arXiv subjects

Semyon Abramyan

Publications and source records attributed to Semyon Abramyan.

5 recordsLinked to original sources

arXiv2026

Graphical composition of mapping spaces between modules of configuration-space-type

In embedding calculus, spaces of embeddings are identified with derived mapping spaces between framed Fulton-MacPherson-type modules (framed configuration spaces). Unfortunately, there are no sufficiently good algebraic models for framed Fulton-MacPherson modules that would allow us to explicitly describe the rational homotopy type of the embedding space. Recently there were several attempts to avoid dealing with the framed versions of Fulton-MacPherson modules by considering framed manifolds, e.g. embeddings modulo immersions $\overline{\mathrm{Emb}}$ in a recent paper by Fresse, Turchin and Willwacher, or embeddings with a deformation of the framing $\widetilde{\mathrm{Emb}}$ in a recent paper by the author. In both cases, the rational homotopy type of the corresponding embedding space has an explicit description in terms of graphs (hairy graph complexes). We construct a combinatorial graphical composition for the composition of embedding spaces $\widetilde{\mathrm{Emb}}$. As a step on our way, we describe the action of the coinduction functor on configuration-space-type $e_n^c$-comodules.

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arXiv2025

Embedding calculus for parallelized manifolds

We study a variant of the embedding functor $\mathop{\mathrm{Emb}}(M, N)$ that incorporates homotopical data from the frame bundle of the target manifold $N$. Given a parallelized $m$-manifold $M$ and an $n$-manifold $N$ equipped with a section of its $m$-frame bundle, we define a modified embedding functor $\widetilde{\mathop{\mathrm{Emb}}}(M, N)$ that interpolates between the standard embedding and a reference framing. Using the manifold calculus of functors, we identify the Taylor tower of $\widetilde{\mathop{\mathrm{Emb}}}(M, N)$ with a mapping space of right modules over the Fulton-MacPherson operad. We prove a convergence theorem under a codimension condition, establishing a weak equivalence between $\widetilde{\mathop{\mathrm{Emb}}}(M, N)$ and its Taylor approximation. Finally, under rationalization, we describe the derived mapping space in terms of a combinatorial hairy graph complex, enabling computational access to the rational homotopy type of the space of embeddings.

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arXiv2021

On homology of the $MSU$ spectrum

We give a complete proof the Novikov isomorphism $\varOmega^{SU}\otimes \mathbb Z[\textstyle\frac12]\cong\mathbb Z[{\textstyle\frac12}][y_2,y_3,\ldots],\quad\mathrm{deg} y_i=2i$, where $\varOmega^{SU}$ is the $SU$-bordism ring. The proof uses the Adams spectral sequence and a description of the comodule structure of $H_*(MSU; \mathbb F_p)$ over the dual Steenrod algebra $\mathfrak A_p^*$ with odd prime $p$, which was also missing in the literature.

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arXiv2019

Higher Whitehead products in moment-angle complexes and substitution of simplicial complexes

We study the question of realisability of iterated higher Whitehead products with a given form of nested brackets by simplicial complexes, using the notion of the moment-angle complex $Z_K$. Namely, we say that a simplicial complex $K$ realises an iterated higher Whitehead product $w$ if $w$ is a nontrivial element of $π_*(Z_K)$. The combinatorial approach to the question of realisability uses the operation of substitution of simplicial complexes: for any iterated higher Whitehead product $w$ we describe a simplicial complex $\partialΔ_w$ that realises $w$. Furthermore, for a particular form of brackets inside $w$, we prove that $\partialΔ_w$ is the smallest complex that realises $w$. We also give a combinatorial criterion for the nontriviality of the product $w$. In the proof of nontriviality we use the Hurewicz image of $w$ in the cellular chains of $Z_K$ and the description of the cohomology product of $Z_K$. The second approach is algebraic: we use the coalgebraic versions of the Koszul and Taylor complex for the face coalgebra of $K$ to describe the canonical cycles corresponding to iterated higher Whitehead products $w$. This gives another criterion for realisability of $w$.

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arXiv2018

Iterated Higher Whitehead products in topology of moment-angle complexes

In this paper we study the topological structure of moment-angle complexes $\mathcal{Z_K}$. We consider two classes of simplicial complexes. The first class $B_Δ$ consists of simplicial complexes $\mathcal{K}$ for which $\mathcal{Z_K}$ is homotopy equivalent to a wedge spheres. The second class $W_Δ$ consists of $\mathcal{K}\in B_Δ$ such that all spheres in the wedge are realized by iterated higher Whitehead products. Buchstaber and Panov asked if it is true that $B_Δ = W_Δ$. In this paper we show that this is not the case. Namely, we give an example of a simplicial complex whose corresponding moment-angle complex is homotopy equivalent to a wedge of spheres, but there is a sphere which cannot be realized by any linear combination of iterated higher Whitehead products. On the other hand we show that class $W_Δ$ is large enough. Namely, we show that the class $W_Δ$ is closed with respect to two explicitly defined operations on simplicial complexes. Then using these operations we prove that there exists a simplicial complex that realizes any given iterated higher Whitehead product. Also we describe the smallest simplicial complex that realizes an iterated product with only two pairs of nested brackets.

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