arXiv subjects

Julia Semikina

Publications and source records attributed to Julia Semikina.

5 recordsLinked to original sources

arXiv2025

Scissors congruence K-theory for equivariant manifolds

We introduce a scissors congruence $K$-theory spectrum which lifts the equivariant scissors congruence groups for compact $G$-manifolds with boundary, and we show that on $π_0$ this is the source of a spectrum level lift of the Burnside ring valued equivariant Euler characteristic of a compact $G$-manifold. We also show that the equivariant scissors congruence groups for varying subgroups assemble into a Mackey functor, which is a shadow of a conjectural higher genuine equivariant structure.

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arXiv2025

Parametrized scissors congruence $K$-theory of manifolds and cobordism categories

We introduce a parametrized version of scissors congruence $K$-theory of manifolds with tangential structure, which includes a topologized version of the scissors congruence $K$-theory of oriented manifolds as a special case. We examine the relation of this $K$-theory spectrum with cut-and-paste invariants, the (parametrized) cobordism category and with (bivariant) algebraic $K$-theory of spaces. We show that the scissors congruence $K$-theory of oriented manifolds agrees on $\pi_0$ with a version of the oriented cobordism category where we allow cobordisms to have free boundaries. Lastly, we show that the spectrum level refinement of the Euler characteristic from the scissors congruence $K$-theory to $K(\mathbb{Z})$ detects on $\pi_1$ the Kervaire semicharacteristic.

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arXiv2024

A K-theory spectrum for cobordism cut and paste groups

Cobordism groups and cut-and-paste groups of manifolds arise from imposing two different relations on the monoid of manifolds under disjoint union. By imposing both relations simultaneously, a cobordism cut and paste group $\overline{\mathrm{SK}}_n$ is defined. In this paper, we extend this definition to manifolds with boundary obtaining $\overline{\mathrm{SK}}^{\partial}_n$ and study the relationship of this group to an appropriately defined cobordism group of manifolds with boundary. The main results are the construction of a spectrum that recovers on $π_0$ the cobordism cut and paste groups of manifolds with boundary, $\overline{\mathrm{SK}}^{\partial}_n$, and a map of spectra that lifts the canonical quotient map $\mathrm{SK}^{\partial}_n \rightarrow \overline{\mathrm{SK}}^{\partial}_n$.

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arXiv2021

Cut and paste invariants of manifolds via algebraic K-theory

Recent work of Jonathan Campbell and Inna Zakharevich has focused on building machinery for studying scissors congruence problems via algebraic $K$-theory, and applying these tools to studying the Grothendieck ring of varieties. In this paper we give a new application of their framework: we construct a $K$-space that recovers the classical $\mathrm{SK}$ ("schneiden und kleben," German for "cut and paste") groups for manifolds on $π_0$, and we construct a derived version of the Euler characteristic.

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arXiv2017

A solvable counterexample to the Hambleton-Taylor-Williams Conjecture

I. Hambleton, L. Taylor and B. Williams conjectured a general formula in spirit of H. Lenstra for the decomposition of $G_n(RG)$ for any finite group $G$ and noetherian ring $R.$ The conjectured decomposition was shown to hold for some large classes of finite groups. D. Webb and D. Yao discovered that the conjecture failed for the symmetric group $S_5$, but remarked that it still might be reasonable to expect the HTW-decomposition for solvable groups. In this paper we show that the solvable group $\mathrm{SL}(2,\mathbb{F}_3)$ is also a counterexample to the conjectured HTW-decomposition. Furthermore, we prove that for any finite group $G$ the rank of $G_1(\mathbb{Z}G)$ does not exceed the rank of the expression in the HTW-decomposition.

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