arXiv subjects

Victor Turchin

Publications and source records attributed to Victor Turchin.

16 recordsLinked to original sources

arXiv2025

Goussarov-Polyak-Viro type formulas for $(4k-1)$-dimensional knots and links in $\mathbb{R}^{6k}$

We produce combinatorial formulas for invariants of smooth embeddings of $(2\ell-1)$-spheres into $\mathbb{R}^{3\ell}$ for $\ell\geq 2$. Furthermore, we obtain such a formula for the Haefliger invariant, which classifies smooth knots $S^{4k-1}\hookrightarrow \mathbb{R}^{6k}$ up to isotopy. Our approach is similar in spirit to the work of Goussarov, Polyak, and Viro expressing finite-type invariants of classical knots in terms of Gauss diagrams. We similarly project higher dimensional knots and links onto a hyperplane and study the preimages of the sets of double and singular points in the embedded spheres. As an auxiliary result, we show that the space of $n$-dimensional braids with $k$ strands in $\mathbb{R}^{n+q}$ is a homotopy retract of the space of long links $\underset{k}{\sqcup}\mathbb{R}^n\hookrightarrow\mathbb{R}^{n+q}$ for $q\geq 3$, thus proving a conjecture of Komendarczyk, Koytcheff and Voli\'c.

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arXiv2024

Mapping spaces of Swiss cheese operads

We show that the color restriction map $\mathrm{Op}^h({\mathcal{SC}}_m,{\mathcal{SC}}_n)\to \mathrm{Op}^h({\mathcal E}_{m-1},{\mathcal E}_{n-1})$ from the derived mapping space of Swiss cheese operads to that of little discs operads, is a weak homotopy equivalence. We explain how this can help in the study of disc concordance embedding spaces.

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arXiv2024

On the smoothing theory delooping of disc diffeomorphism and embedding spaces

The celebrated Morlet-Burghelea-Lashof-Kirby-Siebenmann smoothing theory theorem states that the group $\mathrm{Diff}_\partial(D^n)$ of diffeomorphisms of a disc $D^n$ relative to the boundary is equivalent to $\Omega^{n+1}\left(\mathrm{PL}_n/\mathrm{O}_n\right)$ for any $n\geq 1$ and to $\Omega^{n+1}\left(\mathrm{TOP}_n/\mathrm{O}_n\right)$ for $n\neq 4$. We revise smoothing theory results to show that the delooping generalizes to different versions of disc smooth embedding spaces relative to the boundary, namely the usual embeddings, those modulo immersions, and framed embeddings. The latter spaces deloop as $\mathrm{Emb}_\partial^{fr}(D^m,D^n)\simeq\Omega^{m+1}\left(\mathrm{O}_n\backslash\!\!\backslash\mathrm{PL}_n/\mathrm{PL}_{n,m}\right)\simeq \Omega^{m+1}\left(\mathrm{O}_n\backslash\!\!\backslash\mathrm{TOP}_n/\mathrm{TOP}_{n,m}\right)$ for any $n\geq m\geq 1$ ($n\neq 4$ for the second equivalence), where the left-hand side in the case $n-m=2$ or $(n,m)=(4,3)$ should be replaced by the union of the path-components of $\mathrm{PL}$-trivial knots (framing being disregarded). Moreover, we show that for $n\neq 4$, the delooping is compatible with the Budney $E_{m+1}$-action. We use this delooping to combine the Hatcher $\mathrm{O}_{m+1}$-action and the Budney $E_{m+1}$-action into a framed little discs operad $E_{m+1}^{\mathrm{O}_{m+1}}$-action on $\mathrm{Emb}_\partial^{fr}(D^m,D^n)$.

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arXiv2021

On the homotopy type of the spaces of spherical knots in $R^n$

We study the spaces of embeddings $S^m\hookrightarrow R^n$ and those of long embeddings $R^m\hookrightarrow R^n$, i.e. embeddings of a fixed behavior outside a compact set. More precisely we look at the homotopy fiber of the inclusion of these spaces to the spaces of immersions. We find a natural fiber sequence relating these spaces. We also compare the $L_\infty$-algebras of diagrams that encode their rational homotopy type, when the codimension $n-m\geq 3$.

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arXiv2021

Projective and Reedy model category structures for (infinitesimal) bimodules over an operad

We construct and study projective and Reedy model category structures for bimodules and infinitesimal bimodules over topological operads. Both model structures produce the same homotopy categories. For the model categories in question, we build explicit cofibrant and fibrant replacements. We show that these categories are right proper and under some conditions left proper. We also study the extension/restriction adjunctions.

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arXiv2021

On the delooping of (framed) embedding spaces

It is known that the bimodule derived mapping spaces between two operads have a delooping in terms of the operadic mapping space. We show a relative version of that statement. The result has applications to the spaces of disc embeddings fixed near the boundary and framed disc embeddings.

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arXiv2020

On the rational homotopy type of embedding spaces of manifolds in $R^n$

We study the spaces of embeddings of manifolds in a Euclidean space. More precisely we look at the homotopy fiber of the inclusion of these spaces to the spaces of immersions. As a main result we express the rational homotopy type of connected components of those embedding spaces through combinatorially defined $L_\infty$-algebras of diagrams.

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arXiv2019

Delooping the functor calculus tower

We study a connection between mapping spaces of bimodules and of infinitesimal bimodules over an operad. As main application and motivation of our work, we produce an explicit delooping of the manifold calculus tower associated to the space of smooth maps $D^{m}\rightarrow D^{n}$ of discs, $n\geq m$, avoiding any given multisingularity and coinciding with the standard inclusion near $\partial D^{m}$. In particular, we give a new proof of the delooping of the space of disc embeddings in terms of little discs operads maps with the advantage that it can be applied to more general mapping spaces.

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arXiv2018

The homotopy theory of operad subcategories

We study the subcategory of topological operads $P$ such that $P(0) = *$ (the category of unitary operads in our terminology). We use that this category inherits a model structure, like the category of all operads in topological spaces, and that the embedding functor of this subcategory of unitary operads into the category of all operads admits a left Quillen adjoint. We prove that the derived functor of this left Quillen adjoint functor induces a left inverse of the derived functor of our category embedding at the homotopy category level. We deduce from this result that the derived mapping spaces associated to our model category of unitary operads are homotopy equivalent to the standard derived operad mapping spaces, which we form in the model category of all operads in topological spaces. We prove that analogous statements hold for the subcategory of $k$-truncated unitary operads within the model category of all $k$-truncated operads, for any fixed arity bound $k\geq 1$, where a $k$-truncated operad denotes an operad that is defined up to arity $k$.

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arXiv2018

Euler characteristics for spaces of string links and the modular envelope of $L_\infty$

We make calculations in graph homology which further understanding of the topology of spaces of string links, in particular calculating the Euler characteristics of finite-dimensional summands in their homology and homotopy. In doing so, we also determine the supercharacter of the symmetric group action on the positive arity components of the modular envelope of $L_\infty$.

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arXiv2017

Rational homology and homotopy of high dimensional string links

Arone and the second author showed that when the dimensions are in the stable range, the rational homology and homotopy of the high dimensional anologues of spaces of long knots can be calculated as the homology of a direct sum of finite graph-complexes that they described explicitly. They also showed that these homology and homotopy groups can be interpreted as the higher order Hochschild homology also called Hochschild-Pirashvili homology. In this paper, we generalize all these results to high dimensional analogues of spaces of string links. The methods of our paper are applicable in the range when the ambient dimension is at least twice the maximal dimension of a link component plus two, which in particular guarantees that the spaces under the study are connected. However, we conjecture that our homotopy graph-complex computes the rational homotopy groups of links spaces always when codimension is greater than two, i.e. always when the Goodwillie-Weiss calculus is applicable. Using Haefliger\rq{}s approach to calculate the groups of isotopy classes of higher dimensional links, we confirm our cojecture at the level of π_0.

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arXiv2017

Cycle index sum for non-k-equal configurations

We compute the cycle index sum of the symmetric group action on the homology of the configuration spaces of points in a Euclidean space with the condition that no $k$ of them are equal.

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arXiv2017

The rational homotopy of mapping spaces of E${}_n$ operads

We express the rational homotopy type of the mapping spaces $\mathrm{Map}^h(\mathsf D_m,\mathsf D_n^{\mathbb Q})$ of the little discs operads in terms of graph complexes. Using known facts about the graph homology this allows us to compute the rational homotopy groups in low degrees, and construct infinite series of non-trivial homotopy classes in higher degrees. Furthermore we show that for $n-m>2$, the spaces $\mathrm{Map}^h(\mathsf D_m,\mathsf D_n^{\mathbb Q})$ and $\mathrm{Map}^h(\mathsf D_m,\mathsf D_n)$ are simply connected and rationally equivalent. As application we determine the rational homotopy type of the deloopings of spaces of long embeddings. Some of the results hold also for mapping spaces $\mathrm{Map}_{\leq k}^h(\mathsf D_m,\mathsf D_n^{\mathbb Q})$, $\mathrm{Map}_{\leq k}^h(\mathsf D_m,\mathsf D_n)$, $n-m\geq 2$, of the truncated little discs operads, which allows one to determine rationally the delooping of the Goodwillie-Weiss tower for the spaces of long embeddings.

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arXiv2016

Hochschild-Pirashvili homology on suspensions and representations of $Out(F_n)$

We show that the Hochschild-Pirashvili homology on any suspension admits the so called Hodge splitting. For a map between suspensions $f\colon ΣY\to ΣZ$, the induced map in the Hochschild-Pirashvili homology preserves this splitting if $f$ is a suspension. If $f$ is not a suspension, we show that the splitting is preserved only as a filtration. As a special case, we obtain that the Hochschild-Pirashvili homology on wedges of circles produces new representations of $Out(F_n)$ that do not factor in general through $GL(n,Z)$. The obtained representations are naturally filtered in such a way that the action on the graded quotients does factor through $GL(n,Z)$.

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arXiv2016

Commutative hairy graphs and representations of $Out(F_r)$

We express the hairy graph complexes computing the rational homotopy groups of long embeddings (modulo immersion) of R^m in R^n as "decorated" graph complexes associated to certain representations of the outer automorphism groups of free groups. This interpretation gives rise to a natural spectral sequence, which allows us to shed some light on the structure of the hairy graph cohomology. We also explain briefly the connection to the deformation theory of the little discs operads and some conclusions that this brings.

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arXiv2013

Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots

We continue our investigation of spaces of long embeddings (long embeddings are high-dimensional analogues of long knots). In previous work we showed that when the dimensions are in the stable range, the rational homology groups of these spaces can be calculated as the homology of a direct sum of certain finite graph-complexes, which we described explicitly. In this paper, we establish a similar result for the rational homotopy groups of these spaces. We also put emphasis on different ways how the calculations can be done. In particular we describe three different graph-complexes computing the rational homotopy of spaces of long embeddings. We also compute the generating functions of the Euler characteristics of the summands in the homological splitting.

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