Simplicial Objects Research Articles

Simplicial degrees of functors

The main result of this paper is to show that if G∗ is a simplicial group of finite length, then HnG∗ also has finite length. Here the length of a simplicial group means the length of the corresponding Moore normalization and HnG∗ is the simplicial abelian group given by [k][map ]HnGk. A similar fact is true if we replace G∗ by a simplicial ring and we take the algebraic K-functors instead of group homology.The origin of such results goes back to the classical paper of Dold and Puppe (see Hilfsatz 4·23 of [7]), where the following was proved: letformula herebe a functor between abelian categories of degree d, meaning that the (d+1)st cross-effect functor in the sense of Eilenberg and MacLane [9] vanishes. Then l(TX∗)[les ]dl(X∗) for any simplicial object X∗ in A. Here l(X∗) denotes the length of X∗. Actually, this property characterizes the degree of functors in the abelian case (see Lemma 3·6).One can modify the notion of the cross-effects in the nonabelian framework (see [2]) and define the notion of degree of functors. One can also use the above inequality to define the simplicial degree in the nonabelian set-up. However in this way we get generally different invariants and the aim of this paper is to establish the relationship between the different notions. We show that many classical functors arising in homological algebra and K-theory have finite simplicial degree. Our Conjecture 4·7 claims that this should be always the case.We remark that if the Moore normalization of a simplicial group G∗ is zero in dimensions >k then πiG∗=0 for i>k as well. Therefore, for a functor T of simplicial degree d, one hasformula hereOur work can therefore be considered as a new general method proving vanishing results. Here is a sample application of the main result.

An axiomatic approach to equivariant cohomology theories

This paper presents a translation of a theorem of Cartan into an equivariant setting. This work is largely based on the study of the homotopical algebra in the sense of Quillen of the category of simplicial objects over the category of rationalO g-vector spaces. The application is a solution to the equivariant commutative cochain problem. This solution is slightly better than the solution obtained earlier by Triantafillou in that the transformation groupG need not be finite.

The cobar construction as a Hopf algebra

We show that the integral cobar construction ΩC * X of a 1-reduced simplicial set X has the structure of a homotopy Hopf algebra. This leads to a rational differential Lie algebra ℒ(X) given by a differential d ℒ on the free Lie algebra L(s −1 * X). The universal enveloping algebra Uℒ(X) is naturally isomorphic to ΩC * X⊗ℚ. Let X be a 1-reduced simplicial set and let s −1 * X be the desuspension of the reduced chain complex * X of X. The cobar construction is given by a differential d Ω on the tensor algebra T(s −1 * X) defined over ℤ. The homology of ΩC * X coincides with the homology of the loop space Ω|X|. A formula for d Ω was given by Adams [Ad]. We show that the cobar construction ΩC * X has over ℤ the additional natural structure of a “homotopy Hopf algebra” consisting of a coassociative diagonal and a derivation homotopy E:?ψ≃ψ. Explicit formulas for ψ and E are given in (2.9). The result is based on the geometric cobar construction in [B]. Using the fundamental theory of Anick [A] we derive from the data (ΩC * X,ψ,E) a differential Lie algebra which is defined by a differential d ℒ on the free rational Lie algebra L(s −1 * X). The homology of ℒ(X) coincides with the rational homotopy groups of Ω|X|. Moreover we describe a cocommutative diagonal together with a derivation homotopy H:ψ⊗ℚ≃φ and a natural iso- morphism of differential Hopf algebras Formulas for φ, H and β are obtained by an induction. A map f:X→Y between 1-reduced simplicial sets induces the map f *=L(s −1 * f): L(s −1 * X)→L(s −1 * Y) between free Lie algebras which is compatible with the differential d ℒ. Hence X↦ℒ(X) yields a canonical functor from 1-reduced simplicial sets to free differential Lie algebras. This functor induces an isomorphism of rational homotopy categories. In fact, ℒ is up to homotopy naturally equivalent to the functor λ=NPG of Quillen [Q] which is the composite of four intricate functors. We describe in [B”] inductively a sequence of elements β(2),β(3),… which determine the differential d ℒ and we compute the quadratic part of d ℒ. A complete formula for d ℒ, however, is highly intricate since it can be considered as a kind of “simplicial form” of the classical Hausdorff formula; see (4.9) (4). We cannot expect a simple form of the differential d ℒ since the homology of ℒ(X) yields the rational homotopy groups of X; that is This supplements the result of Kan [K] who gave a combinatorial description of π*(X) in terms of the free simplicial group GX. Hence π*(X) requires computations in free groups while the formula for π*(X)⊗ℚ requires only the computation of the homology of a chain complex of rational vector spaces. We point out that ℒ(X) is a chain complex of finite type if X has only finitely many non degenerate simplices. Hence in this case d ℒ yields a finite formula for π n (X)⊗ℚ. Such a formula was not yet achieved in the literature. For example using the Sullivan method one obtains π*(X)⊗ℚ only implicitly by computing a minimal model of the De Rham algebra A(X); see Bousfield-Gugenheim [BG]. Here A(X) is defined explicitly in terms of X but is not of finite type if X has only finitely many non degenerate simplices so that A(X) does not give a finite formula for π n (X)⊗ℚ. In this case the minimal model M X of A(X) is of finite type and hence also gives a finite formula for π n (X)⊗ℚ. There is, however, no explicit formula for M X in terms of X in the literature and M X is not natural in X. On the other hand also the model λ(X) of Quillen is considerably larger than ℒ(X) above and is only defined as the normalization of a simplicial object; see (4.9) (4) below.

Simplicial objects in algebraic topology, by P. May. Pp 161. £19-25. 1993. ISBN 0-226-51181-2 (University of Chicago Press)

Simplicial objects in algebraic topology, by P. May. Pp 161. £19-25. 1993. ISBN 0-226-51181-2 (University of Chicago Press)

Odd primary Steenrod operations in first-quadrant spectral sequences

This paper defines two kinds of Steenrod operations in the spectral sequence of a bisimplical mod p coalgebra and shows them to be a complete list of all such possible Steenrod operations. These operations are compatible with the differentials and with Steenrod operations on the total complex. A general rule is given for computing the operations on E2. A generalization of the Kudo transgres- sion theorem is also proved, placing it in a larger and more natural setting. 1. Introduction. In 1955, Massey (9) asked if it were possible to introduce Steenrod squares and/ith powers into the spectral sequence of a fibre space. Since that time several people have contributed to the solution of this problem, notably Vazquez (22), Araki (1), Kristensen (5), Kuo (7), Rector (15), Smith (18), Singer (16), Mori ((12) and (13)), and Dwyer (4). This paper will define mod p Steenrod operations on a class of spectral sequences; its sequel will define higher divided powers in certain spectral sequences. These operations are shown to be a complete list of all Steenrod-type operations on spectral sequences. This extends to the odd prime case the work of Singer (16) on Steenrod squares in first-quadrant spectral sequences and the work of Dwyer (4) on higher divided squares in second quadrant spectral sequences. Mori ((12) and (13)) has also obtained most of our results for the odd prime case of the Eilenberg-Moore spectral sequence through different methods. In the course of these papers the ambiguity regarding indeterminacy in the definition of the operations in (16), (12), and (13) is resolved, and the question asked by Singer (16) regarding the existence of horizontal operations in the spectral sequence of a mixed simplicial object is answered. A generalization of the Kudo transgression theorem (6) is also proved, placing the usual Kudo transgression theorem in a larger and more natural setting. The principal applications of these results are to the Eilenberg-Moore and the Serre spectral sequences as well as to the change-of-rings spectral sequence as outlined by Singer in (17). Let A be the category of finite ordered sets and nondecreasing maps (10, p. 4) and let Aop be the opposite category. A bisimplicial coalgebra is a covariant functor from

In Defence of Eponymy

The polemic against eponymy in the Old Intelligencer (Volume 2, No. 4, p. 204) deserves rebuttal by all those who do not share its authors' bleak philosophy of a rigid and depersonalised mathematics fragmented into narrow specialities. The claim that eponyms "are in themselves meaningless, for they possess no descriptive content" certainly does not apply to the many cases where mention of mathematicians' names is sufficient to stimulate the associations of their fields of work, their schools, the theorems they proved, the problems they raised, and the ideas they introduced or popularised. To deny descriptive content here is to misunderstand the way language functions. On the other hand, the proposal that "a common word should be selected which has one or more connotations suggestive of the mathematical concept to be named, and this common word should then be assigned a precise technical meaning" has many disadvantages. One of the most serious from the (Atiyah! [ 1 ]) point of view of mathematics as a unified subject is that a single mathematical concept may appear in various guises in various branches of pure and applied mathematics, so that an appropriate vernacular description of one of its manifestations may be quite inappropriate, confusing, or meaningless as a description of the underlying idea in any of its other manifestations. A minor but typical example from personal experience concerns the choice of the eponymous term "Mal'cev varieties" for the classes of algebras studied in [2]. These classes are of interest to universal algebraists because congruences on such algebras are mutually permutable, to structural anthropologists because they model the logic of analogy, and to simplicial topologists because they are the classes of algebras in which all simplicial objects have the (eponymous) Kan property. The "Algebra Universalis" school's name "permutable varieties" for such classes (cf. [3]), fully in accord with Henwood and Rival's proposal, is totally baffling and devoid of descriptive power for topologists or structuralists unacquainted with the notion of a congruence on an algebra. About the author

Subjective geometry and geometric psychology

“Subjective geometry” is a term coined by Weintraub and Krantz to describe the distortion imposed upon geometric patterns by the visual system itself—so-called optical illusions. The latter are widely regarded as being generated by misplaced “constancy” effects, i.e., they are regarded as stemming from the invariance of an object's appearance under wide variations in viewing conditions, such as obliquity, rotations, etc. The invariances represented by these constancies—shape constancy, size constancy, etc.—are spatiotemporal invariants of certain Lie subgroups of P 4 ( R ) ⊕ CO (1, 3) ⊕ GL (4, R ) that govern Euclidean and non-Euclidean geometry. Tha Euclidean subgroups describe a Cyclopean visual world; the non-Euclidean, a binocular (bipolar) world of hyperbolic nature, according to the work of Luneburg, Blank, Indow, and others. The visual field of view is itself a geometric object involvingnot only “figure” and “ground” but also visual contours (orbits of the Lie groups involved), linear perspective, interposition, and contact and symplectic structures. The retina and “cortical retina” are both covered by a family of “circular-surround” cellular response fields (of a “Mexican hat” nature) which constitute an atlas for the visual manifold S . Upon this manifold are defined certain equivariant vector bundles that account for constancy phenomena and certain jet bundles, arising out of the vector bundles by prolongation, that generate the differential invariants characterizing higher form perception. The resultant theory of perceptual-cognitive processing has been termed “geometric psychology,” in analogy to MacLane's “geometrical mechanics” and Brockett–Hermann–Mayne's “geometry of systems,” the mathematical structure being very similar in all three instances. Functorial maps from the category G v FB(S) of equivariant fibre bundles to the simplicial category and the category of simplicial objects complete the theory by extending the perceptual system to cognitive phenomena and information-processing psychology.

On three types of simplicial objects

On three types of simplicial objects

RESOLUTIONS CORRESPONDING TO CLOSED MAPPINGS

We consider certain resolutions defined for any closed mapping (they determine certain new spectral sequences). We first prove the naturality of the resolutions in their arguments. Next, we show that when the coefficient domain is a ring, the term is a sheaf of rings, and all the for are -modules. This allows us to determine typical conditions for the resolution to be soft. It is shown that the resolution is a simplicial object in the sense of Eilenberg?Zilber, and a direct definition is given for multiplication in . For completeness, an outline is given for the proof of the following fact (communicated to the author by A. V. Zarelua): in the case of a regular finite-sheeted covering, the spectral sequence corresponding to the resolution is isomorphic to the Cartan spectral sequence. Bibliography: 8 items.

A general theory of relative regular neighborhoods

Regular neighborhoods of polyhedra in manifolds have been discussed in many papers, most notably, in the present context, in [W], [H-Z], [H] and [S]. We study regular neighborhoods of X mod Y in Z where (Z, X, Y) is an arbitrary triad of finite dimensional, locally compact polyhedra. When applied to manifolds our theory generalizes previous results in that we make no compactness assumptions and place no restrictions on the polyhedra X and Y to be considered. The paper is organized as follows: 1. Definitions and notation 2. Simplicial preliminaries 3. The uniqueness theorem 4. The characterization of collared subpolyhedra 5. Cone-retracting and collaring regular neighborhoods 6. The stellar neighborhood theorem 7. Relationships between regular neighborhoods 8. Collapsibility and collapsible retractions 9. Regular neighborhoods via collapsibility. V is defined to be a regular neighborhood of X mod Y in Z if there exists a full triangulation (J, K, L; h) of (Z, X, Y) in which V underlies a relative first derived neighborhood, V=hN(K-L, J'). Thus, on the one hand, a given regular neighborhood can be presented in an explicit manner with a fixed triangulation. On the other hand, regular neighborhoods are piecewise linear rather than simplicial objects (there exists a triangulation) and there are uncountably many regular neighborhoods of X mod Y in Z implicit in the choice of triangulating complex and triangulating homeomorphism. We develop the theory by first exploiting the seeming concreteness (??2-5) and then developing the implicit generality (??6-9). After presenting the basic simplicial data we turn immediately to the uniqueness theorem (3.1). (Existence is automatic.) This asserts that, given two regular neighborhoods of X mod Y in Z, there is an ambient piecewise linear isotopy taking one onto the other and keeping X u Y fixed. The proof is quite direct and carries a great deal of extra information. For example (3.4) if Z is an n-manifold the isotopy can be realized by a sequence of 2n moves relative to X u Y.