Recent zbMATH articles in MSC 55Phttps://www.zbmath.org/atom/cc/55P2021-04-16T16:22:00+00:00WerkzeugGenera of the torsion-free polyhedra.https://www.zbmath.org/1456.550052021-04-16T16:22:00+00:00"Kolesnyk, P. O."https://www.zbmath.org/authors/?q=ai:kolesnyk.p-oSummary: We study the genera of polyhedra (finite cell complexes), i.e., the classes of polyhedra such that all their localizations are stably homotopically equivalent. More precisely, we describe the genera of the torsion-free polyhedra of dimensions not greater than 11. In particular, we find the number of stable homotopy classes in these genera.The trace of the local \(\mathbb{A}^1\)-degree.https://www.zbmath.org/1456.140272021-04-16T16:22:00+00:00"Brazelton, Thomas"https://www.zbmath.org/authors/?q=ai:brazelton.thomas"Burklund, Robert"https://www.zbmath.org/authors/?q=ai:burklund.robert"McKean, Stephen"https://www.zbmath.org/authors/?q=ai:mckean.stephen"Montoro, Michael"https://www.zbmath.org/authors/?q=ai:montoro.michael"Opie, Morgan"https://www.zbmath.org/authors/?q=ai:opie.morganThe theory of \(\mathbb{A}^1\)-enumerative geometry is a relatively young topic, defined as an application of the \(\mathbb{A}^1\)-homotopy theory of \textit{F. Morel} and \textit{V. Voevodsky} [Publ. Math., Inst. Hautes Étud. Sci. 90, 45--143 (1999; Zbl 0983.14007)] to enumerative geometry over arbitrary base fields \(\kappa\); see [\textit{B. Williams} and \textit{K. Wickelgren}, in: Handbook of homotopy theory. Boca Raton, FL: CRC Press. 42 p. (2020; Zbl 07303335)].
In this short article, the authors study the local \(\mathbb{A}^1\)-degree, a foundational tool in \(\mathbb{A}^1\)-enumerative geometry. Analogous to the classical Brouwer degree of a continuous map \(\mathbb{R}^n\to \mathbb{R}^n\) with an isolated zero at the origin taking values in the integers, the local \(\mathbb{A}^1\)-degree is an invariant of a map \(f\colon\mathbb{A}^n_\kappa\to \mathbb{A}^n_\kappa\) at an isolated zero \(p\) taking values in the Grothendieck--Witt group \(\mathrm{GW}(\kappa)\) of the field \(\kappa\). The main theorem [Theorem 1.3] states that if the map \(\kappa\to\kappa(p)\) from the residue field of \(p\) is separable and of finite degree, then the local \(\mathbb{A}^1\)-degree of \(f\) can be computed as the image of the local \(\mathbb{A}^1\)-degree of the base change of \(f\) over \(\kappa(p)\) under the natural transfer map of Grothendieck-Witt groups. This result is somewhat inspired by Morel's original work in defining the local \(\mathbb{A}^1\)-degree [\textit{F. Morel}, in: Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22--30, 2006. Volume II: Invited lectures. Zürich: European Mathematical Society (EMS). 1035--1059 (2006; Zbl 1097.14014)] and more recent work by \textit{J. L. Kass} and \textit{K. Wickelgren} [Duke Math. J. 168, No. 3, 429--469 (2019; Zbl 1412.14014)]. The authors also obtain a corollary of Theorem 1.3, generalising a statement of Kass and Wickelgren [loc. cit.] relating the local \(\mathbb{A}^1\)-degree with the Scheja-Storch bilinear form.
The authors begin in Section 2 by reviewing the necessary background in \(\mathbb{A}^1\)-homotopy theory, including the construction of the local \(\mathbb{A}^1\)-degree, and some of the six functor formalism of \(\mathbb{A}^1\)-stable homotopy theory, as detailed by \textit{M. Hoyois} [Algebr. Geom. Topol. 14, No. 6, 3603--3658 (2014; Zbl 1351.14013)], for example. The six functor formalism allows for interpretations of the various maps defining the local \(\mathbb{A}^1\)-degree and the transfer of Grothendieck-Witt groups internally in \(\mathcal{SH}(\kappa)\), the stable motivic homotopy category of \(\kappa\).
In Section 3, the authors prove Theorem 1.3. To do this, a commutative diagram [Diagram (3)] is constructed in the unstable motivic homotopy category of \(\kappa\). Upon passage to the stable category \(\mathcal{SH}(\kappa)\), Diagram (3) yields Diagram (4) containing the local \(\mathbb{A}^1\)-degree as an endomorphism of the motivic sphere \(\mathbb{P}^n_\kappa/\mathbb{P}^{n-1}_\kappa\). Using the purity theorem of Morel-Voevodsky and an analysis of certain motivic Thom spaces, various arrows in Diagram (4) are recast and shown to be invertible; see Diagram (7). Theorem 1.3 now follows from Diagram (7) and the fact that \(\kappa\to \kappa(p)\) is finite and separable. As a final paragraph, the authors show the Scheja-Storch form agrees with the local \(\mathbb{A}^1\)-degree when \(\kappa\to\kappa(p)\) is separable and of finite degree, extending a previous result of Wickelgren and Kass [loc. cit.], where one assumes that \(p\) is \(\kappa\)-rational.
Reviewer: Jack Davies (Utrecht)Spectral algebra models of unstable \(v_n\)-periodic homotopy theory.https://www.zbmath.org/1456.550072021-04-16T16:22:00+00:00"Behrens, Mark"https://www.zbmath.org/authors/?q=ai:behrens.mark-joseph"Rezk, Charles"https://www.zbmath.org/authors/?q=ai:rezk.charlesLet \(X\) be a topological space and let \({\mathbb{S}}^X\) denote the Spanier-Whitehead
dual of \(X\). This is a commutative ring spectrum. A long standing question in topology is
to what extent one can recover \(X\) from \({\mathbb{S}}^X\). Classical rational homotopy
theory implies that one can essentially recover the rational homotopy type of \(X\) from
the rational homotopy type of \({\mathbb{S}}^X\). Moreover, there exists an algebra over the
Lie operad which is a kind of a Koszul dual to \({\mathbb{S}}^X\) whose rational homotopy groups
are isomorphic to the rational homotopy groups of \(X\).
A theorem of \textit{M. A. Mandell} [Topology 40, No. 1, 43--94 (2001; Zbl 0974.55004)] gives a partial \(p\)-adic analogue of these results. Mandell showed that the
\(p\)-adic homotopy type of \(X\) is essentially determined by the \(p\)-adic homotopy type of
\({\mathbb{S}}^X\). However, there is no known \(p\)-adic analogue of Lie algebra models
for homotopy types.
In this paper the authors develop a chromatic analogue of these theories. Namely, they show
that the topological André-Quillen cohomology spectrum of \({\mathbb{S}}^X\) serves
as a kind of Lie algebra model for the \(K(n)\)-localization of \(X\). More precisely, they show that
there is a natural map
\[
c^{K(n)}_X\colon \Phi_{K(n)}(X)\to\mathrm{TAQ}_{{\mathbb{S}}_{K(n)}}({\mathbb{S}}_{K(n)}^X).
\]
Here \(K(n)\) is Morava \(K\)-theory, \({\mathbb{S}}_{K(n)}\) denotes the \(K(n)\)-localization
of the sphere spectrum and \(\Phi\) is the Bousfield-Kuhn functor. The authors prove that
\(c^{K(n)}_X\) induces an isomorphism on homotopy groups when \(X\) is a sphere or a
product of spheres or most generally, when the Goodwillie tower of the identity converges in
\(v_n\)-periodic homotopy for \(X\).
For the entire collection see [Zbl 07174496].
Reviewer: Marja Kankaanrinta (Helsinki)Results on the homotopy type of the spaces of locally convex curves on $\mathbb{S}^3$.https://www.zbmath.org/1456.570242021-04-16T16:22:00+00:00"Alves, Emília"https://www.zbmath.org/authors/?q=ai:alves.emilia"Saldanha, Nicolau C."https://www.zbmath.org/authors/?q=ai:saldanha.nicolau-corcaoA smooth curve \(\gamma:[0,1]\to \mathbb{S}^3\) in \(4\)-dimensional Euclidean space \(\mathbb{R}^4\)
with image on the sphere \(\mathbb{S}^3\), is said to be locally convex, if the set of vectors
\({\gamma}(t),{\gamma}'(t), {\gamma}''(t),{\gamma}'''(t)\) is a positive basis in \(\mathbb{R}^4\)
for all \(t\in[0,1]\). By the Gram-Schmidt procedure, we can turn this basis into an orthonormal basis and thereby associate a Frenet frame curve \(\mathcal{F}_{\gamma}: [0,1]\to SO_4\) in the special orthogonal
group \(SO_4\) to a locally convex curve.
For any matrix \(Q\in SO_4\), let \(\mathcal{L}\mathbb{S}^3(Q)\) denote the space of all locally
convex curves \(\gamma:[0,1]\to \mathbb{S}^3\) where \(\mathcal{F}_{\gamma}(0)=I\) (the identity matrix)
and \(\mathcal{F}_{\gamma}(1)=Q\). It was proved by [\textit{N. C. Saldanha} and \textit{B. Shapiro}, J. Singul. 4, 1--22 (2012; Zbl 1292.58002)] that there are at most 3 different homeomorphism types among the path components of \(\mathcal{L}\mathbb{S}^3(Q)\). But otherwise very little seems to be known about these components and their generalizations to higher dimensional spheres \(\mathbb{S}^n\).
In the present paper the authors prove several interesting theorems on the homotopy and homology of these path components, in particular for the case \(Q=-I\). The results are technical and cannot be given in detail.
Reviewer: Vagn Lundsgaard Hansen (Lyngby)Truncation of unitary operads.https://www.zbmath.org/1456.180152021-04-16T16:22:00+00:00"Bao, Yan-Hong"https://www.zbmath.org/authors/?q=ai:bao.yanhong"Ye, Yu"https://www.zbmath.org/authors/?q=ai:ye.yu"Zhang, James J."https://www.zbmath.org/authors/?q=ai:zhang.james-yiming|zhang.james-jUnitary operads over a fixed field \(\Bbbk\) are those one-dimensional in arities zero and one; let \(Op_+\) denote the category of such.
A unitary operad \(P\) has a natural family of restriction operators \(P(n) \to P(s)\) for \(s \leq n\), obtained by composing with the \(0\)-ary operation in \(n-s\) places.
These restrictions are used to define a sequence of ideals \(\vphantom{\Upsilon}^k\Upsilon\) of \(P\) by intersecting some of their kernels.
An analogue of the Gelfand-Kirillov dimension of \(\Bbbk\)-algebras is given.
Namely, the GK dimension of a (locally finite-dimensional) operad \(P\) is defined by the formula
\[
\mathrm{GKdim}(P) = \limsup_{n\to \infty} \left( \log_n \left( \sum_{i=0}^n \dim_{\Bbbk} P(i)\right) \right).
\]
The authors also introduce and study other invariants, including the exponent, the signature, and the Hilbert series.
Many of the results are concerned with `2-unitary operads', which are related to `operads with multiplication' in the sense of Gerstenhaber-Voronov
[\textit{M. Gerstenhaber} and \textit{A. Voronov}, Int. Math. Res. Not. 1995, No. 3, 141--153 (1995; Zbl 0827.18004)].
More precisely, a \textbf{2-unitary} operad is an object of the comma category \(Mag\downarrow Op_+\) of operads under the magma operad, that is a unitary operad with a specified non-associative multiplication \(\mu\) (there are also associative and commutative variants of this notion).
If \(P\) is 2-unitary and has finite GK dimension, then the GK dimension must be an integer.
Further, this GK dimension can be characterized by the vanishing of the ideals \(\vphantom{\Upsilon}^k\Upsilon\).
The authors classify the 2-unitary operads of small GK dimension.
There is only a single 2-unitary operad of GK dimension 1, namely the commutative operad \(Com\).
Every augmented \(\Bbbk\)-algebra \(\Lambda\) generates a 2-unitary operad which is \(\Lambda\) in arity one and which has dimension greater than one in all higher arities so long as \(\Lambda\) is different from \(\Bbbk\).
This construction is part of an equivalence of categories between 2-unitary operads with GK dimension at most 2 and finite-dimensional, augmented \(\Bbbk\)-algebras.
The quotient operads of the associative operad \(Ass\) with fixed GK dimension at most 4 are classified when \(\Bbbk\) has characteristic zero.
There are only three such quotient operads, namely \(Ass / \vphantom{\Upsilon}^k\Upsilon\) for \(k=1,3,4\) of GK dimension \(k\).
In particular, there is no quotient operad of \(Ass\) with GK dimension 2.
It is shown that the situation is more complicated for \(k=5\).
There are several other interesting directions in the paper as well, including characterizations of artinian, semiprime operads (in the reduced, unitary, or 2-unitary cases), the fact that every signature can be realized by an object of \(Com\downarrow Op_+\), and the relationship between the GK dimension of operads and of their algebras.
Reviewer: Philip Hackney (Lafayette)Rational homotopy of mapping spaces between complex Grassmannians.https://www.zbmath.org/1456.550062021-04-16T16:22:00+00:00"Gatsinzi, Jean Baptiste"https://www.zbmath.org/authors/?q=ai:gatsinzi.jean-baptiste"Otieno, Paul Antony"https://www.zbmath.org/authors/?q=ai:otieno.paul-antony"Onyango-Otieno, Vitalis"https://www.zbmath.org/authors/?q=ai:onyango-otieno.vitalisThe authors use Sullivan models to describe the rational homotopy type of the component of the inclusion in the mapping space \(\mathrm{map}(Gr(2,n),Gr(2,n+r);i_n)\). They show that the homotopy type of \(\mathrm{map}(Gr(2,n),Gr(2,n+r);i_n)\) is that of a product of odd spheres. They go on to show that the cohomology algebra of \(\mathrm{map}(Gr(2,n),Gr(2,n+r);i_{n},r)\) for \(r>3n-6\) contains a polynomial algebra over a generator of degree \(2\).
Reviewer: Rugare Kwashira (Johannesburg)