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Malkhaz Bakuradze

Publications and source records attributed to Malkhaz Bakuradze.

16 recordsLinked to original sources
arXiv2025

On equivariant embeddings of G-bundles

For a compact group G, we give a sufficient condition for embedding one G-equivariant vector bundle into another one and for a stable isomorphism between two such bundles to imply an isomorphism. Our criteria involve multiplicities of irreducible representations of stabiliser groups. We also apply our result to ordinary nonequivariant vector bundles over the fields of quaternions, real and complex numbers and to ``real'' and ``quaternionic'' vector bundles. Our results apply to the classification of symmetry-protected topological phases of matter, providing computable bounds on the number of energy bands required to distinguish robust from fragile topological phases.

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arXiv2025

Isomorphism and stable isomorphism in "real" and "quaternionic" K-theory

We find lower bounds on the rank of a "real" vector bundle over an involutive space, such that "real" vector bundles of higher rank have a trivial summand and such that a stable isomorphism for such bundles implies ordinary isomorphism. We prove similar lower bounds also for "quaternionic" bundles. These estimates have consequences for the classification of topological insulators with time-reversal symmetry.

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arXiv2023

Complex cobordism modulo $c_1$-spherical cobordism and related genera

We prove that the ideal in complex cobordism ring $\MU^*$ generated by the polynomial generators $S=(x_1, x_k, k\geq 3)$ of $c_1$-spherical cobordism ring $W^*$, viewed as elements in $\MU^*$ by forgetful map is prime. 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 $Σ\subseteq S$ is also regular in $\MU^*$ and therefore gives a multiplicative complex oriented cohomology theory $\MU^*_Σ(-)$. The generators of $W^*[1/2]$ can be specified in such a way that for $Σ=(x_k, k\geq 3)$ the corresponding cohomology is identical to the Abel cohomology, previously constructed in \cite{BUSATO}. Another example corresponding to $Σ=(x_k, k\geq 5)$ gives the coefficient ring of the universal Buchstaber formal group law after tensored by $\mathbb{Z}[1/2]$, i.e., is identical to the scalar ring of the Krichever-Hoehn complex elliptic genus \cite{KR}, \cite{H}.

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arXiv2022

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.

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arXiv2019

On addition theorems related to elliptic integrals

This paper provides some explicit formulas related to addition theorems for elliptic integrals $\int_0^x dt/R(t)$, where $R(t)$ is the square root from a polynomial of degree 4. These integrals are related to complex elliptic genera and are motivated by Euler's addition theorem for elliptic integrals of the first kind.

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arXiv2016

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 $n=1$.

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arXiv2016

Morava K-theory rings of the extensions of C_2 by the products of cyclic 2-groups

In \cite{SCH1} Schuster proved that $mod$ 2 Morava $K$-theory $K(s)^*(BG)$ is evenly generated for all groups $G$ of order 32. There exist 51 non-isomorphic groups of order 32. In \cite{H}, these groups are numbered by $1, \cdots ,51$. For the groups $G_{38},\cdots, G_{41}$, that fit in the title, the explicit ring structure is determined in \cite{BJ}. In particular, $K(s)^*(BG)$ is the quotient of a polynomial ring in 6 variables over $K(s)^*(pt)$ by an ideal generated by explicit polynomials. In this article we present some calculations using the same arguments in combination with a theorem of \cite{B0} on good groups in the sense of Hopkins-Kuhn-Ravenel. In particular, we consider the groups $G_{36},G_{37}$, each isomorphic to a semidirect product $(C_4\times C_2\times C_2)\rtimes C_2$, the group $G_{34}\cong (C_4\times C_4)\rtimes C_2$ and its non-split version $G_{35}$. For these groups the action of $C_2$ is diagonal, i.e., simpler than for the groups $G_{38},\cdots, G_{41}$, however the rings $K(s)^*(BG)$ have the same complexity.

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arXiv2016

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.

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arXiv2014

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.

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arXiv2013

K^*(BG) rings for groups $G=G_{38},...,G_{41}$ of order 32

B. Schuster \cite{SCH1} proved that the $mod$ 2 Morava $K$-theory $K(s)^*(BG)$ is evenly generated for all groups $G$ of order 32. For the four groups $G$ with the numbers 38, 39, 40 and 41 in the Hall-Senior list \cite{H}, the ring $K(2)^*(BG)$ has been shown to be generated as a $K(2)^*$-module by transferred Euler classes. In this paper, we show this for arbitrary $s$ and compute the ring structure of $K(s)^*(BG)$. Namely, we show that $K(s)^*(BG)$ is the quotient of a polynomial ring in 6 variables over $K(s)^*(pt)$ by an ideal for which we list explicit generators.

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arXiv2013

Computing the Krichever genus

Let $ψ$ denote the genus that corresponds to the formal group law having invariant differential $ω(t)$ equal to $\sqrt{1+p_1t+p_2t^2+p_3t^3+p_4t^4}$ and let $κ$ 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-Höhn genus $ϕ_{KH}$ is the composition $ψ\circ κ^{-1}$. 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 $A_{ij}$, $i,j\geq 3$.

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arXiv2003

Transfer and complex oriented cohomology rings

For finite coverings we elucidate the interaction between transferred Chern classes and Chern classes of transferred bundles. This involves computing the ring structure for the complex oriented cohomology of various homotopy orbit spaces. In turn these results provide universal examples for computing the stable Euler classes (i.e. Tr^*(1)) and transferred Chern classes for p-fold covers. Applications to the classifying spaces of p-groups are given.

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