##
**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?

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 |