19 research outputs found

### 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 $c_1$-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 $MSU_*[1/2]$ 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$^*[1/2]$ modulo MSU$^*[1/2]$ 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 $C_2$ by $C_{2^{n+1}}\times C_{2^{n+1}}$ 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