J Group Research Articles

The transfer is functorial

We prove that the Becker–Gottlieb transfer is functorial up to homotopy, for all fibrations with finitely dominated fibers. This resolves a lingering foundational question about the transfer, which was originally defined in the late 1970s in order to simplify the proof of the Adams conjecture. Our approach differs from previous attempts in that we closely emulate the geometric argument in the case of a smooth fiber bundle. This leads to a “multiplicative” description of the transfer, different from the standard presentation as the trace of a diagonal map.

Nielsen type numbers for periodic points of fibre preserving maps

In 1995 Heath, Keppelmann and Wong studied a Nielsen type number NF(f,p) for a fibre preserving map f of fibration p. This number is a lower bound for the least number of fixed points within the fibre homotopy class of f. In this paper we generalize these ideas from fixed point theory to periodic point theory, and define two Nielsen type numbers NPnF(f,p) and NΦnF(f,p) for periodic points of a fibre preserving map. These numbers, which can be thought of as the dual of periodic Nielsen type numbers for a relative map due to Heath, Schirmer and You, are bigger than the ordinary Nielsen type numbers. The definition of NPnF(f,p) and NΦnF(f,p) is reminiscent of the naive type addition formulae for periodic points of fibre preserving maps due to Heath and Keppelmann. Some calculations and examples are given.

Free and nearly free curves from conic pencils

We construct some infinite series of free and nearly free curves using pencils of conics with a base locus of cardinality at most two. These curves have an interesting topology, e.g. a high degree Alexander polynomial that can be explicitly determined, a Milnor fiber homotopy equivalent to a bouquet of circles, or an irreducible translated component in the characteristic variety of their complement. Monodromy eigenspaces in the first cohomology group of the corresponding Milnor fibers are also described in terms of explicit differential forms.

Relative K0 and relative cycle class map

Let F:A→B be an exact functor between small exact categories. We study the zeroth homotopy group K0(F) of the homotopy fiber of the map K(A)→K(B) between K-theory spectra. Under the assumption that F is a cofinal and that B is split exact, we give an explicit description of K0(F) in terms of the triangulated functor Db(A)→Db(B) between the derived categories.We apply it to the pair (X,D) of a scheme X and an affine closed subscheme D of X, and get a description of the relative K0-group K0(X,D) in terms of perfect complexes; it is generated by pairs of two perfect complexes of X together with quasi-isomorphisms along D. This description makes it possible to assign a cycle class in K0(X,D) to a cycle on X not meeting D in an intuitive way. When X is a separated regular scheme of finite type over a field and D is an affine effective Cartier divisor on X, we prove that the cycle classes induce a surjective group homomorphism from the Chow group with modulus CH⁎(X|D) defined by Binda–Saito to a suitable subquotient of K0(X,D).

A homotopy theoretical generalisation of the Bestvina–Brady construction

By using the notion of polyhedral products (X_,A_)K, we recognise the Bestvina–Brady construction [4] as the fundamental group of the homotopy fibre of (S1,⁎)L→S1, where L is a flag complex. We generalise their construction by studying the homotopy fibre F of (S1,⁎)L→(S1,⁎)K for an arbitrary simplicial complex L and K an (m−1)-dimensional simplex. For a particular class of simplicial complexes L, we describe the homology of F, its fixed points, and maximal invariant quotients for coordinate subgroups of Zm. This generalises the work of Leary and Saadetoğlu [13] who studied the case when m=1.

A survey of the homotopy properties of inclusion of certain types of configuration spaces into the Cartesian product

Let X be a topological space. In this survey the authors consider several types of configuration spaces, namely, the classical (usual) configuration spaces F n (X) and D n (X), the orbit configuration spaces F G n (X) and F G n (X)/S n with respect to a free action of a group G on X, and the graph configuration spaces FГ n (X) and FГ n (X)/H, where Г is a graph and H is a suitable subgroup of the symmetric group S n . The ordered configuration spaces F n (X), F G n (X), FГ n (X) are all subsets of the n-fold Cartesian product n Π 1 X of X with itself, and satisfy F G n (X) → F n (X) → FГ n (X) → n Π 1 X. If A denotes one of these configuration spaces, the authors analyse the difference between A and n Π 1 X from a topological and homotopical point of view. The principal results known in the literature concern the usual configuration spaces. The authors are particularly interested in the homomorphism on the level of the homotopy groups of the spaces induced by the inclusion ι: A → n Π 1 X, the homotopy type of the homotopy fibre I ι of the map ι via certain constructions on various spaces that depend on X, and the long exact sequence in homotopy of the fibration involving I ι and arising from the inclusion ι. In this respect, if X is either a surface without boundary, in particular if X is the 2-sphere or the real projective plane, or a space whose universal covering is contractible, or an orbit space S k /G of the k-dimensional sphere by a free action of a Lie group G, the authors present recent results obtained by themselves for the first case, and in collaboration with Golasinski for the second and third cases. The authors also briefly indicate some older results relative to the homotopy of these spaces that are related to the problems of interest. In order to motivate various questions, for the remaining types of configuration spaces, a few of their basic properties are described and proved. A list of open questions and problems is given at the end of the paper.

Clinical application of supraglottic jet ventilation during painless fiberbronchoscopy

Objective To evaluate the efficacy and safety of supraglottic jet ventilation and modified laryngeal mask airway during painless fiberbronchoscopy. Methods Eighty patients undergoing painless fiberbronchoscopy were randomly divided into 2 groups(n=40), receiving supraglottic jet ventilation (J group) and modified laryngeal mask airway (L group), respectively. The outcomes of the procedures were recorded, including success rate of the first tubing insertion, the time spent in successful positioning of the tube, the stability of the tube, and related complications. Furthermore, MAP, HR, SpO2 and PETCO2 were monitored before anesthesia(T0), immediately after inserting laryngeal mask airway(or nasopharyngeal ventilation)(T1) or fiberbronchoscope through the glottis(T2), and 3 min(T3) after and by the end of fiberbronchoscopy. Results The success rate of the first tubing insertion and exposure condition in J group(100%, 97.5%) are better than L group(81.1%, 81.1%)(P 0.05). The occurrence of sore throat in J group(81.1%) is less frequent than that in L group(12.5%)(P 0.05). Conclusions Supraglottic jet ventilation for painless fiberbronchoscopy is a safer and more effective procedure than modified laryngeal mask airway. Key words: Supraglottic; Jet ventilation; Fiberbronchoscopy; Laryngeal mask airway

On the topological contents of η–invariants

We discuss an universal bordism invariant obtained from the Atiyah-Patodi-Singer eta-invariant from the analytic and homotopy theoretic point of view. Classical invariants like the Adams e-invariant, $\rho$-invariants and $String$-bordism invariants are derived as special cases. The main results are a secondary index theorem about the coincidence of the analytic and topological constructions and intrinsic expressions for the bordism invariants.

Function spaces and classifying spaces of algebras over a prop

The goal of this paper is to prove that the classifying spaces of categories of algebras governed by a prop can be determined by using function spaces on the category of props. We first consider a function space of props to define the moduli space of algebra structures over this prop on an object of the base category. Then we mainly prove that this moduli space is the homotopy fiber of a forgetful map of classifying spaces, generalizing to the prop setting a theorem of Rezk. The crux of our proof lies in the construction of certain universal diagrams in categories of algebras over a prop. We introduce a general method to carry out such constructions in a functorial way.

On the homotopy fibre of the inclusion map $$F_n(X)\hookrightarrow \prod _1^nX$$ F n ( X ) ↪ ∏ 1 n X for some orbit spaces X

Under certain conditions, we describe the homotopy type of the homotopy fibre of the inclusion map \(F_n(X)\hookrightarrow \prod _1^nX\) for the nth configuration space \(F_n(X)\) of a topological manifold X without boundary such that \(\mathrm{{dim}}(X)\ge 3\). We then apply our results to the cases where either the universal covering of X is contractible or X is an orbit space \(\mathbb {S}^k/G\) of a tame, free action of a Lie group G on the k-sphere \(\mathbb {S}^k\). If the group G is finite and k is odd, we give a full description of the long exact sequence in homotopy of the homotopy fibration of the inclusion map \(F_n(\mathbb {S}^k/G)\hookrightarrow \prod _1^n\mathbb {S}^k/G\).

Witt, GW, K-theory of quasi-projective schemes

In this article, we prove some results on Witt, Grothendieck–Witt (GW) and K-theory of noetherian quasi-projective schemes X, over affine schemes Spec(A). For integers k≥0, let CMk(X) denote the category of coherent OX-modules F, with locally free dimension dimV(X)⁡(F)=k=grade(F). We prove that there is an equivalence Db(CMk(X))→Dk(V(X)) of the derived categories. It follows that there is a sequence of zig-zag maps K(CMk+1(X))⟶K(CMk(X))⟶∐x∈X(k)K(CMk(Xx)) of the K-theory spectra that is a homotopy fibration. In fact, this is analogous to the homotopy fiber sequence of the G-theory spaces of Quillen (see proof of [16, Theorem 5.4]). We also establish similar homotopy fibrations of GW-spectra and GW-bispectra, by application of the same equivalence theorem.

Towers and Fibered Products of Model Structures

Given a left Quillen presheaf of localized model structures, we study the homotopy limit model structure on the associated category of sections. We focus specifically on towers and fibered products (pullbacks) of model categories. As applications we consider Postnikov towers of model categories, chromatic towers of spectra and Bousfield arithmetic squares of spectra. For stable model categories, we show that the homotopy fiber of a stable left Bousfield localization is a stable right Bousfield localization.

Homotopy theory in toric topology

In toric topology one associates with each simplicial complex on vertices two key spaces, the Davis–Januszkiewicz space and the moment-angle complex , which are related by a homotopy fibration . A great deal of work has been done to study the properties of and , their generalizations to polyhedral products, and applications to algebra, combinatorics, and geometry. Chap. 1 surveys some of the main results in the homotopy theory of these spaces. Chap. 2 breaks new ground by initiating a study of the map . It is shown that, for a certain family of simplicial complexes , the map is a sum of higher and iterated Whitehead products. Bibliography: 49 titles.

On the rational homotopical nilpotency index of principal bundles

Let Aut(p) denote the space of all self-fibre homotopy equivalences of a principal G-bundle \(p: E\rightarrow X\) of simply connected CW complexes with E finite. When G is a compact connected topological group, we show that there exists an inequality $$\begin{aligned} n-\mathrm{N}(p)\le \mathrm{Hnil}_{\mathbb {Q}}(\mathrm{{Aut}}(p)_0)\le n \end{aligned}$$ for any space X, where n is the number of non-trivial rational homotopy groups of G and \(\mathrm{N}(p)\) is defined in Sect. 2. In particular, \(\mathrm{Hnil}_{\mathbb {Q}}(\mathrm{{Aut}}(p)_{0})=n\) if p is a fibre homotopy trivial bundle and X is finite.

Homotopy excision and cellularity

Consider a push-out diagram of spaces C B, construct the homotopy push-out, and then the homotopy pull-back of the diagram one gets by forgetting the initial object A. We compare the difference between A and this homotopy pull-back. This difference is measured in terms of the homotopy fibers of the original maps. Restricting our attention to the connectivity of these maps, we recover the classical Blakers-Massey Theorem.

Efficacy of cannabinoid-2 receptor activation in preventing liver ischemia-reperfusion injury in mice

Objective To investitate the efficacy of cannabinoid-2 receptor (CB2R) activation in preventing liver ischemia-reperfusion (I/R) injury in mice.Methods Forty-eight male C57BL/6J mice,weighing 20-30 g,were divided into 4 groups (n =12 each):sham operation group (group S),group I/R,CB2R agonist JWH133 group (group J),and CB2R agonist JWH133 + CB2R antagonist SR144528 group (group JS).Liver I/ R was produced by blocking the hepatic artery and portal vein for 30 min followed by 45 min of reperfusion in anesthetized mice.At 60 min before ischemia,JWH133 20 mg/kg was injected intraperitoneally in group J,and JWH133 20 mg/kg and SR144528 30 mg/kg were injected intraperitoneally in group JS.The liver was removed at 45 min of reperfusion for determination of tumor necrosis factor-α (TNF-α),macrophage inflammatory protein-1α (MIP-1α) and MIP-2 contents (using ELISA),and expression of TNF-α,MIP-1α,MIP-2 and intercellular adhesion molecule-1 (ICAM-1) mRNA (by RT-PCR) in the liver tissues and for microscopic examination of the pathological changes.Results Compared with group S,the contents of TNF-α,MIP-1α and MIP-2 were significantly increased,and the expression of TNF-α,MIP-1α,MIP-2 and ICAM-1 mRNA was up-regulated in J and JS groups.Compared with group I/R,the contents of TNF-α,MIP-1α and MIP-2 were significantly decreased,and the expression of TNF-α,MIP-1α,MIP-2 and ICAM-1 mRNA was down-regulated in J group,and no significant change was found in the parameters mentioned above in JS group.Compared with group J,the contents ofTNF-α,MIP-1α and MIP-2 were significantly increased,and the expression of TNF-α,MIP-1α,MIP-2 and ICAM-1 mRNA was up-regulated in JS group.The pathological changes of liver tissues were significantly attenuated in group J as compared with I/R and JS groups.Conclusion CB2R activation is effective in preventing liver I/R injury in mice and the mechanism is related to inhibitioni of inflammatory responses in liver tissues. Key words: Receptor, annabinoid, CB2; Reperfusion injury ; Liver

Nonabelian higher derived brackets

Let M be a graded Lie algebra, together with graded Lie subalgebras L and A such that as a graded space M is the direct sum of L and A, and A is abelian. Let D be a degree one derivation of M squaring to zero and sending L into itself, then Voronov's construction of higher derived brackets associates to D an L∞ structure on A[−1]. It is known, and it follows from the results of this paper, that the resulting L∞ algebra is a weak model for the homotopy fiber of the inclusion of differential graded Lie algebras i:(L,D,[⋅,⋅])→(M,D,[⋅,⋅]). We prove this fact using homotopical transfer of L∞ structures, in this way we also extend Voronov's construction when the assumption that A is abelian is dropped: the resulting formulas involve Bernoulli numbers. In the last section we consider some example and some further application.

Properties of Selick's filtration of the double suspension E2

To help study the double suspension [Formula: see text] when localised at a prime p, Selick filtered Ω2S2n+1 by H-spaces which geometrically realise a natural Hopf algebra filtration of H*(Ω2S2n+1;ℤ/p). Later, Gray showed that the fiber Wn of E2 has an integral classifying space BWn and there is a homotopy fibration [Formula: see text]. In this paper we correspondingly filter BWn in a manner compatible with Selick's filtration and the homotopy fibration [Formula: see text], study the multiplicative properties and homotopy exponents of the spaces in the filtrations, and use the filtrations to filter exponent information for the homotopy groups of S2n+1. Our results link three seemingly different in nature classical homotopy fibrations given by Toda, Selick and Gray and make them special cases of a systematic whole. In addition we construct a spectral sequence which converges to the homotopy groups of BWn.

On the classification of fibrations

We identify the homotopy type of the moduli of maps with a given homotopy type of the base and the homotopy fiber. A new model for the space of weak equivalences and its classifying space is given.

M-Theory with Framed Corners and Tertiary Index Invariants

The study of the partition function in M-theory involves the use of index theory on a twelve-dimensional bounding manifold. In eleven dimensions, viewed as a boundary, this is given by secondary index invariants such as the Atiyah-Patodi-Singer eta-invariant, the Chern-Simons invariant, or the Adams e-invariant. If the eleven-dimensional manifold itself has a boundary, the resulting ten-dimensional manifold can be viewed as a codimension two corner. The partition function in this context has been studied by the author in relation to index theory for manifolds with corners, essentially on the product of two intervals. In this paper, we focus on the case of framed manifolds (which are automatically Spin) and provide a formulation of the refined partition function using a tertiary index invariant, namely the f-invariant introduced by Laures within elliptic cohomology. We describe the context globally, connecting the various spaces and theories around M-theory, and providing a physical realization and interpretation of some ingredients appearing in the constructions due to Bunke-Naumann and Bodecker. The formulation leads to a natural interpretation of anomalies using corners and uncovers some resulting constraints in the heterotic corner. The analysis for type IIA leads to a physical identification of various components of eta-forms appearing in the formula for the phase of the partition function.