Computing the Krichever genus

Let $\psi$ denote the genus that corresponds to the formal group law having invariant differential $\omega(t)$ equal to $\sqrt{1+p_1t+p_2t^2+p_3t^3+p_4t^4}$ and let $\kappa$ classify the formal group law strictly isomorphic to the universal formal group law under strict isomorphism x\CP(x). We prove that on the rational complex bordism ring the Krichever-H\"ohn genus $\phi_{KH}$ is the composition $\psi\circ \kappa^{-1}$. We construct certain elements $A_{ij}$ in the Lazard ring and give an alternative definition of the universal Krichever formal group law. We conclude that the coefficient ring of the universal Krichever formal group law is the quotient of the Lazard ring by the ideal generated by all $A_{ij}$, $i,j\geq 3$.Comment: 6 pages, revised Journal of Homotopy and Related Structures, 201

Polynomial behavior of the Honda formal group law

This note provides the calculation of the formal group law $F(x,y)$ in modulo $p$ Morava $K$-theory at prime $p$ and $s>1$ as an element in $K(s)^*[x][[y]]$ and one application to relevant examples.Comment: 4 pages, submitted in Journal of Homotopy and Related Structure

Complex cobordism modulo c1c_1-spherical cobordism and related genera

We prove that a sequence $S=(x_1, x_k, k\geq 3)$ of polynomial generators of $c_1$-spherical cobordism ring $W_*$, viewed as a sequance in the complex cobordism ring \MU_* by forgetful map, is regular. Using the Baas-Sullivan theory of cobordism with singularities we define a commutative complex oriented cohomology theory \MU^*_S(-), complex cobordism modulo $c_1$-spherical cobordism, with the coefficient ring \MU_*/S. Then any $\Sigma\subseteq S$ is also regular in \MU^* and therefore gives a multiplicative complex oriented cohomology theory \MU^*_{\Sigma}(-). The generators of $W_*$ can be specified in such a way that for $\Sigma=(x_k, k\geq 3)$ the corresponding cohomology is identical to the Abel cohomology, previously constructed in \cite{BUSATO}. Another example corresponding to $\Sigma=(x_k, k\geq 5)$ is classified by the Krichever-Hoehn complex elliptic genus \cite{KR}, \cite{H} modulo torsion.Comment: 10 page

Polynomial generators of MSU∗[1/2]MSU_*[1/2] related to to classifying maps of certain formal group laws

This note provides a set of polynomial generators of $MSU_*[1/2]$ defined by the formal group law in spherical cobordism. One aspect is to obtain the genera on $MSU_*[1/2]$ with values in polynomial ring as the restrictions of the classifying map of the Abel formal group law and the Buchstaber formal group law. The latter is associated with the Krichever-Hoehn complex elliptic genus.Comment: 11 page

Complex cobordism MU∗[1/2]^*[1/2] modulo MSU∗[1/2]^*[1/2] and related genera

This paper presents a commutative complex oriented cohomology theory with coefficients the quotient ring of complex cobordism MU$^*[1/2]$ modulo the ideal generated by any subsequence of any polynomial generators in special unitary cobordism MSU$^*[1/2]$ viewed as elements in MU$^*[1/2]$ by forgetful map.Comment: 5 page

Some explicit expressions concerning formal group laws

This paper provides some explicit expressions concerning the formal group laws of the Brown-Peterson cohomology, the cohomology theory obtained from Brown-Peterson theory by killing all but one Witt symbol, the Morava $K$-theory and the Abel cohomology.Comment: 11 page

On vanishing of all fourfold products of the Ray classes in symplectic cobordism

This note provides certain computations with transfer associated with projective bundles of Spin vector bundles. One aspect is to revise the proof of the main result of \cite{B} which says that all fourfold products of the Ray classes are zero in symplectic cobordism.Comment: 7 page

All extensions of C2C_2 by C2n+1×C2n+1C_{2^{n+1}}\times C_{2^{n+1}} are good

Let $C_{m}$ be a cyclic group of order $m$. We prove that if the group $G$ fits into an extension $1\to C_{2^{n+1}}^2\to G\to C_2\to 1$ then $G$ is good in the sense of Hopkins-Kuhn-Ravenel, i.e., $K(s)^*(BG)$ is evenly generated by transfers of Euler classes of complex representations of subgroups of $G$. Previously this fact was known for $n=1$.Comment: 12 page