J Group Research Articles

Homotopy theory of non-symmetric operads, II: Change of base category and left properness

In this paper we study spaces of algebras over an operad (non-symmetric) in symmetric monoidal model categories. We first compute the homotopy fiber of the forgetful functor sending an algebra to its underlying object, extending a result of Rezk. We then apply this computation to the construction of geometric moduli stacks of algebras over an operad in a homotopical algebraic geometry context in the sense of To\"en and Vezzosi. We show under mild hypotheses that the moduli stack of unital associative algebras is a Zariski open substack of the moduli stack of non-necessarily unital associative algebras. The classical analogue for finite-dimensional vector spaces was noticed by Gabriel.

Deloopings of the spaces of long embeddings

The homotopy fiber of the inclusion from the long embedding space to the long immersion space is known to be an iterated based loop space (if the codimension is greater than two). In this paper we deloop the homotopy fiber to obtain the topological Stiefel manifold, combining results of Lashof and of Lees. We also give a delooping of the long embedding space, which can be regarded as a version of Morlet-Burghelea-Lashof's delooping of the diffeomorphism group of the disk relative to the boundary. As a corollary, we show that the homotopy fiber is weakly equivalent to a space on which the framed little disks operad acts possibly nontrivially, and hence its rational homology is a (higher) BV-algebra in a stable range of dimensions.

Fibrations of topological stacks

In this note we define fibrations of topological stacks and establish their main properties. When restricted to topological spaces, our notion of fibration coincides with the classical one. We prove various standard results about fibrations (long exact sequence for homotopy groups, Leray–Serre and Eilenberg–Moore spectral sequences, etc.). We prove various criteria for a morphism of topological stacks to be a fibration, and use these to produce examples of fibrations. We prove that every morphism of topological stacks factors through a fibration and construct the homotopy fiber of a morphism of topological stacks. As an immediate consequence of the machinery we develop, we also prove van Kampenʼs theorem for fundamental groups of topological stacks.

Quillen's work on the Adams Conjecture

AbstractIn the 1960's and 1970's, the Adams Conjecture figured prominently both in homotopy theory and in geometric topology. Quillen sketched one way to attack the conjecture and then proved it with an entirely different line of argument. Both of his approaches led to spectacular and beautiful new mathematics.

A method for establishment of rat model with pulmonary contusion

Objective To create an easy and specific rat model of isolated bilateral pulmonary contusion and determine the maximal sublethal injury energy.Methods Injury energy was produced by free falling weights and passed through a designed precordial shield to rats' bilateral posterolateral chest wall.The rats were divided into 2.1 J,2.4 J,2.7 J and 3.0 J groups,according to the volume of injury energy.Percentage of lung contusion volume in bilateral lung was measured by blood gas analysis and three dimensional CT (3DCT) at four hours post-injury to assess lung injury severity after contusion.Pathological examination of heart and lung tissue was performed to confirm pulmonary contusion and rule out myocardial contusion.Results Death rate in 3.0 J and 2.4 J groups was 33％ and 11％,respectively.PaO2 in 2.7 J group was significantly lower than that in 2.4 J group (P ＜ 0.01),but pulmonary contusion percentage in 2.7 J group was significantly higher than that in 2.4 J group (P ＜0.01).All groups showed negative correlation between PaO2 and pulmonary contusion percentage measured by 3DCT (R2 =0.762).Hemorrhage,atelectasis and neutrophil infiltration were documented in lung biopsy.No evidence of myofiber break was recorded in heart biopsy.Conclusion This method can duplicate satisfactory models of isolated bilateral pulmonary contusion and 2.7 J can be regarded as the maximal sublethral injury energy. Key words: Lung; Contusions; Imaging,three-dimensional

Rational algebraicK–theory of topologicalK–theory

We show that after rationalization there is a homotopy fiber sequence BBU -> K(ku) -> K(Z). We interpret this as a correspondence between the virtual 2-vector bundles over a space X and their associated anomaly bundles over the free loop space LX. We also rationally compute K(KU) by using the localization sequence, and K(MU) by a method that applies to all connective S-algebras.

The moduli space of thickenings

This paper studies the homotopy type of the moduli space of compact n-manifold thickenings of a finite complex. The main result computes the homotopy fibers of the stabilization map from n-thickenings to (n+1)-thickenings in a wide range. In particular, we obtain an EHP-type sequence for thickenings extending work of C.T.C. Wall.

The Fibre of a Pinch Map in a Model Category

In the category of pointed topological spaces, let F be the homotopy fibre of the pinching map X ∪ CA → X ∪ CA / X from the mapping cone on a cofibration A → X onto the suspension of A. Gray (Proc Lond Math Soc (3) 26:497–520, 1973) proved that F is weakly homotopy equivalent to the reduced product (X, A)∞. In this paper we prove an analogue of this phenomenon in a model category, under suitable conditions including a cube axiom.

1-loop graphs and configuration space integral for embedding spaces

AbstractWe will construct differential forms on the embedding spaces Emb(j, n) for n − j ≥ 2 using configuration space integral associated with 1-loop graphs, and show that some linear combinations of these forms are closed in some dimensions. There are other dimensions in which we can show the closedness if we replace Emb(j, n) by Emb(j, n), the homotopy fiber of the inclusion Emb(j, n) ↪ Imm(j, n). We also show that the closed forms obtained give rise to nontrivial cohomology classes, evaluating them on some cycles of Emb(j, n) and Emb(j, n). In particular we obtain nontrivial cohomology classes (for example, in H3(Emb(2, 5))) of higher degrees than those of the first nonvanishing homotopy groups.

Bounded homotopy theory and the K-theory of weighted complexes

Given a bounding class B, we construct a bounded refinement BK(−) of Quillen’s K-theory functor from rings to spaces. As defined, BK(−) is a functor from weighted rings to spaces, and is equipped with a comparison map BK → K induced by “forgetting control.” In contrast to the situation with B-bounded cohomology, there is a functorial splitting BK(−) ≅ K(−)×BK rel(−) where BK rel(−) is the homotopy fiber of the comparison map.

Homotopy limits of model categories and more general homotopy theories

Generalizing a definition of homotopy fiber products of model categories, we give a definition of the homotopy limit of a diagram of left Quillen functors between model categories. As has been previously shown for homotopy fiber products, we prove that such a homotopy limit does in fact correspond to the usual homotopy limit, when we work in a more general model for homotopy theories in which they can be regarded as objects of a model category.

Homotopy fiber products of homotopy theories

Given an appropriate diagram of left Quillen functors between model categories, one can define a notion of homotopy fiber product, but one might ask if it is really the correct one. Here, we show that this homotopy pullback is well-behaved with respect to translating it into the setting of more general homotopy theories, given by complete Segal spaces, where we have well-defined homotopy pullbacks.

Upper and Lower Bounds of the (co)chain Type Level of a Space

We establish an upper bound for the cochain type level of the total space of a pull-back fibration. It explains to us why the numerical invariants for principal bundles over the sphere are less than or equal to two. Moreover computational examples of the levels of path spaces and Borel constructions, including biquotient spaces and Davis-Januszkiewicz spaces, are presented. We also show that the chain type level of the homotopy fibre of a map is greater than the E-category in the sense of Kahl, which is an algebraic approximation of the Lusternik-Schnirelmann category of the map. The inequality fits between the grade and the projective dimension of the cohomology of the homotopy fibre.

The study of graft offset in the anterior cruciate ligament reconstruction

Objective To investigate the impact of graft position shift on anterior cruciate ligament reconstruction induced by femoral fixation of interference screw. Methods Nineteen fresh cadaveric knees were used and assigned to three groups. 1） Study of graft position shift： 5 knees were randomly selected, interference screws of 7 mm, 8 mm and 9 mm were used in autologous tendon fixation, then the graft position shift were measured. 2） Study of isometry： 7 knees were randomly divided into the isometric reconstruction group （D group）. In the D group, Retrobutton, interference screw and interference screw in location-corrected bone tunnel were used respectively as fixation. The isometry of grafts was evaluated. 3） Study of tibia anterior translation： 7 knees were randomly divided into the anatomic reconstruction group （J group）. In the J group,the tibia anterior translation was measured in four different conditions in the same joint： intact knee joint,knee joint without ACL, ACL anatomic reconstruction by interference screw fixation, and ACL anatomic reconstruction by interference screw fixation with corrected bone tunnel location. Results 1） With 7 mm, 8mm and 9 mm interference screw fixation, graft position shift were （2.36±0.11） mm, （2.72±0.06） mm and （3.00±0.06） mm respectively. 2） Graft length change： graft length change in Retrobutton group and corrected bone tunnel group were less than 3 mm, while graft length change in those fixed with interference screw were stretched in more than 3 mm. 3） Study of tibia anterior translation： there was no difference among the intact group, the anatomic group and the corrected group at 0° and 15°. However, the difference was found between the intact group and other groups at 30°、60° and 90° of flexion, as well as between these two reconstructed methods at 20° joint flexion （P＜0.05）. Conclusion In both isometric and anatomic ACL reconstruction with interference screw, the graft is pushed tightly toward the femoral tunnel wall, which shifts the graft away from the desired position. In our study we find out that the corrected location of the femoral bone tunnel significantly improves the isometry of ACL reconstruction and anatomic reconstruction. Key words: Anterior cruciate ligament; Transplants; Exercise; Biomechanics

2-primary Anick fibrations

Cohen conjectured that for r>=2 there is a space T^2n+1(2^r) and a homotopy fibration sequence Loop^2 S^2n+1 --> S^2n-1 --> T^2n+1(2^r) --> Loop S^2n+1 with the property that the left map composed with the double suspension, Loop^2 S^2n+1 --> S^2n-1 --> Loop^2 S^2n+1, is homotopic to the 2^r-power map. We positively resolve this conjecture when r>=3. Several preliminary results are also proved which are of interest in their own right.

L-infinity maps and twistings

We give a construction of an L-infinity map from any L-infinity algebra into its truncated Chevalley-Eilenberg complex as well as its cyclic and A-infinity analogues. This map fits with the inclusion into the full Chevalley-Eilenberg complex (or its respective analogues) to form a homotopy fiber sequence of L-infinity-algebras. Application to deformation theory and graph homology are given. We employ the machinery of Maurer-Cartan functors in L-infinity and A-infinity algebras and associated twistings which should be of independent interest.

Homotopy fiber sequences induced by 2-functors

This paper contains some contributions to the study of the relationship between 2-categories and the homotopy types of their classifying spaces. Mainly, generalizations are given of both Quillen’s Theorem B and Thomason’s Homotopy Colimit Theorem to 2-functors.

Hermitian K-theory and 2-regularity for totally real number fields

We completely determine the 2-primary torsion subgroups of the hermitian K-groups of rings of 2-integers in totally real 2-regular number fields. The result is almost periodic with period 8. Moreover, the 2-regular case is precisely the class of totally real number fields that have homotopy cartesian “Bökstedt square”, relating the K-theory of the 2-integers to that of the fields of real and complex numbers and finite fields. We also identify the homotopy fibers of the forgetful and hyperbolic maps relating hermitian and algebraic K-theory. The result is then exactly periodic of period 8 in the orthogonal case. In both the orthogonal and symplectic cases, we prove a 2-primary hermitian homotopy limit conjecture for these rings.

Adams Operations on Higher Arithmetic $K$-theory

We construct Adams operations on the rational higher arithmetic K -groups of a proper arithmetic variety. The definition applies to the higher arithmetic K -groups given by Takeda as well as to the groups suggested by Deligne and Soulé, by means of the homotopy groups of the homotopy fiber of the regulator map. They are compatible with the Adams operations on algebraic K -theory. The definition relies on the chain morphism representing Adams operations in higher algebraic K -theory given previously by the author. It is shown that this chain morphism commutes strictly with the representative of the Beilinson regulator given by Burgos and Wang.

Bundles, cohomology and truncated symmetric polynomials

The cohomology of the classifying space BU(n) of the unitary group can be identified with the the ring of symmetric polynomials on n variables by restricting to the cohomology of BT , where T\subset U(n) is a maximal torus. In this paper we explore the situation where BT = (\ CP^{\infty})^n is replaced by a product of finite dimensional projective spaces (\ CP^d)^n , fitting into an associated bundle U(n)\times_T (\ S^{2d+1})^n\to (\ CP^d)^n\to BU(n). We establish a purely algebraic version of this problem by exhibiting an explicit system of generators for the ideal of truncated symmetric polynomials. We use this algebraic result to give a precise descriptions of the kernel of the homomorphism in cohomology induced by the natural map (\ CP^d)^n\to BU(n) . We also calculate the cohomology of the homotopy fiber of the natural map E S_n\times_{S_n}(\ CP^d)^n\to BU(n) .