Recent zbMATH articles in MSC 55Nhttps://zbmath.org/atom/cc/55N2024-07-05T15:31:27.292712ZWerkzeugSimultaneously vanishing higher derived limitshttps://zbmath.org/1535.032432024-07-05T15:31:27.292712Z"Bergfalk, Jeffrey"https://zbmath.org/authors/?q=ai:bergfalk.jeffrey"Lambie-Hanson, Chris"https://zbmath.org/authors/?q=ai:lambie-hanson.chrisSummary: In [Trans. Am. Math. Soc. 307, No. 2, 725--744 (1988; Zbl 0648.55007)], \textit{S. Mardešić} and \textit{A. V. Prasolov} isolated an inverse system \(\boldsymbol{A}\) with the property that the additivity of strong homology on any class of spaces which includes the closed subsets of Euclidean space would entail that \(\lim^n\boldsymbol{A}\) (the \(n\)th derived limit of \(\boldsymbol{A}\)) vanishes for every \(n>0\). Since that time, the question of whether it is consistent with the \(\mathsf{ZFC}\) axioms that \(\lim^n \boldsymbol{A}=0\) for every \(n>0\) has remained open. It remains possible as well that this condition in fact implies that strong homology is additive on the category of metric spaces.
We show that assuming the existence of a weakly compact cardinal, it is indeed consistent with the \(\mathsf{ZFC}\) axioms that \(\lim^n \boldsymbol{A}=0\) for all \(n>0\). We show this via a finite-support iteration of Hechler forcings which is of weakly compact length. More precisely, we show that in any forcing extension by this iteration, a condition equivalent to \(\lim^n\boldsymbol{A}=0\) will hold for each \(n>0\). This condition is of interest in its own right; namely, it is the triviality of every coherent \(n\)-dimensional family of certain specified sorts of partial functions \(\mathbb{N}^2\to \mathbb{Z}\) which are indexed in turn by \(n\)-tuples of functions \(f:\mathbb{N}\to \mathbb{N}\). The triviality and coherence in question here generalise the classical and well-studied case of \(n=1\).Decomposition of pointwise finite-dimensional \(\mathbb{S}^1\) persistence moduleshttps://zbmath.org/1535.160192024-07-05T15:31:27.292712Z"Hanson, Eric J."https://zbmath.org/authors/?q=ai:hanson.eric-j"Rock, Job Daisie"https://zbmath.org/authors/?q=ai:rock.job-daisieFrom the authors' abstract: Persistent homology is an essential tool in topological data analysis. The authors ``prove that [...] pointwise finite-dimensional persistence modules indexed by \(\mathbb{S}^1\) decompose uniquely, up to isomorphism, into the direct sum of a bar code and finitely-many Jordan cells. In the language of representation theory, this is a direct sum of string modules and band modules. Persistence modules indexed on \(\mathbb{S}^1\) have also been called angle-valued or circular persistence modules''. Either a cyclic order or partial order on \(\mathbb{S}^1\) is allowed. The authors ``also show that a pointwise finite-dimensional \(\mathbb{S}^1\) persistence module is indecomposable if and only if it is a bar or Jordan cell''. Finally, the authors ``classify the isomorphism classes of such indecomposable modules''.
Reviewer: Mee Seong Im (Annapolis)On bialgebras, comodules, descent data and Thom spectra in \(\infty\)-categorieshttps://zbmath.org/1535.160412024-07-05T15:31:27.292712Z"Beardsley, Jonathan"https://zbmath.org/authors/?q=ai:beardsley.jonathanThe paper studies \(\infty\)-categorical coalgebras and bialgebras which are not necessarily commutative nor cocommutative as well as their \(\infty\)-categories of modules and comodules.
The authors establish several results for coalgebraic structure in \(\infty\)-categories, specifically with connections to the spectral noncommutative geometry of cobordism theories. It is proved that the categories of comodules and modules over a bialgebra always admit suitably structured monoidal structures in which the tensor product is taken in the ambient category.
The authors give two examples of higher coalgebraic structure. First, they prove that for a map of \(\mathbb{E}_n\)-ring spectra \(\phi: A \to B\), the associated \(\infty\)-category of descent data is equivalent to the \(\infty\)-category of comodules over \(B \otimes_A B\), the so-called descent coring. Secondly, they show that Thom spectra are canonically equipped with a highly structured comodule structure which is equivalent to the \(\infty\)-categorical Thom diagonal of
\textit{M. Ando} et al. [J. Topol. 7, No. 3, 869--893 (2014; Zbl 1312.55011)]. They also show that this highly structured diagonal decomposes the Thom isomorphism for an oriented Thom spectrum in the expected way indicating that Thom spectra are good examples of spectral noncommutative torsors.
Reviewer: Mahender Singh (S.A.S. Nagar)Characterising actions on trees yielding non-trivial quasimorphismshttps://zbmath.org/1535.201372024-07-05T15:31:27.292712Z"Iozzi, Alessandra"https://zbmath.org/authors/?q=ai:iozzi.alessandra"Pagliantini, Cristina"https://zbmath.org/authors/?q=ai:pagliantini.cristina"Sisto, Alessandro"https://zbmath.org/authors/?q=ai:sisto.alessandroSummary: Using a cocycle defined by \textit{N. Monod} and \textit{Y. Shalom} [J. Differ. Geom. 67, No. 3, 395--455 (2004; Zbl 1127.53035)] we introduce the \textit{median} quasimorphisms for groups acting on trees. Then we characterise actions on trees that give rise to non-trivial median quasimorphisms. Roughly speaking, either the action is highly transitive on geodesics, or it fixes a point in the boundary, or there exists an infinite family of non-trivial median quasimorphisms. In particular, in the last case the second bounded cohomology of the group is infinite dimensional as a vector space. As an application, we show that a cocompact lattice in the automorphism group of a product of trees has only trivial quasimorphisms if and only if the closures of the projections on each of the two factors are locally \(\infty\)-transitive.Note on homeomorphic spaces into open subsets of compact polyhedrahttps://zbmath.org/1535.550092024-07-05T15:31:27.292712Z"Gómez, F."https://zbmath.org/authors/?q=ai:gomez.federico|gomez.fernando-palacios|gomez.felix-pedro-quispe|gomez.fabio|gomez.francisco-j|gomez.florian(no abstract)\(K\)-theory of real Grassmann manifoldshttps://zbmath.org/1535.550102024-07-05T15:31:27.292712Z"Podder, Sudeep"https://zbmath.org/authors/?q=ai:podder.sudeep"Sankaran, Parameswaran"https://zbmath.org/authors/?q=ai:sankaran.parameswaranSummary: Let \(G_{n,k}\) denote the real Grassmann manifold of \(k\)-dimensional vector subspaces of \(\mathbb{R}^n\). We compute the complex \(K\)-ring of \(G_{n,k}\), up to a small indeterminacy, for all values of \(n,k\) where \(2\leqslant k\leqslant n-2\). When \(n\equiv 0\pmod 4\), \(k\equiv 1\pmod 2\), we use the Hodgkin spectral sequence to determine the \(K\)-ring completely.A unified view on the functorial nerve theorem and its variationshttps://zbmath.org/1535.550112024-07-05T15:31:27.292712Z"Bauer, Ulrich"https://zbmath.org/authors/?q=ai:bauer.ulrich"Kerber, Michael"https://zbmath.org/authors/?q=ai:kerber.michael"Roll, Fabian"https://zbmath.org/authors/?q=ai:roll.fabian"Rolle, Alexander"https://zbmath.org/authors/?q=ai:rolle.alexanderIn this paper, the authors present together several different functional nerve theorems under various assumptions. The nerve theorem is a fundamental part of the algebraic topology tool box. It plays a very important part in computational and applied algebraic topology. In particular, in topological data analysis, practitioners regularly require an appropriate functorial nerve theorem, and often such a theorem in proven in a specific context. This paper presents several versions of the nerve theorem, and unifies them into a theorem which implies several of these variants. Due to the functorial nature of these theorems, the authors present the necessary category theory to prove the theorems, as well as the unified version.
Beginning with `nerve theorems for covers by closed convex sets in Euclidean spaces', which they prove using fundamental techniques. After this, they prove nerve theorems for `covers of simplicial complexes by subcomplexes', before finally presenting a `unified nerve theorem', which absorbs many versions of the nerve theorem. To prove this unified nerve theorem, they use techniques from abstract homotopy theory, in particular model categories.
Reviewer: Yossi Bokor Bleile (Aalborg)Topological invariance of torsion-sensitive intersection homologyhttps://zbmath.org/1535.550122024-07-05T15:31:27.292712Z"Friedman, Greg"https://zbmath.org/authors/?q=ai:friedman.gregCategories of torsion-sensitive perverse sheaves were introduced in [\textit{G. Friedman}, Mich. Math. J. 68, No. 4, 675--726 (2019; Zbl 1439.55008)] in order to unify several intersection homology duality theorems, see [\textit{M. Goresky} and \textit{R. MacPherson}, Topology 19, 135--165 (1980; Zbl 0448.55004), \textit{M. Goresky} and \textit{R. MacPherson}, Invent. Math. 72, 77--129 (1983; Zbl 0529.55007), \textit{M. Goresky} and \textit{P. Siegel}, Comment. Math. Helv. 58, 96--110 (1983; Zbl 0529.55008), \textit{S. E. Cappell} and \textit{J. L. Shaneson}, Ann. Math. (2) 134, No. 2, 325--374 (1991; Zbl 0759.55002)], as special cases of a more general duality theorem in which torsion phenomena are encoded into the sheaf complexes.
The torsion-sensitive-Deligne sheaves are a generalisation of Deligne sheaves on stratified pseudomaniflods, [\textit{A. Beilinson} et al., Faisceaux pervers. Actes du colloque ``Analyse et Topologie sur les Espaces Singuliers''. Partie I. 2nd edition. Paris: Société Mathématique de France (SMF) (2018; Zbl 1390.14055)], since they are the intermediate extensions of torsion-sensitive coefficient systems and they can be characterised by a set of axioms which generalise the ones by Goresky and MacPherson for ordinary Deligne sheaves.
This paper is devoted to the study of the topological invariance (up to quasi-isomorphism) of the torsion-sensitive-Deligne sheaves. For the classical Deligne sheaves this problem was considered in [Goresky and MacPherson, Zbl 0529.55007, \textit{H. C. King}, Topology Appl. 20, 149--160 (1985; Zbl 0568.55003)]. The topological invariance in the torsion sensitive category is achieved in Theorem 3.6; this theorem recovers (with a different proof) the original Goresky-MacPherson theorem [Goresky and MacPherson, loc. cit.], as well as results in [\textit{D. Chataur} et al., Ill. J. Math. 63, No. 1, 127--163 (2019; Zbl 1451.55002), \textit{N. Habegger} and \textit{L. Saper}, Invent. Math. 105, No. 2, 247--272 (1991; Zbl 0759.55003)].
Reviewer: Alessio Cipriani (Verona)A short note on simplicial stratificationshttps://zbmath.org/1535.550132024-07-05T15:31:27.292712Z"Wrazidlo, Dominik J."https://zbmath.org/authors/?q=ai:wrazidlo.dominik-jThe classical approach to generalize Poincaré duality and related structures from manifolds to manifolds with singularities is intersection homology and cohomology. A recent new idea to achieve the same goal is to assign an ``intersection space'' to the singular manifold and consider the ordinary homology and cohomology of it. This spatial version of intersection homology is not the same as the classical version but is related to it.
The first -- somewhat restrictive -- construction of intersection spaces, due to \textit{M. Banagl} [Electron. Res. Announc. Math. Sci. 16, 63--73 (2009; Zbl 1215.55003) and Intersection spaces, spatial homology truncation, and string theory. Dordrecht: Springer (2010; Zbl 1219.55001)], was generalized by \textit{M. Agustín Vicente} and \textit{J. Fernandez de Bobadilla} [Doc. Math. 25, 1653--1725 (2020; Zbl 1457.32082)] to a wide variety of spaces satisfying extra assumptions. The paper under review proves that these assumptions hold in great generality: they hold for triangulated PL pseudomanifolds, for example for complex algebraic varieties.
Reviewer: Richárd Rimányi (Chapel Hill)Weighted (co)homology and weighted Laplacianhttps://zbmath.org/1535.550142024-07-05T15:31:27.292712Z"Wu, Chengyuan"https://zbmath.org/authors/?q=ai:wu.chengyuan"Ren, Shiquan"https://zbmath.org/authors/?q=ai:ren.shiquan"Wu, Jie"https://zbmath.org/authors/?q=ai:wu.jie.2"Xia, Kelin"https://zbmath.org/authors/?q=ai:xia.kelinIn this paper, the authors extend the combinatorial Laplace operator introduced in [\textit{D. Horak} and \textit{J. Jost}, Adv. Math. 244, 303--336 (2013; Zbl 1290.05103)] to weighted simplicial complexes, in turn providing new examples and computations.
For a simplicial complex \(K\), denote by \(d_i\) the \(i\)-th face map. Consider an \(R\)-module \(G\) over a commutative unital ring \(R\). Then, Definition 2.1 in the paper at hand defines a weight function \(\phi\) as a map
\[
\phi\colon K\times K\to R
\]
such that \(\phi(d_i\sigma, d_j d_i \sigma)\phi(\sigma,d_i\sigma)g= \phi(d_j\sigma, d_j d_i \sigma)\phi(\sigma,d_j\sigma)g\) for all \(\sigma\) in \(K\), \(g\in G\), and \(j<i\). This definition is shown to encompass both the definition of the classical differential, and of the weighted boundary of [\textit{R. J. MacG. Dawson}, Cah. Topologie Géom. Différ. Catégoriques 31, No. 3, 229--243 (1990; Zbl 0735.18011)]. The weight function also yields \(\phi\)-weighted cohomology groups. After analyzing \(\phi\)-weighted cohomology theories, the \(\phi\)-weighted Laplacian is introduced in Section 5. The main generalization of Horak and Jost's results on the calculation of the multiplicity of the eigenvalue zero in the Laplacian spectrum is given in Section 6. The theory is then applied in Section 7 to concrete examples as weighted polygons and digraphs.
Reviewer: Luigi Caputi (Torino)External Spanier-Whitehead duality and homology representation theorems for diagram spaceshttps://zbmath.org/1535.550152024-07-05T15:31:27.292712Z"Lackmann, Malte"https://zbmath.org/authors/?q=ai:lackmann.malteRecall that Brown's representability theorem states that any abelian presheaf \(\mathrm{Ho}(\mathbf{CW}_*)^{\mathrm{op}} \to \mathbf{Ab}\) satisfying certain exactness properties is represented by an abelian group object in \(\mathrm{Ho}(\mathbf{CW}_*)\) (see [\textit{E. H. Brown jun.}, Ann. Math. (2) 75, 467--484 (1962; Zbl 0101.40603)]). This result may be used to show that cohomology theories are representable by spectra. To show that homology theories are representable by spectra one requires a strengthening of Brown's result due to \textit{J. F. Adams} [Topology 10, 185--198 (1971; Zbl 0197.19604)], namely that an abelian presheaf \(\mathrm{Ho}(\mathbf{CW}_*)^{\mathrm{op}} \to \mathbf{Ab}\) as above is determined by its restriction to the subcategory of \(\mathrm{Ho}(\mathbf{CW}_*)\) spanned by finite pointed CW-complexes. Then one may use Spanier-Whitehead duality to reduce the question of the representability of homology theories to one about the representability of cohomology theories.
The present article carries out a similar strategy as the one outlined above to prove a representability theorem for homology theories on the category of presheaves \(C^{\mathrm{op}} \to \mathbf{Top}_*\), where \(C\) is a small category. In this more general setting, the author must overcome two hurdles:
\begin{itemize}
\item[1.] The category of \(C\)-indexed spectra is no longer canonically self-dual for general \(C\).
\item[2.] Adams' representability theorem is no longer true for arbitrary \(C\).
\end{itemize}
To deal with the first hurdle the author observes that \([C^{\mathrm{op}}, \mathbf{Spectra}]\) is canonically dual to \([C, \mathbf{Spectra}]\) in the monoidal \(2\)-category with objects small categories, and morphism categories \(\mathrm{Hom}(A,B) = \mathrm{Ho} [A^{\mathrm{op}} \times B, \mathbf{Spectra}]\), and the fact that \([C^{\mathrm{op}}, \mathbf{Spectra}]\) and \([C, \mathbf{Spectra}]\) are generated under colimits by \(C\) and \(C^{\mathrm{op}}\), respectively. Thus, the author is faced with the second hurdle, which is surmounted using a result of Neeman that Adams' representability theorem still goes through for \(C\) countable (see [\textit{A. Neeman}, Topology 36, No. 3, 619--645 (1997; Zbl 0869.55008)]).
Finally, the author specialises to the case of homology theories on \([C^{\mathrm{op}}, \mathbf{Top}_*]\) valued in \(\mathbf{Q}\)-vector spaces, and studies when such homology theories split into simpler pieces.
A large part of the present article is dedicated to proving model categorical results in order to make the above precise.
Reviewer: Adrian Clough (Abu Dhabi)Periodic self maps and thick ideals in the stable motivic homotopy category over \(\mathbb{C}\) at odd primeshttps://zbmath.org/1535.550202024-07-05T15:31:27.292712Z"Stahn, Sven-Torben"https://zbmath.org/authors/?q=ai:stahn.sven-torbenIn this paper, the author studies thick subcategories of the stable motivic homotopy category \(\mathcal{SH}(\mathbb{C})\) over the base field \(\mathbb{C}\), building upon work of Ruth Joachimi [\textit{R. Joachimi}, Springer Proc. Math. Stat. 309, 109--219 (2020; Zbl 1475.55023)] and in relation to the thick subcategory theorem [\textit{M. J. Hopkins} and \textit{J. H. Smith}, Ann. Math. (2) 148, No. 1, 1--49 (1998; Zbl 0924.55010)] of classical stable homotopy theory. More precisely, he works with quasi-finite cellular spectra, locally with respect to an odd prime \(\ell\) or after completion.
Betti realization yields \(R : \mathcal{SH}(\mathbb{C})^{qfin}_{(\ell)} \rightarrow \mathcal{SH}_{(\ell)}^{fin}\) to the localization of the classical stable homotopy category of finite spectra and one also has the `constant sheaf' functor \(c : \mathcal{SH}_{(\ell)}^{fin} \rightarrow \mathcal{SH}(\mathbb{C})^{qfin}_{(\ell)}\). This gives, for \(\mathcal{C}_n \subset \mathcal{SH}_{(\ell)}^{fin}\) the thick subcategory of type \(n\) spectra, the tensor ideals
\[
\mathrm{thickid} (c \mathcal{C}_n) \subseteq R^{-1} (\mathcal{C}_n).
\]
Joachimi pointed out that there are more thick tensor ideals in the motivic setting. Moreover, nilpotence is more delicate: for example, the motivic \(\eta\) is not nilpotent. She defined the thick ideal \(\mathcal{C}_{AK(n)}\) as the spectra for which \(AK(n)_{**}(-)\) vanishes, for \(AK(n)\) the \(n\)th algebraic Morava \(K\)-theory, and showed that \[\mathcal{C}_{AK(n+1)} \subseteq \mathcal{C}_{AK(n)} \subseteq R^{-1} (\mathcal{C}_{n+1}).\]
The author first considers the full subcategory of \(\mathcal{SH}(\mathbb{C})^{qfin}_{(\ell)}\) (or its \(\ell\)-complete variant) admitting a motivic \(v_n\)-self map. Assuming that a motivic nilpotence conjecture holds (this only involves certain bidegrees), he proves that this subcategory is thick.
To construct examples of motivic \(v_n\)-self maps (in the \(\ell\)-complete setting), he considers Joachimi's motivic spectrum \(\mathbb{X}_n\), a motivic counterpart of Smith's type \(n\) spectrum. Joachimi showed that \(AK(s)_{**}(\mathbb{X}_n)\) vanishes for \(s<n\) and is non-zero for \(s=n\). The author shows that \(AK(n)_{**}(\mathbb{X}_n)\) is free over \(AK(n)_{**}\), which allows the calculation of \(AK(n)_{**}(D \mathbb{X}_n \wedge \mathbb{X}_n)\) by Künneth. A motivic Adams spectral sequence adaptation of the classical argument then yields a motivic \(v_n\)-self map.
To deal with non-nilpotence of motivic \(\eta\), it is natural to smash with the cone \(C_\eta\), thus considering the thick ideals \(\mathrm{thickid}(C_\eta) \cap \mathcal{C}_{AK(n)}\). The author proves that \(C_\eta \wedge \mathbb{X}_{n+1}\) lies in \(\mathcal{C}_{AK(n)}\) but not in \(\mathcal{C}_{AK(n+1)}\), showing that these thick ideals are nonzero and distinct.
Finally he shows that the inclusion \(\mathcal{C}_{AK(1)} \subsetneq R^{-1}(\mathcal{C}_2)\) is proper and that \(\mathrm{thickid} (c \mathcal{C}_2)\not \subset \mathcal{C}_{AK(1)}\) (this corrects an assertion of Joachimi's). He indicates that the argument also applies for higher chromatic \(n\).
Reviewer: Geoffrey Powell (Angers)Formality and finiteness in rational homotopy theoryhttps://zbmath.org/1535.550222024-07-05T15:31:27.292712Z"Suciu, Alexander I."https://zbmath.org/authors/?q=ai:suciu.alexander-iThe present work is a very welcomed survey on formality and finiteness in the context of rational homotopy theory.
First, we recall these two central notions. A nilpotent finite \(\mathbb Q\)-type space \(X\) is \textit{formal} if the Sullivan algebra of piecewise linear polynomial rational forms \(A_{\mathrm{PL}}(X)\) is connected to the rational cohomology algebra \(H^*(X;\mathbb Q)\) via a zig-zag of differential graded commutative algebras (cdga's, henceforth)
\[
A_{\mathrm{PL}}(X) \xleftarrow{\simeq} \bullet \xrightarrow{\simeq} H^*(X;\mathbb Q).
\]
Here, the cohomology algebra carries the trivial differential. In other words: \(X\) is formal if the algebras \(A_{\mathrm{PL}}(X)\) and \(H^*(X;\mathbb Q)\) represent the same homotopy type in the corresponding homotopy category of cdga's. Formal spaces form one of the most celebrated classes of spaces since the notion was introduced, due to their relative simplicity with respect to arbitrary homotopy types.
Finiteness refers to the possibility of approximating an object of interest by another object of the same type with convenient finiteness hypotheses. For example, a \(q\)-finite space is a path-connected topological space \(X\) homotopy equivalent to a finite CW-complex whose skeleton stabilizes at stage \(q\). Similarly, a cdga \((A,d)\) is \(q\)-finite if it is connected (i.e., \(A^0=k\) is the ground field) and each degree \(n\) component \(A^n\) is a finite dimensional \(k\)-vector space for all \(n\leq q\).
This survey collects many of the foundational results on these topics, as well as many of the results of the prolific author and his collaborators on them. In any case, the theory has developed too broadly for a survey, or even a book, to contain most of what is known. Along the text, the reader will find abundant examples to illustrate the results or to show that some of the statements are sharp. The text also poses some conjectures and open questions on the topics.
The paper is divided into five main sections that we briefly summarize.
Part I presents the basics on cdga's and their homotopy theory, formality and its variants (such as intrinsic and partial formality), and the first basic results on formality. These include the behavior of formality under field extension, formality of cdga maps, the relationship to Massey products, the descent property, the theory of positive weight decompositions, minimal models, and Poincaré duality.
Part II deals with several of the Lie algebras involved in this theory, discussing some of their properties and interconnections. The reader should not confuse this section to be the study of formality of differential graded Lie algebras (i.e., the Quillen side of rational homotopy theory), the defining characteristic of \textit{coformal} spaces. This section includes graded and filtered, Malcev, and holonomy Lie algebras. Some of these Lie algebras arise as the associated Lie algebra to a group, whose graded pieces are the successive quotients of the lower central series of the group tensored with the ground field, and whose Lie bracket is induced from the commutators of the group. In some other instances, these Lie algebras are related to group cohomology, or are constructed from some special cdga's (that might arise as a model of a topological space).
Part III contains the basics of rational homotopy theory of spaces, such as completions, rationalizations, and algebraic models for spaces and groups. The main focus is on the formality and finiteness properties of such models. Important results and constructions are given for polyhedral products, configuration spaces, right-angled Artin groups, among many others.
The remaining two parts are slightly more specialized, and touch upon topics mostly on algebraic geometry.
Part IV deals with Alexander invariants and the cohomology jump loci of spaces and suitable algebraic models. It further connects the characteristic and resonance varieties to various formality and finiteness properties previously established.
Part V applies the theory presented to the study of Kähler manifolds and smooth, quasi-projective varieties, compact Lie group actions on manifolds, and closed, orientable 3-manifolds.
Reviewer: José Manuel Moreno Fernández (Málaga)Linking in tree-manifoldshttps://zbmath.org/1535.570432024-07-05T15:31:27.292712Z"Wang, Xueqi"https://zbmath.org/authors/?q=ai:wang.xueqi|wang.xueqi.1Let \(T\) be a tree whose vertices are \(2m\)-dimensional oriented vector fields over the sphere \(S^{2m}\). Plumbing, as described in [\textit{W. Browder}, Surgery on simply-connected manifolds. Berlin-Heidelberg-New York: Springer-Verlag (1972; Zbl 0239.57016)], gives an oriented manifold \(W(T)\) whose boundary \(M(T)\) is called a tree manifold. The fibers of the associated sphere bundles to the vertices of \(T\) provide embedded \((2m-1)\)-dimensional spheres in \(M(T)\). In the case when \(M(T)\) is a rational homology sphere, the author calculates the linking number of these spheres. This paper corrects earlier work of \textit{T. tom Dieck} [Geom. Dedicata 34, No. 1, 57--65 (1990; Zbl 0759.57016)].
Reviewer: James Hebda (St. Louis)