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)


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