J Group Research Articles

Differential graded versus simplicial categories

We establish a connection between differential graded and simplicial categories by constructing a three-step zig-zag of Quillen adjunctions relating the homotopy theories of the two. In an intermediate step, we extend the Dold–Kan correspondence to a Quillen equivalence between categories enriched over non-negatively graded complexes and categories enriched over simplicial modules. As an application, we obtain a simple calculation of Simpson's homotopy fiber, which is known to be a key step in the construction of a moduli stack of perfect complexes on a smooth projective variety.

On Algebraic K-theory Categorical Groups

Homotopy categorical groups of any pointed space are defined via the fundamental groupoid of iterated loop spaces. This notion allows, paralleling the group case, to introduce the notion of K-categorical groups $\mathbb{K}_iR$ of any ring R. We also show the existence of a fundamental categorical crossed module associated to any fibre homotopy sequence and then, $\mathbb{K}_1R$ and $\mathbb{K}_2R$ are characterized, respectively, as the homotopy cokernel and kernel of the fundamental categorical crossed module associated to the fibre homotopy sequence $FR\xrightarrow{{d_{R} }}BGLR\xrightarrow{{q_{R} }}BGLR^{ + } $ As consequence, the 3th level of the Postnikov tower of the K-theory spectrum of R is classified by this categorical crossed module.

The compare of propofol,fentanyl plus local anesthesia and Ma saddle used in PPH patients

Objective To investigate two different ways of procedure for prolapse and hemorrhoids (PPH) and the clinical effects of postoperative recovery.Methods 60 patients with hemorrhoids were randomly divided into A,J two groups,A group of saddle Ma manner;J group using local anesthesia plus intravenous anesthesia.Perioperative observa-tion of MAP,HR and SpO2 changes in the effect of intraoperative and postoperative compli- cations of surgery,anesthesia,hospital patients and hospital costs of the time the results of a comparative analysis.Results intravenous anesthesia drugs push in the group after 3minMAP significantly lower than the basis of value,P＜O.O1 and SpO2 lower than the value of the foundation;urinary retention compared two groups of intravenous anesthesia was significantly less than the saddle Ma group,P＜0.01;after vertigo Compared two groups of intravenous anesthesia group than the saddle Ma group,P＜O.05.Narcotic two hours compared statistically significant difference,P＜0.01.Group general anesthesia in hospital longer than a shorter saddle Ma Group at a cost less.Conclusions the popularity of day surgery and anesthesia in PPH patients,satisfied with the effect of anesthesia in surgery,postoperative recovery can be discharged from hospital earlier,saving the cost of health care. Key words: Propofol; Fentanyl; Local anesthesia; SaddleMa; Procedure for prolapse and hemorrhoids

Derivatives of embedding functors I: the stable case

For smooth manifolds $M$ and $N$, let $\Ebar(M, N)$ be the homotopy fiber of the map $\Emb(M, N)\longrightarrow \Imm(M, N)$. Consider the functor from the category of Euclidean spaces to the category of spectra, defined by the formula $V\mapsto \Sigma^\infty\Ebar(M, N\times V)$. In this paper, we describe the Taylor polynomials of this functor, in the sense of M. Weiss' orthogonal calculus, in the case when $N$ is a nice open submanifold of a Euclidean space. This leads to a description of the derivatives of this functor when $N$ is a tame stably parallelizable manifold (we believe that the parallelizability assumption is not essential). Our construction involves a certain space of rooted forests (or, equivalently, a space of partitions) with leaves marked by points in $M$, and a certain ``homotopy bundle of spectra'' over this space of trees. The $n$-th derivative is then described as the ``spectrum of restricted sections'' of this bundle. This is the first in a series of two papers. In the second part, we will give an analogous description of the derivatives of the functor $\Ebar(M, N\times V)$, involving a similar construction with certain spaces of connected graphs (instead of forests) with points marked in $M$.

Rational homotopy of the polyhedral product functor

Let (X, *) be a pointed CW-complex, K be a simplicial complex on n vertices and X-K be the associated polyhedral power. In this paper, we construct a Sullivan model of XK from K and from a model of X. Let F(K, X) be the homotopy fiber of the inclusion X-K -> X-n. Recent results of Grbic and Theriault, on one side, and of Denham and Suciu, on the other side, show the diversity of the possible homotopy types for F( K, X). Here, we prove that the corresponding map between Sullivan models is Golod attached, generalizing a result of J. Backelin. This property is deduced from the existence of a succession of vibrations whose fibers are suspensions. We consider also the Lusternik-Schnirelmann category of XK. In the case that cat X-n = n cat X, we prove that cat X-K = (cat X)(1 + dimK). Finally, we mention that this work is written in the case of a sequence of pairs, (X) under bar = (X-i, A(i)) 1 <= i <= n, as in a recent work of Bahri, Bendersky, Cohen and Gitler.

On fibering and splitting of 5-manifolds over the circle

Our main result is a generalization of Cappell's 5-dimensional splitting theorem. As an application, we analyze, up to internal s-cobordism, the smoothable splitting and fibering problems for certain 5-manifolds mapping to the circle. For example, these maps may have homotopy fibers which are in the class of finite connected sums of certain geometric 4-manifolds. Most of these homotopy fibers have non-vanishing second mod 2 homology and have fundamental groups of exponential growth, which are not known to be tractable by Freedman–Quinn topological surgery. Indeed, our key technique is topological cobordism, which may not be the trace of surgeries.

Loop structures in Taylor towers

We study spaces of natural transformations between homogeneous functors in Goodwillie’s calculus of homotopy functors and in Weiss’s orthogonal calculus. We give a description of such spaces of natural transformations in terms of the homotopy fixed point construction. Our main application uses this description in combination with the Segal Conjecture to obtain a delooping theorem for connecting maps in the Goodwillie tower of the identity and in the Weiss tower of BU.V/. The interest in such deloopings stems from conjectures made by the first and the third author [4] that these towers provide a source of contracting homotopies for certain projective chain complexes of spectra. 55P65; 55P47, 18G55 1 Introduction and notation In this paper, we study homogeneous functors in the sense of Weiss’s orthogonal calculus [13]. More precisely, we calculate the space of natural transformations between such functors, and we give a few examples and applications. Our main application is a delooping theorem for connecting maps in the Goodwillie tower of the identity functor for pointed spaces and in the Weiss tower of the functor V7!BU.V/. The motivation for this delooping theorem will be discussed later in this introduction. To state our results, we summarize the usual notation for this context. Let F be a functor from pointed spaces or from finite-dimensional real or complex vector spaces to pointed spaces or spectra. Goodwillie and Weiss calculus assign to such a functor F a “Taylor” tower of functors ! Pn F! Pn 1 F! ; together with a natural map from F to the homotopy inverse limit of the tower that is often a weak homotopy equivalence. The homotopy fiber of the map Pn F! Pn 1 F is customarily denoted Dn F and is referred to as “the nth homogeneous layer of F ,” while Pn F is referred to as “the nth Taylor polynomial of F .” Depending on whether

The $3$-primary classifying space of the fiber of the double suspension

Gray showed that the homotopy fiber W n of the double suspension S 2n-1 → E2 Ω 2 S 2n+1 has an integral classifying space BW n , which fits in a homotopy fibration S 2n-1 → E2 Ω 2 S 2n+1 → υ BW n . In addition, after localizing at an odd prime p, BW n is an H-space and if p ≥ 5, then BW n is homotopy associative and homotopy commutative, and v is an H-map. We positively resolve a conjecture of Gray's that the same multiplicative properties hold for p = 3 as well. We go on to give some exponent consequences.

An algebraic model for the loop space homology of a homotopy fiber

Let F denote the homotopy fiber of a map f:K-->L of 2-reduced simplicial sets. Using as input data the strongly homotopy coalgebra structure of the chain complexes of K and L, we construct a small, explicit chain algebra, the homology of which is isomorphic as a graded algebra to the homology of GF, the simplicial (Kan) loop group on F. To construct this model, we develop machinery for modeling the homotopy fiber of a morphism of chain Hopf algebras. Essential to our construction is a generalization of the operadic description of the category DCSH of chain coalgebras and of strongly homotopy coalgebra maps given in (math.AT/0505559) to strongly homotopy morphisms of comodules over Hopf algebras. This operadic description is expressed in terms of a general theory of monoidal structures in categories with morphism sets parametrized by co-rings, which we elaborate here.

An upper bound for the 3-primary homotopy exponent of the exceptional Lie group $E_7$

A new homotopy fibration is constructed at the prime 3 which shows that the quotient group $E_{7}/F_{4}$ is spherically resolved. This is then used to show that the 3-primary homotopy exponent of $E_{7}$ is bounded above by $3^{23}$, which is at most four powers of 3 from being optimal.

Calculus of functors, operad formality, and rational homology of embedding spaces

Let M be a smooth manifold and V a Euclidean space. Let \( \overline{{{\text{Emb}}}} \)(M,V) be the homotopy fiber of the map Emb(M,V) → Imm(M,V). This paper is about the rational homology of \( \overline{{{\text{Emb}}}} \)(M,V). We study it by applying embedding calculus and orthogonal calculus to the bifunctor (M,V)↦ HQ ∧ \( \overline{{{\text{Emb}}}} \)(M,V)+. Our main theorem states that if $$ \dim V \geqslant 2{\text{ED}}{\left( M \right)} + 1 $$ (where ED(M) is the embedding dimension of M), the Taylor tower in the sense of orthogonal calculus (henceforward called “the orthogonal tower”) of this functor splits as a product of its layers. Equivalently, the rational homology spectral sequence associated with the tower collapses at E1. In the case of knot embeddings, this spectral sequence coincides with the Vassiliev spectral sequence. The main ingredients in the proof are embedding calculus and Kontsevich's theorem on the formality of the little balls operad. We write explicit formulas for the layers in the orthogonal tower of the functor $$ HQ \wedge \overline{{{\text{Emb}}}} {\left( {M,V} \right)}_{ + }. $$ The formulas show, in particular, that the (rational) homotopy type of the layers of the orthogonal tower is determined by the (rational) homotopy type of M. This, together with our rational splitting theorem, implies that, under the above assumption on codimension, rational homology equivalences of manifolds induce isomorphisms between the rational homology groups of \( \overline{{{\text{Emb}}}} \)(–,V).

Nilpotency of unstable K-theory

In the preceding papers [H. Hamanaka, A. Kono, On [ X , U ( n ) ] , when dim X is 2 n, J. Math. Kyoto Univ. 43 (2) (2003) 333–348; H. Hamanaka, On [ X , U ( n ) ] , when dim X is 2 n + 1 , J. Math. Kyoto Univ. 44 (3) (2004) 655–667; H. Hamanaka, Adams e-invariant, Toda bracket and [ X , U ( n ) ] , J. Math. Kyoto Univ. 43 (4) (2003) 815–828], the group structure of the homotopy set [ X , U ( n ) ] with the pointwise multiplication is studied, where X is a finite CW-complex and U ( n ) is the unitary group. It is seen that nil [ X , U ( n ) ] = 2 for some X with its dimension 2 n, and, when dim X = 2 n + 1 and n is even, [ X , U ( n ) ] is expressed as the two stage central extension of an Abelian group, i.e., nil [ X , U ( n ) ] ⩽ 3 . In this paper, we consider the nilpotency class of [ X , U ( n ) ] , especially, for given k, the maximum of the nil [ X , U ( n ) ] under the condition dim X ⩽ 2 n + k is estimated and determined for k = 0 , 1 , 2 .

Reduced K-theory of Azumaya algebras

In the theory of central simple algebras, often we are dealing with abelian groups which arise from the kernel or co-kernel of functors which respect transfer maps (for example K-functors). Since a central simple algebra splits and the functors above are “trivial” in the split case, one can prove certain calculus on these functors. The common examples are kernel or co-kernel of the maps K i ( F ) → K i ( D ) , where K i are Quillen K-groups, D is a division algebra and F its center, or the homotopy fiber arising from the long exact sequence of above map, or the reduced Whitehead group SK 1 . In this note we introduce an abstract functor over the category of Azumaya algebras which covers all the functors mentioned above and prove the usual calculus for it. This, for example, immediately shows that K-theory of an Azumaya algebra over a local ring is “almost” the same as K-theory of the base ring. The main result is to prove that reduced K-theory of an Azumaya algebra over a Henselian ring coincides with reduced K-theory of its residue central simple algebra. The note ends with some calculation trying to determine the homotopy fibers mentioned above.

Negative K-theory of derived categories

We define negative K-groups for exact categories and for ``derived categories'' in the framework of Frobenius pairs, generalizing definitions of Bass, Karoubi, Carter, Pedersen-Weibel and Thomason. We prove localization and vanishing theorems for these groups. Devissage (for noetherian abelian categories), additivity, and resolution hold. We show that the first negative K-group of an abelian category vanishes, and that, in general, negative K-groups of a noetherian abelian category vanish. Our methods yield an explicit non-connective delooping of the K-theory of exact categories and chain complexes, generalizing constructions of Wagoner and Pedersen-Weibel. Extending a theorem of Auslander and Sherman, we discuss the K-theory homotopy fiber of e⊕→ e and its implications for negative K-groups. In the appendix, we replace Waldhausen's cylinder functor by a slightly weaker form of non-functorial factorization which is still sufficient to prove his approximation and fibration theorems.

UNIVERSAL SPACES OF TWO-CELL COMPLEXES AND THEIR EXPONENT BOUNDS

Let P2n+1 be a two-cell complex which is formed by attaching a (2n + 1)-cell to a 2m-sphere by a suspension map. We construct a universal space U for P2n+1 in the category of homotopy associative, homotopy commutative H-spaces. By universal, we mean that U is homotopy associative, homotopy commutative and has the property that any map f: P2n+1 → Y to a homotopy associative, homotopy commutative H-space Y extends to a uniquely determined H-map f̄: U → Y. We then prove upper and lower bounds of the H-homotopy exponent of U. In the case of a mod pr, Moore space U is the homotopy fibre S2n+1{pr} of the pr-power map on S2n+1, and we reproduce Neisendorfer's result that S2n+1{pr} is homotopy associative, homotopy commutative and that the pr-power map on S2n+1{pr} is null homotopic.

Knots, operads, and double loop spaces

We show that the space of long knots in an Euclidean space of dimension larger than three is a double loop space, proving a conjecture by Sinha. We also construct a double loop space structure on framed long knots, and show that the map forgetting the framing is not a double loop map in odd dimension. However, there is always such a map in the reverse direction expressing the double loop space of framed long knots as a semidirect product. A similar compatible decomposition holds for the homotopy fiber of the inclusion of long knots into immersions. We also show via string topology that the space of closed knots in a sphere, suitably desuspended, admits an action of the little 2-discs operad in the category of spectra. A fundamental tool is the McClure-Smith cosimplicial machinery, that produces double loop spaces out of topological operads with multiplication.

Fiber products, Poincaré duality and 𝐴_{∞}-ring spectra

For a Poincaré duality space X d X^d and a map X → B X \to B , consider the homotopy fiber product X × B X X \times ^B X . If X X is orientable with respect to a multiplicative cohomology theory E E , then, after suitably regrading, it is shown that the E E -homology of X × B X X \times ^B X has the structure of a graded associative algebra. When X → B X \to B is the diagonal map of a manifold X X , one recovers a result of Chas and Sullivan about the homology of the unbased loop space L X LX .

Fibrator properties of manifolds

Approximate fibrations form a useful class of maps, in part, because they provide computable relationships involving the domain, image and homotopy fiber. Fibrator properties, which pertain to and depend upon the homotopy fiber, allow instant recognition of approximate fibrations. This is a survey of results about those fibrator properties.

Relative group completions

We extend arbitrary group completions to the category of pairs ( G , N ) where G is a group and N is a normal subgroup of G. Relative localizations are special cases. Our construction is a group-theoretical analogue of fibrewise completion and fibrewise localization in homotopy theory, and generalizes earlier work of Hilton and others on relative localization at primes. We use our approach to find conditions under which factoring out group radicals preserves exactness. This has implications in the study of the effect of plus-constructions on homotopy fibre sequences.

Postnikov factorizations at infinity

We have developed Postnikov sections for Brown–Grossman homotopy groups and for Steenrod homotopy groups in the category of exterior spaces, which is an extension of the proper category. The homotopy fibre of a fibration in the factorization associated with Brown–Grossman groups is an Eilenberg–Mac Lane exterior space for this type of groups and it has two non-trivial consecutive Steenrod homotopy groups. For a space which is first countable at infinity, one of these groups is given by the inverse limit of the homotopy groups of the neighbourhoods at infinity, the other group is isomorphic to the first derived of the inverse limit of this system of groups. In the factorization associated with Steenrod groups the homotopy fibre is an Eilenberg–Mac Lane exterior space for this type of groups and it has two non-trivial consecutive Brown–Grossman homotopy groups. We also obtain a mix factorization containing both kinds of previous factorizations and having homotopy fibres which are Eilenberg–Mac Lane exterior spaces for both kinds of groups. Given a compact metric space embedded in the Hilbert cube, its open neighbourhoods provide the Hilbert cube the structure of an exterior space and the homotopy fibres of the factorizations above are Eilenberg–Mac Lane exterior spaces with respect to inward (or approaching) Quigley groups.