Recent zbMATH articles in MSC 55Phttps://zbmath.org/atom/cc/55P2024-02-28T19:32:02.718555ZWerkzeugAlgebraic cobordism and étale cohomologyhttps://zbmath.org/1527.140422024-02-28T19:32:02.718555Z"Elmanto, Elden"https://zbmath.org/authors/?q=ai:elmanto.elden"Levine, Marc"https://zbmath.org/authors/?q=ai:levine.marc-n"Spitzweck, Markus"https://zbmath.org/authors/?q=ai:spitzweck.markus"Østvær, Paul Arne"https://zbmath.org/authors/?q=ai:ostvaer.paul-arneThe motivating question in this work, Question 1.1, is understanding the difference between \(SH(S)\) and \(SH_{\text{ét}}(S)\), the stable motivic homotopy categories constructed in the Nisnevich and étale topologies. The motivating example is Thomason's result [\textit{R. W. Thomason}, Ann. Sci. Éc. Norm. Supér. (4) 18, 437--552 (1985; Zbl 0596.14012)] that, under certain hypotheses, Bott periodic mod \(\ell^v\) algebraic \(K\)-theory satisfies étale hyperdescent. The main theorem, Theorem 1.6, is that under suitable hypotheses, for any \(MGL\)-module \(E\) (where \(MGL\) denotes algebraic cobordism), there exists a Bott element in \(MGL/\ell^v\) such that Bott periodic mod \(\ell^v\) \(E\)-homology satisfies étale hyperdescent, and moreover, identifies with the étale localization of \(E\). This theorem is upgraded to an integral statement, at least after inverting certain primes, in Theorem 1.10.
The proof uses the slice spectral sequence and its étale analogue to reduce from modules over \(MGL\) to mod \(\ell^v\) motivic cohomology, which can then be approached directly or by appealing to the Bloch-Kato Conjecture. After reviewing some preliminary material in Section 2, the authors develop the ``slice comparison paradigm'' relating the ordinary and étale slice spectral sequences in Section 3 and prove convergence of the étale slice spectral sequence in the cases of interest in Section 4. Section 5 contains some requisite results on the forgetful functor from \(SH_{\text{ét}}(S)\) to \(SH(S)\), which along with the results of the previous section, are used to prove all of the main theorems in Section 6. Section 7 gathers further applications and consequences of the main theorems, including interesting results concerning étale-local module categories, compatibility of base change with étale localization, the construction of an étale hyperdescent spectral sequence, and cellularity of mod \(\ell^v\) étale localization.
The results and techniques in this article are very interesting and give rise to many natural questions. In fact, after the publication of this paper, a strengthening of the main theorem was proven by \textit{T. Bachmann} et al. [``Stable motivic invariants are eventually étale local'', J. Eur. Math. Soc. (to appear)], which proves the main result for the motivic sphere spectrum. Further, the slice comparison paradigm is set up to allow comparison between other topologies (not just the Nisnevich and étale topologies), and should have further interesting applications. Finally, the appendices contain many useful discussions, results, and recollections which could serve as a helpful reference for other researchers in motivic homotopy theory.
Reviewer: James D. Quigley (Ithaca)Fibrant resolutions for motivic Thom spectrahttps://zbmath.org/1527.140482024-02-28T19:32:02.718555Z"Garkusha, Grigory"https://zbmath.org/authors/?q=ai:garkusha.grigory"Neshitov, Alexander"https://zbmath.org/authors/?q=ai:neshitov.alexanderThe article provides models with specific properties of important \(T\)-spectra in the Morel-Voevodsky motivic stable homotopy category over a field \(k\); here \(T:=\mathbb{A}^1/\mathbb{A}^1-\{0\}\simeq \mathbb{P}^1\) is the Thom space of the trivial line bundle over the field \(k\). More precisely, the authors use Voevodsky's theory of framed correspondences, as elaborated by Garkusha-Panin and others, to construct almost fibrant models for suitable Thom spectra, that is, (symmetric) \(T\)-spectra built out of infinite sequences of pairs of smooth \(k\)-varieties with contractible alternating group actions. Prominent examples of the latter are Voevodsky's algebraic cobordism spectrum \(\mathrm{MGL}\) and its variants, as well as the sphere spectrum over \(k\). The fibrancy property implies that it is in principle simpler to compute homotopy classes of maps into these almost fibrant models. Hence these specific models should allow one for example to produce computations of homotopy groups (or sheaves of such) for algebraic cobordism and the sphere spectrum. Marc Levine's generalization of the Suslin-Voevodsky rigidity theorem from [\textit{M. Levine}, J. Topol. 7, No. 2, 327--362 (2014; Zbl 1333.14021)] is then applied to compute homotopy groups with finite coefficients for Thom spectra over an algebraically closed subfield of the complex numbers via topological realization.
Reviewer: Oliver Röndigs (Osnabrück)The universal operad acting on loop spaces, and generalisationshttps://zbmath.org/1527.180192024-02-28T19:32:02.718555Z"Cheng, Eugenia"https://zbmath.org/authors/?q=ai:cheng.eugenia"Trimble, Todd"https://zbmath.org/authors/?q=ai:trimble.todd-hThe principal objective in this paper is to show that the operads used by Trimble [What are ``fundamental-groupoids'' ?, Seminar at DPMMS, Cambridge, 24 August 1999] and \textit{M. A. Batanin} [Adv. Math. 136, No. 1, 39--103 (1998; Zbl 0912.18006)] in their definitions of \(n\)-category have a nice universal property. The authors take the view that the main purpose of identifying this universal property is to help find smaller non-universal operads for practical use. One such ``practical use'' is the modelling of homotopy types.
There are at least two ways of using operads in higher-dimensional algebra. Trimble proceeds inductively, using a classical operad at each dimension. Batanin on the other hand parametrizes all dimensions at once, using a more general form of operad called \textit{globular operad} in which the arities of operations are no longer just natural numbers but globular pasing diagrams. In both cases, \(\omega\)-categories are defined, with the \(\omega\)-groupoids being identified among them afterwards.
The synopsis of the paper goes as follows.
\begin{itemize}
\item[\S 1] establishes the main universality theorem in sufficient generality to cover key examples.
\item[\S 2] proves some technical results needed to address size issues when constructing internal homs.
\item[\S 3] addresses operads acting on loop spaces, which serves to emphasize the unwieldly nature of the universal operad in question as well as the importance of finding non-universal ones for calculations, as is done in the theory of loop spaces.
\item[\S 4] addresses the motivating example, namely, globular operads for discussing the fundamental \(\omega\)-operads of spaces, exploiting [\textit{E. Cheng}, Homology Homotopy Appl. 13, No. 2, 217--248 (2011; Zbl 1255.18006)] in which it is established that every Trimble \(n\)-category is a Batanin \(n\)-category. Part of the proof produces a functor
\[
\left\{ \begin{array} [c]{c} \text{classical operads}\\
\text{acting on path spaces} \end{array} \right\} \rightarrow\left\{ \begin{array} [c]{c} \text{globular operads}\\
\text{acting on }\omega\text{-path spces} \end{array} \right\}
\]
Even the universal operad on the left yields a non-universal operad on the right, and applying the functor to non-universal operads on the left gives further non-universal examples on the right. \end{itemize}
Reviewer: Hirokazu Nishimura (Tsukuba)Labelled cospan categories and properadshttps://zbmath.org/1527.180202024-02-28T19:32:02.718555Z"Beardsley, Jonathan"https://zbmath.org/authors/?q=ai:beardsley.jonathan"Hackney, Philip"https://zbmath.org/authors/?q=ai:hackney.philipProperads are generalizations of operads that allow operations with multiple inputs and multiple outputs, with composition along connected graphs. A \textit{labelled cospan category} in the sense of Steinebrunner is a symmetric monoidal functor \(\pi \colon C \to \mathbf{Csp}\) satisfying some axioms. Here \(\mathbf{Csp}\) is the category with finite sets as objects, and a morphism from \(A\) to \(B\) is an equivalence class of cospans \(A \to C \leftarrow B\). The authors prove the following conjecture of Steinebrunner.
{Theorem A}. The 2-category of properads is biequivalent to the 2-category of labelled cospan categories.
The proof of Theorem A uses the following variant.
{Theorem B}. There is a strict 2-equivalence between the 2-category of properads and the 2-category of strict labelled cospan categories.
Reviewer: Donald Yau (Newark)The Hurewicz theorem for cubical homologyhttps://zbmath.org/1527.550052024-02-28T19:32:02.718555Z"Carranza, Daniel"https://zbmath.org/authors/?q=ai:carranza.daniel"Kapulkin, Krzysztof"https://zbmath.org/authors/?q=ai:kapulkin.krzysztof"Tonks, Andrew"https://zbmath.org/authors/?q=ai:tonks.andrewSummary: We give an elementary proof of the Hurewicz theorem relating homotopy and homology groups of a cubical Kan complex. Our approach is based on the notion of a loop space of a cubical set, developed in a companion paper [\textit{D. Carranza} and \textit{C. Kapulkin}, ``Homotopy groups of cubical sets'', Preprint, \url{arXiv:2202.03511}] by the first two authors.Obstruction theory and the level \(n\) elliptic genushttps://zbmath.org/1527.550072024-02-28T19:32:02.718555Z"Senger, Andrew"https://zbmath.org/authors/?q=ai:senger.andrewComplex oriented cohomology theories are the pillars of chromatic homotopy theory. A complex orientation can be identified with a map of homotopy commutative rings from the complex cobordism spectrum MU and it is natural to ask whether this map lifts to a map of commutative ring spectra. This question has been studied before in other contexts exhibiting geometric underpinnings.
The author of the paper under review provides new existence and uniqueness results for commutative ring spectrum orientations. As an application, the author proves that the orientation known as the Hirzebruch level \(n\) elliptic genus lifts uniquely up to homotopy to a map of commutative ring spectra. The approach uses the obstruction theory developed by \textit{M. J. Hopkins} and \textit{T. Lawson} [Math. Z. 290, No. 1--2, 83--101 (2018; Zbl 1417.55012)] as well as the Ando criterion, which has to do with the compatiblity of the complex orientation with power operations. The paper is elegant and a pleasure to read.
Reviewer: Gabriel Angelini-Knoll (Paris)Eilenberg Mac Lane spectra as \(p\)-cyclonic Thom Spectrahttps://zbmath.org/1527.550102024-02-28T19:32:02.718555Z"Levy, Ishan"https://zbmath.org/authors/?q=ai:levy.ishanThe famous result of \textit{M. Mahowald} [Topology 16, 249--256 (1977; Zbl 0357.55020)] shows that if we start with the nontrivial virtual bundle on a circle \(\mathbb{S}^1\to BO\) and extend it to a double loop map \(\mu_2\colon \Omega^2 \mathbb{S}^3\to BO\), then the Thom spectrum \((\Omega^2 \mathbb{S}^3)^{\mu_2}\) is equivalent to the Eilenberg-MacLane spectrum \(H \mathbb{F}_2\) as an \(\mathbb{E}_2\)-ring. This result was extended to the equivariant setting by \textit{M. Behrens} and \textit{D. Wilson} [Proc. Am. Math. Soc. 146, No. 11, 5003--5012 (2018; Zbl 1409.55010)], who showed that if \(\rho\) is the regular representation of the group \(C_2\), then there is an equivalence of \(\mathbb{E}_\rho\)-algebras \(H\underline{\mathbb{F}_2}\) and \((\Omega^\rho\mathbb{S}^{\rho+1})^{\tilde{\mu}}\), where the latter is the Thom spectrum associated to an equivariant analogue of the map \(\mu_2\). Generalizations to odd primes were done by Hopkins in the non-equivariant case and \textit{J. Hahn} and \textit{D. Wilson} [Geom. Topol. 24, No. 6, 2709--2748 (2020; Zbl 1459.55008)], who extended the previous result of Behrens-Wilson to cyclic \(p\)-groups and their faithful representations.
In the paper under review the author extends the result of Hahn-Wilson by considering faithful representations \(\lambda\colon G\to O(2)\) for a \(p\)-group \(G\) and shows that there is an equivalence of \(\mathbb{E}_\lambda\)-algebras between the Thom spectrum \(X_{\lambda,p}\), arising in a similar way as in results of Mahowald, Behrens-Wilson and Hahn-Wilson, and the Eilenberg-MacLane spectrum \(H\underline{\mathbb{F}_p}\). Moreover, it is shown that this is the best possible result -- namely, if there is a representation \(V\colon G\to O(n)\) and an equivalence \(X_{V,p}\simeq \mathbb{F}_p\), then \(G\) has to be a \(p\)-group, \(V\) is faithful and \(n=2\).
These results are achieved by employing the notion of \emph{cyclonic spectra}, introduced in [\textit{C. Barwick} and \textit{S. Glasman}, ``Cyclonic spectra, cyclotomic spectra, and a conjecture of Kaledin'', Preprint, \url{arXiv:1602.02163}] and by using techniques of \emph{equivariant factorization homology} [\textit{A. Horev}, ``Genuine equivariant factorization homology'', Preprint, \url{arXiv:1910.07226}]. In particular, the author introduces a module structure on \(\mathbb{E}_V\)-algebras over a certain object constructed via equivariant factorization homology and further uses it in the proofs of main theorems.
Reviewer: Igor Sikora (Ankara)Étale cohomology, purity and formality with torsion coefficientshttps://zbmath.org/1527.550112024-02-28T19:32:02.718555Z"Cirici, Joana"https://zbmath.org/authors/?q=ai:cirici.joana"Horel, Geoffroy"https://zbmath.org/authors/?q=ai:horel.geoffroyThe authors investigate the connection between formality, a notion that originates in rational homotopy theory, and (fractional) purity of the mixed Hodge structure on étale cohomology. More precisely, they study algebras over operads in the category of schemes and the consequences of the purity of the étale cohomology on formality of such algebras. The study of this kind of connection is in the wake of the striking result of Deligne-Griffiths-Morgan-Sullivan on the formality of Kähler manifolds [\textit{P. Deligne} et al., Invent. Math. 29, 245--274 (1975; Zbl 0312.55011)] and follows the authors' earlier investigation [\textit{J. Cirici} and \textit{G. Horel}, Ann. Sci. Éc. Norm. Supér. (4) 53, No. 4, 1071--1104 (2020; Zbl 1457.32080)].
One of the main technical notions used by the authors is the following. Given a scheme \(X\) over a \(p\)-adic field \(K\) with residue field \(\mathbb{F}_q\), and some rational number \(\alpha\) such that \(0 < \alpha < h\) (where \(h\) is the order of \(q\) in \(\mathbb{F}_\ell\) with \(\ell \neq p\)), the authors say that \(H^n_{et}(X_{\bar{K}}; \mathbb{F}_\ell)\) is a \textit{pure Tate module} of weight \(\alpha n\) if the only eigenvalue of the Frobenius is \(q^{\alpha n}\) (or the cohomology group vanishes if \(\alpha n\) is not an integer).
The first main (homological) result of the article is that if \(X\) is a \(P\)-algebra, where \(P\) is some operad satisfying a technical condition, if:
\[
\text{for every color } c \text{ of } P, \; H^n_{et}(X(c)_{\bar{K}}; \mathbb{F}_\ell) \text{ is pure of weight } \alpha n,
\]
then \(C_*(X_{an}; \mathbb{F}_\ell)\) is \(N\)-formal, where \(N = \lfloor (h-1)/\alpha \rfloor\) and \(X_{an}\) is the underlying complex analytic space of \(X \times_K \mathbb{C}\).
The authors' second main (cohomological) result is that, given a scheme \(X\) satisfying the purity condition as above, then \(C^*_{\mathrm{sing}}(X_{an}; \mathbb{F}_\ell)\) is \(N\)-formal. More, \(k\)-fold Massey products in \(H^*(X_{an}; \mathbb{F}_\ell)\) vanish if \(\alpha(k-2)/h\) is not an integer, and if \(H^*(X_{an}; \mathbb{F}_\ell)\) vanishes up to degree \(r\), then there are no nontrivial Massey products of to degree \(\lceil hr/\alpha + 2r+1 \rceil\).
This has several important consequences. The authors apply their theorems to the cyclic operad \(\bar{\mathcal{M}}_{0, \bullet}\) of compactified moduli spaces of curves of genus \(0\), and show that it is \(2(\ell-2)\)-formal over \(\mathbb{F}_{\ell}\). Using the action of the Grothendieck-Teichmüller group, they can also apply their methods to the little disks operad and show that it is \((\ell-2)\)-formal over \(\mathbb{F}_\ell\), which implies formality of the \((\ell-1)\)-truncated operad (and this result is sharp). It then follows from their cohomological result that the configuration space of \(m\) points in \(\mathbb{C}^d\) is fully formal over \(\mathbb{F}_\ell\) if \(\ell \geq (m-1)d+2\).
Reviewer: Najib Idrissi (Paris)Iterated integrals with values in Azumaya algebrashttps://zbmath.org/1527.550122024-02-28T19:32:02.718555Z"Glass, Cheyne"https://zbmath.org/authors/?q=ai:glass.cheyne-j"Tradler, Thomas"https://zbmath.org/authors/?q=ai:tradler.thomas"Zeinalian, Mahmoud"https://zbmath.org/authors/?q=ai:zeinalian.mahmoudIn general, Chen's iterated integrals with values in a bundle of algebras fail to have a well defined product. By making use of parallel transport, the authors generalize Chen's iterated integrals to the setting of forms with values in an Azumaya algebra and prove that if \(M\) is simply connected, then there exist generalizations of the two sided bar construction, and the Hochschild complex along with iterated integrals to forms on the path and loop space respectively, which form a commutative diagram induced by the inclusion \(LM\longrightarrow PM\). In the last section they note that in the case of a finite dimensional vector bundle, endomorphism-bundle valued forms can be related to real-valued forms by the usual trace map.
Reviewer: Jean-Claude Thomas (Angers)Good covers for vortex nerve cell complexes. Free group presentation of intersecting nested cycles in planar CW spaceshttps://zbmath.org/1527.550132024-02-28T19:32:02.718555Z"Peters, J. F."https://zbmath.org/authors/?q=ai:peters.john-f|peters.james-f-iiiSummary: This paper introduces good covers for cell complexes in the form of what are known as path nerve complexes in a planar Whitehead CW space together with their Rotman free group presentations. A \textbf{path} is a mapping \(h:[0,1]\to K\) over a space \(K\). Paths provide the backbone of homotopy theory introduced by J.H.C. Whitehead during the 1930s. A \textbf{path nerve complex} is a collection of sets of path-connected points (path images) that have nonempty intersection.
A space \(X\) has a good cover, provided \(X=\bigcup E\) for subsets \(E\) in \(X\) and the space is contractible. This form of good cover was introduced by K. Tanaka in 2021. The focus here is on intersecting path cycles (sequences of paths attached to each other in vortexes with nonempty interiors) that form a path nerve. A path nerve results from the nonvoid intersection of a collection of path cycles. The geometric realization of a path cycle is a 1-cycle. A 1-cycle is a finite sequence of path-connected 0-cells (vertexes) with no end vertex and with a nonvoid interior. A 1-cycle has the structure of a path cycle in which sequences of paths are replaced by edges.
A group \(G(V,+)\) containing a basis \(\mathcal{B}\) is \textit{free}, provided every member of \(V\) can be written as a linear combination of elements (generators) of the basis \(\mathcal{B}\subset V\). Let \(\bigtriangleup\) be the members \(v\) of \(V\), each written as a linear combination of the basis elements of \(\mathcal{B}\). A presentation of \(G(V,+)\) is a mapping \(\mathcal{B}\times\bigtriangleup\to G(\bigg\{v\in V:=\sum_{\substack{k\in\mathbb{Z} g\in\mathcal{B}}}{kg}\bigg\},+)\). This form of presentation of structures was introduced by J. J. Rotman during the 1960s as part of a study of groups. The main results in this paper are (1) Every path triangle cluster has a free group presentation, (2) Every path triangle cluster has a free group presentation, (3) Every path vortex has a free group presentation, (4) Every path vortex nerve has a free group presentation, (5) A vortex nerve and the union of the sets in the nerve have the same homotopy type and (6) Every path triangulaton of a cell complex has a good cover.
For the entire collection see [Zbl 1522.34005].Cohomology algebra of mapping spaces between quaternion Grassmannianshttps://zbmath.org/1527.550142024-02-28T19:32:02.718555Z"Maphane, Oteng"https://zbmath.org/authors/?q=ai:maphane.otengSummary: Let \(G_{k,n}(\mathbb{H} )\) for \(2 \leq\) k<n denote the quaternion Grassmann manifold of k-dimensional vector subspaces of \(\mathbb{H}^n\). In this paper we compute, in terms of the Sullivan models, the rational cohomology algebra of the component of the inclusion i: \(G_{k,n}(\mathbb{H}) \rightarrow G_{k,n+r}(\mathbb{H} )\) in the space of mappings from \(G_{k,n}(\mathbb{H} )\) to \(G_{k,n+r}(\mathbb{H} )\) for r \(\geq 1\) and, more generally, we show that the cohomology of Map \((G_{k,n}(\mathbb{H}),G_{k,n+r}(\mathbb{H} )\);i) contains a truncated algebra \(\mathbb{Q} \)[x]\(x_4^{r+n+k^2-nk}\) for n \(\geq 4\).Metastable complex vector bundles over complex projective spaceshttps://zbmath.org/1527.550152024-02-28T19:32:02.718555Z"Hu, Yang"https://zbmath.org/authors/?q=ai:hu.yangIn [Trans. Am. Math. Soc. 347, No. 10, 3743--3796 (1995; Zbl 0866.55020)], \textit{M. Weiss} developed orthogonal calculus, which is a functor calculus for studying functors from the category of finite-dimensional real inner product spaces to the category of pointed topological spaces. Unitary calculus is a unitary analogue of orthogonal calculus. In this paper, the author applies Weiss's unitary calculus to enumerate certain unstable topological complex vector bundles over complex projective spaces.
Let \(\mathrm{Vect}_r^0({\mathbb{C}}P^l)\) denote the pointed set of isomorphism classes of rank \(r\) complex vector bundles over the complex projective space \({\mathbb{C}}P^l\) whose Chern classes all vanish. The author uses unitary calculus to prove that if \(l>2\) and \(\frac{1}{2}\leq r\leq l-1\), then there is an isomorphism
\[
\mathrm{Vect}_r^0({\mathbb{C}}P^l)\cong \{ {\mathbb{C}}P^l, \Sigma {\mathbb{C}}P^\infty_r\},
\]
where \({\mathbb{C}}P^\infty_r\) is the stunted projective space \({\mathbb{C}}P^\infty/{\mathbb{C}}P^{r-1}\), and \(\{ {\mathbb{C}}P^l, \Sigma {\mathbb{C}}P^\infty_r\}\) denotes stable homotopy classes of maps. He then uses this isomorphism to prove the following results: Let \(l>2\). Then the cardinality of \(\mathrm{Vect}_{l-1}^0({\mathbb{C}}P^l)\) equals \(2\) if \(l\) is odd, and \(1\) if \(l\) is even. Assume then \(l>3\). He calculates the cardinality \(\phi(l)\) of \(\mathrm{Vect}_{l-2}^0({\mathbb{C}}P^l)\), and shows that the numbers \(\phi(l)\) exhibit a \(24\)-fold periodic behavior.
Reviewer: Marja Kankaanrinta (Helsinki)Monoids related to self homotopy equivalences of fibred producthttps://zbmath.org/1527.550182024-02-28T19:32:02.718555Z"Dutta, Gopal Chandra"https://zbmath.org/authors/?q=ai:dutta.gopal-chandra"Sen, Debasis"https://zbmath.org/authors/?q=ai:sen.debasisThe group of homotopy classes of self-homotopy equivalences \(\mbox{Aut}(X)\) of a connected based space \(X\) has been studied extensively for a long time (e.g., [\textit{M. Arkowitz}, Lect. Notes Math. 1425, 170--203 (1990; Zbl 0713.55004) and \textit{J. W. Rutter}, Spaces of homotopy self-equivalences. A survey. Berlin: Springer (1997; Zbl 0889.55004)]). Various subgroups of \(\mbox{Aut}(X)\) have also been studied: e.g., the subgroup that consists of the self-homotopy equivalences that induce the identity homomorphism on homotopy groups up to a certain range (cf. [\textit{M. Arkowitz} and \textit{K.-i. Maruyama}, Topology Appl. 87, No. 2, 133--154 (1998; Zbl 0929.55008) and \textit{K. Tsukiyama}, Hiroshima Math. J. 5, 215--222 (1975; Zbl 0309.55015)]). Motivated by this, \textit{H. W. Choi} and \textit{K. Y. Lee} [Topology Appl. 181, 104--111 (2015; Zbl 1307.55002)] introduced a sequence of monoids related to \(\mbox{Aut}(X)\) and studied two invariants: self-closeness number and self-length. \par The authors introduce a nested sequence of monoids
\[
\mathcal{U}^k_\sharp(X)= \{f \in [X, X]^B_B;\,\pi_i(f) : \pi_i(X)\stackrel{\approx}{\to}\pi_i(X),\; 0\le i \le k\}
\]
related to self-homotopy equivalences of fibrewise pointed spaces, such that the limit is the group of homotopy classes of fibrewise pointed self-equivalences. This monoid is explored for fibred products in terms of individual spaces. Furthermore, two related invariants associated to these monoids: self-closeness number and self-length are studied.
Reviewer: Marek Golasiński (Olsztyn)Invertible phases of matter with spatial symmetryhttps://zbmath.org/1527.811012024-02-28T19:32:02.718555Z"Freed, Daniel S."https://zbmath.org/authors/?q=ai:freed.daniel-s"Hopkins, Michael J."https://zbmath.org/authors/?q=ai:hopkins.michael-jSummary: We propose a general formula for the group of invertible topological phases on a space \(Y\), possibly equipped with the action of a group \(G\). Our formula applies to arbitrary symmetry types. When \(Y\) is Euclidean space and \(G\) a crystallographic group, the term `topological crystalline phases' is sometimes used for these phases of matter.Corrigendum to: ``Numerical solutions for the \(f(R)\)-Klein-Gordon system''https://zbmath.org/1527.830462024-02-28T19:32:02.718555Z"Beckering Vinckers, Ulrich K."https://zbmath.org/authors/?q=ai:vinckers.ulrich-k-beckering"de la Cruz-Dombriz, Álvaro"https://zbmath.org/authors/?q=ai:de-la-cruz-dombriz.alvaro"Pollney, Denis"https://zbmath.org/authors/?q=ai:pollney.denisFrom the text: Here, the authors correct five typographical errors and one computational error.
Corrigendum to the authors' paper [ibid. 40, No. 17, Article ID 175009, 30 p. (2023; Zbl 1519.83066)].