifold. Then, a cohomology theory $$h^*(.)$$ is called C- oriented if in this theory there exist Chern characteristic classes $$c^ h_ i(\xi)$$, $$i=0,1,...$$, of a complex vector bundle. The theory of complex cobordism $$MU^*(.)$$ is a universal C-oriented cohomology theory as for every C-oriented cohomology theory $$h^*(.)$$ there exists a natural multiplicative transformation $$\mu_ h: MU^*(.)\to h^*(.)$$ such that $$\mu_ hc_ i^{MU}(\xi)=c^ h_ i(\xi)$$ and therefore $$\mu_ hF_{MU}(u,v)=f_ h(u,v)$$, where $$f_ h(u,v)=c^ h_ i(\xi_ 1\otimes \xi_ 2)$$ are the formal groups in the theory $$h^*(.)$$ and $$\xi$$ $$\to CP(\infty)$$ is the universal fibration. For a C-oriented theory $$h^*(.)$$ the formulae $$\sigma h(\xi)=\exp (bg(c^ h_ 1(\epsilon))$$, $$ch_ h(c^ h_ 1(\xi))=\bar g(c^ H_ 1(\xi))$$ give the Chern-Novikov character [S. P. Novikov, Izv. Akad. Nauk SSSR, Ser. Mat. 31, 855-951 (1967; Zbl 0169.545); V. M. Bukhshtaber, Supplement in the book of V. P. Snaith [Algebraicheskij cobordism i K-teorya”, 227-248 (Mir, 1983); Translation from Mem. Am. Math. Soc. 221 (1979; Zbl 0413.55004)] $$ch_ h: h^*(.)\to H^*(.;h^*(pt)\otimes Q)$$, where B is some Q-algebra, $$g(t)$$, $$\bar g(t)$$ are logarithm and exponent of the formal group $$f_ h(u,v)$$ and $$H^*(.)$$, $$K^*(.)$$ are, respectively, classical cohomology and K-theory. If $$H^*(pt)$$ does not have torsion then it is convenient to describe the geometrical contruction with Chern classes in the theory $$h^*(.)$$ in terms of these characters. N. Ray [Adv. Math. 61, 49-100 (1986; Zbl 0631.55002); Proc. Symp. Durham/Engl. 1985, Lond. Math. Soc. Lect. Note Ser. 117, 195-238 (1987; Zbl 0651.55004)] compared every formal group f(u,v) with the logarithm g(u) of a series of binomial polynomials [see R. Mullin and G.-C. Rota, Graph theory appl., Proc Adv. Sem. Wisconsin, Madison 1969, 167-213 (1970; Zbl 0259.12001)], with the help of the generating function $\sum_{n\geq 0}p_ n(x)t^ n=\exp (xg(t)),$ and Ray obtained new proofs for many important formulas in algebraic topology as some corollaries of the corresponding results. The homomorphism (Hirzebruch’s genus) $$\mu: \Omega_ U\to R$$ of the classified formal group f(u,v) over the ring R provides the latter structure as $$\Omega_ U$$-module. By using the tensor product of $$\Omega_ U$$-modules the authors define the homotopical functor $$MU^*_ f(.)=MU^*(.)\otimes_{\Omega_ U}R$$. If this functor satisfies the exactness axiom, then it is a cohomology theory.
The natural generalization of Toda’s genus is the two-parameter genus [see I. M. Krichever, Izv. Akad. Nauk SSSR, Ser. Mat. 38, 1289-1304 (1974; Zbl 0315.57021)] $T_{\alpha,\beta}: \Omega_ U\to Z[\alpha,\beta]$ which corresponds to the formal group $$f(u,v)=(u+v+\alpha uv)/(1-\beta uv)$$. The exponent of this formal group is $\bar g(u)=\frac{\exp (\epsilon_ 1u)-\exp (\epsilon_ 2u)}{\epsilon_ 1\exp (\epsilon_ 1u)-\epsilon_ 2\exp (\epsilon_ 2u)},$ where $$\epsilon_ 1$$, $$\epsilon_ 2$$ are the solutions of the equation $$\epsilon^ 2-\alpha \epsilon +\beta =0$$, which verifies the functional equation $(2)\quad \bar g(u+v)=\frac{\bar g(u)+\bar g(v)+\alpha \bar g(u)\bar g(v)}{1-\beta \bar g(u)\bar g(v)}.$ An important generalization of Toda’s genus connected with the virtual arithmetical genus [see F. Hirzebruch, Topological methods in algebraic geometry (1966; Zbl 0138.420); I. M. Krichever, Izv. Akad. Nauk SSSR, Ser. Mat. 40, 828-844 (1976; Zbl 0341.57024)] is the two-parameter genus $\chi_{\alpha,\beta}: \Omega_ U\to Q[\alpha,\beta],$ with the exponent $(3)\quad \bar g(u)=\frac{\exp (\alpha u)-\exp (\beta u)}{\alpha -\beta}.$ Formula (3) gives the general solution of the functional equation $(4)\quad \bar g(u+v)=\bar g(u)\theta (v)+\bar g(v)\theta (u),\quad \bar g(0)=0,\quad \bar g'(0)=1$ obtained by N. H. Abel [Oeuvres complètes I (Christiania, 1881), pp. 1-10]. S. Ochanine [Topology 26, 143-151 (1987; Zbl 0626.57014)] investigated the Hirzebruch genus (elliptic genus), corresponding to the Euler formal group over the ring $(5)\quad {\mathbb{Z}}[1/2][\gamma,\epsilon]:\quad f(u,v)=\frac{u\sqrt{R(v)}+v\sqrt{R(u)}}{1-\epsilon u^ 2v^ 2},\quad R(u)=1-2\delta u^ 2+\epsilon u^ 4,$ whose logarithm is the following elliptic integral: $$g(u)=\int^{u}_{0}R(u)^{-1/2}du$$ and the exponent $$\bar g(u)$$ is the Jacobi series $$(1/\lambda)sn(\lambda u,k)$$ for suitable $$\lambda$$, $$k$$. Formula (5) is equivalent to the addition theorem for the series $$\bar g(u)$$: $(6)\quad g(u+v)=\frac{\bar g(u)\sqrt{R(\bar g(v))}+\bar g(v)\sqrt{R(\bar g(u))}}{1-\epsilon \bar g(u)^ 2\bar g(v)^ 2}.$ Recently, F. Hirzebruch [Differential geomerical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser. Ser. C 250, 37-63 (1988; Zbl 0667.32009)] established some relations between elliptic genera and the genera $$T_{\alpha,\beta}$$, $$\chi_{\alpha,\beta}$$. Namely, Hirzebruch studied the genera $$\phi(N,k,\ell,\tau)$$ for which the exponent corresponding to the formal groups $$\bar g(N,k,\ell,\tau,u)$$ is an elliptic function of argument $$\tau\in {\mathbb{C}}$$, Im $$\tau>0$$, where $$0\leq k<N$$, $$0\leq \ell <N.$$
In the present paper the authors study the relations between the above formal groups on the basis that their exponents satisfy the following functional equation $\bar g(u+v)=(\bar g(u)\theta (v)+\bar g(v)\theta (u))\psi (g(u)g(v)),$ which generalizes the functional equations (2), (4), (6). It is proved that the ring $$\Lambda$$ of the coefficients of formal groups (Abel universal formal groups) $$f(u,v)=u+v+\alpha_ 1uv+\sum_{n\geq 2}\alpha_ n(uv^ n+vu^ n)$$ does not have torsion and therefore, according to (1), $$\Lambda =\Omega_ U/J$$, where J is the ideal generated by [H(i,j)]-[CP(1)][CP(i-1)][CP(j-1)].
On the basis of these results the authors solve the following question: under which localization of the ring $$\Lambda$$ the functor $$MU^*_ f(.)$$ is a cohomological theory?
Reviewer: I.Pop (Iaşi)

### MSC:

 55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology 14L05 Formal groups, $$p$$-divisible groups
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