Sullivan Conjecture Research Articles

On Seymour's and Sullivan's second neighbourhood conjectures

AbstractFor a vertex of a digraph, (, respectively) is the number of vertices at distance 1 from (to, respectively) and is the number of vertices at distance 2 from . In 1995, Seymour conjectured that for any oriented graph there exists a vertex such that . In 2006, Sullivan conjectured that there exists a vertex in such that . We give a sufficient condition in terms of the number of transitive triangles for an oriented graph to satisfy Sullivan's conjecture. In particular, this implies that Sullivan's conjecture holds for all orientations of planar graphs and triangle‐free graphs. An oriented graph is an oriented split graph if the vertices of can be partitioned into vertex sets and such that is an independent set and induces a tournament. We also show that the two conjectures hold for some families of oriented split graphs, in particular, when induces a regular or an almost regular tournament.

Homotopical Presentations and Calculations of Algebraic $K_0$-Groups for Rings of Continuous Functions

Let K_0(C_{\Bbb F}(X)) = K_0\circ C_{\Bbb F}(X) be the K_0 -group of the ring C_{\Bbb F}(X) of {\Bbb F} -valued continuous functions on a topological space X , where {\Bbb F} is the field of real or complex numbers or the quaternion algebra. It is known that the functor K_0\circ C_{\Bbb F} is representable on the category of compact Hausdorff spaces. It is a homotopy functor which is not representable on the category of topological spaces. Making use of the compactly-bounded homotopy set, which is a variant of the homotopy set, the functor K_0\circ C_{\Bbb F} has a homotopical presentation by the product of the ring of integers {\Bbb Z} and the infinite Grassmannian G_{\infty}(\Bbb F ) . This presentation makes it possible to calculate the groups K_0(C_{\Bbb F}(X)) explicitly for some infinite dimensional complexes X by use of the results of H. Miller on Sullivan conjecture.

Milnor–Wood inequalities for manifolds locally isometric to a product of hyperbolic planes

This Note describes sharp Milnor–Wood inequalities for the Euler number of flat oriented vector bundles over closed Riemannian manifolds locally isometric to products of hyperbolic planes. One consequence is that such manifolds do not admit an affine structure, confirming Chern–Sullivan's conjecture in this case. The manifolds under consideration are of particular interest, since in contrary to some other locally symmetric spaces they do admit interesting flat vector bundles in the corresponding dimension. When the manifold is irreducible and of higher rank, it is shown that flat oriented vector bundles are determined completely by the sign of the Euler number. To cite this article: M. Bucher, T. Gelander, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

HOMOTOPY FIXED POINT SETS AND ACTIONS ON HOMOGENEOUS SPACES OF p-COMPACT GROUPS

We generalize a result of Dror Farjoun and Zabrodsky on the relationship between fixed point sets and homotopy fixed point sets, which is related to the generalized Sullivan Conjecture. As an application, we discuss extension problems considering actions on homogeneous spaces of p-compact groups.

Degenerate fibres in the Stone-Cech compactification of the universal bundle of a finite group

Applied to a continuous surjection π: E → B of completely regular Hausdorff spaces E and B, the Stone-Cech compactification functor β yields a surjection βπ: βE → βB. For an n-fold covering map π, we show that the fibres of βπ, while never containing more than n points, may degenerate to sets of cardinality properly dividing n. In the special case of the universal bundle π: EG → BG of a p-group G, we show more precisely that every possible type of G-orbit occurs among the fibres of βπ. To prove this, we use a weak form of the so-called generalized Sullivan conjecture.

Constructing Compacta of Different Extensional Dimensions

AbstractApplying the Sullivan conjecture we construct compacta of certain cohomological and extensional dimensions.

Homotopy Lie groups

Homotopy Lie groups, recently invented by W.G. Dwyer and C.W. Wilkerson [13], represent the culmination of a long evolution. The basic philosophy behind the process was formulated almost 25 years ago by Rector [32, 33] in his vision of a homotopy theoretic incarnation of Lie group theory. What was then technically impossible has now become feasible thanks to modern advances such as Miller’s proof of the Sullivan conjecture [25] and Lannes’s division functors [22]. Today, with Dwyer and Wilkerson’s implementation of Rector’s vision, the tantalizing classification theorem seems to be within grasp. Supported by motivating examples and clarifying exercises, this guide quickly leads, without ignoring the context or the proof strategy, from classical finite loop spaces to the important definitions and striking results of this new theory.

Infinite dimensional compacta having cohomological dimension two: an application of the Sullivan conjecture

IN [S], A. N. Dranishnikov produced examples of infinite dimensional metric compacta that have integral cohomological dimension equal to three. These examples established the distinct nature of these classical dimension theories, settling the problem, posed by P. S. Alexandroff in 1932 [l], of whether the integral cohomological dimension of a compact metric space is the same as its covering dimension. The techniques used in [S] shed little light on whether or not there could be such an example having integral cohomological dimension two. The infinite dimensionality of the examples in [S] is, ultimately, detected using the vanishing of the reduced complex K-theory with Z/p coefficients (p a prime) of an Eilenberg-MacLane complex K(Z, 3). While reduced complex K-theory with Z/p coefficients of an Eilenberg-MacLane complex K(Z, n) vanishes for n 2 3, the same is not true for K(Z, 2)). (The reader is referred to [2] and [4] for details of these K-theoretic assertions.) The specific nature of K-theory itself plays no direct role in the analyses in [S]. The essential feature is that it is a generalized cohomology theory for which K(Z, 3) behaves as a point; i.e., the reduced K-theory of K(Z, 3) is trivial. Actually, the latter is not true for complex K-theory itself but is valid when Z/p coefficients (p a prime) are used (see [2], [4]). The absence of readily available generalized cohomology theory for which K(Z, 2) is known to behave as a point requires an alternate approach to that in [S] in order to produce an infinite dimensional compact metric space having integral cohomological dimension equal to two. (In addition, the generalized cohomology theory would have to have the property that, in each dimension, its value on a finite complex is a finite group, the latter is true for K-theory with jinite coeficients.) An overview of the approach providing an alternate method to that in [S] and producing cohomological dimension two examples is:

On Stable Maps From Eilenberg-Maclane Spaces to Classifying Spaces

0. Introduction. In recent years, there has been a general interest in the problem of understanding the homotopy type of function spaces Map (X, Y) where X is an infinite complex. The special case where X is the classifying space of a finite group gives the celebrated Segal's conjecture [4] when Y = lim fQnn = QS' and Sullivan's conjecture [10] when Y is a finite complex. One would like to know whether the affirmative solution to these conjectures can also be applied to the study of Map (X, Y) for other infinite complexes X. The philosophy behind this approach to the problem has already manifested itself in [5], [6], [7] and [8] among others to give interesting results. In this note, we will utilize the solution to Segal's conjecture to compute stable maps from Eilenberg-MacLane spaces to classifying spaces. As an application, we will also describe the homotopy type of Map (X, Y) at a prime p where X = K(1T, n), wT finite, n > 2 and Y = Q(G), G = 0, U, Sp. The original motivation behind this paper was due to Anderson & Hodgkin [1]. There they proved vanishing results on the topological K-theory of higher Eilenberg-MacLane spaces using Atyiah's theorem on the K-theory of the classifying spaces of compact Lie groups. We shall prove that the same result holds for stable cohomotopy groups. The author is especially grateful to Professor Gunnar Carlsson for pointing out [1] and how Segal's conjecture comes into play.

Strongly torsion generated groups

It has long been known that the integral homology of a non-trivial finite group must be non-zero in infinitely many dimensions 17. Recent work on the Sullivan conjecture in homotopy theory has made it possible to extend this result to locally finite groups. For more general groups with torsion it becomes more difficult to make such a strong statement. Nevertheless we prove that when a non-perfect group is generated by torsion elements its integral homology must also be non-zero in infinitely many dimensions. Remarkably, this result is best possible, in that for perfect torsion generated groups all (finite or infinite) sequences of abelian groups are shown below to be attainable as higher homology groups.

Equivariant stable homotopy and Sullivan's conjecture

Let G be a p-group, and let X be a G-complex. Let EG denote a contractible space on which G acts freely. By the "homotopy fixed point set" of X, we mean the fixed point set F(EG, X) G, where F(EG, X) denotes the function space of all maps EG ~ X, equipped with a G-action by 9f = 9f9-1. If we let * denote the one point space with trivial G-action, we may also consider the G-space F ( , ,X ) ; it is canonically G-homeomorphic to X. The G-map E G ~ , induces a map ~/: X G ~F( , , X) G. In [19], D. Sullivan proposed the following.

Fibrewise completion and unstable Adams spectral sequences

We describe a tower of spaces whose inverse limit is a “fiberwise completion” of a fibrationE →B, and study the resulting spectral sequence converging to the homotopy groups of the space of lifts of a mapX →B. This is used to give a proof of the “generalized Sullivan conjecture”.

On phantom maps and a theorem of H. Miller

A mapf:X →Y is a phantom map if any composition off with a map from a finite complex intoX is null homotopic. The proof of the Sullivan conjecture by H. Miller enables us to understand more deeply this phenomena. We prove, among other things, that any map from a space with finitely many non-vanishing homotopy groups into a finite complex is phantom and that any fibration over a 2-connected space with finitely many non-vanishing homotopy groups and with fiber a finite complex is trivial over each skeleton of the base.

Locally finite approximation of Lie groups. II

In an earlier paper [10], we constructed a ‘locally finite approximation away from a given prime p’ of the classifying space BG of a Lie group with finite component group. Such an approximation consists of a locally finite group g and a homotopy class of maps which in particular induces an isomorphism in cohomology with finite coefficients of order prime to p. The usefulness of such a construction is that it reduces various homotopy-theoretic questions concerning the space BG to the corresponding questions concerning Bπ for finite subgroups π. For example, we demonstrated in [10] how H. Miller's proof of the Sullivan conjecture concerning maps from , where π is a finite group and X is a finite-dimensional complex, can be extended to maps BG→X for G a Lie group with finite component group.

A propos de conjectures de Serre et Sullivan

We first give a new proof of a conjecture of J.-P. Serre on the homotopy of finite complexes, which was recently proved by C. McGibbon and J. Neisendorfer. The emphasis is on a property of the mod. 2 homology of certain spaces: their “quasi-boundedness” as right modules over the Steenrod algebra. This property is preserved when one goes from a simply connected space to its loop space and also when one takes a covering of anH-space. Then we show that this notion of quasi-boundedness simplifies H. Miller's proof of D. Sullivan's conjecture on the contractibility of the space of pointed maps from the classifying space of the groupe ℤ/2 into a finite complex.

Correction to "The Sullivan Conjecture on Maps from Classifying Space"

On page 49, I assert that given an unstable coalgebra C over the mod p Steenrod algebra A, C E CA, the module of primitives PC is the suspension of an unstable A-module: E 'PC E U. This is true for p = 2, but certainly false for p > 2: Consider, for instance, the A-coalgebra H* BZP which plays such a large role in this paper. This error makes nonsense of the spectral sequence (2.5), Ext' ( M, 1RP (Cp)) = Ext s +,(E Ma C), which is central to the proof of the Sullivan conjecture. However, we may proceed as follows. Consider, instead of the category U, the category V of A-modules satisfying the modified unstable condition xPt= Ofor lxi < 2pt,

The Sullivan Conjecture on Maps from Classifying Spaces

The Sullivan Conjecture on Maps from Classifying Spaces