Martin Arkowitz

Self-homotopy equivalences which induce the identity on homology, cohomology or homotopy groups

Abstract For a based, 1-connected, finite CW-complex X, we study the following subgroups of the group of homotopy classes of self-homotopy equivalences of X: e ∗ (X) , the subgroup of homotopy classes which induce the identity on homology groups, e ∗ (X) , the subgroup of homotopy classes which induce the identity on cohomology groups and e#dim + r(X), the subgroup of homotopy classes which induce the identity on homotopy groups in dimensions ⩽ dim X + r. We investigate these groups when X is a Moore space and when X is a co-Moore space. We give the structure of the groups in these cases and provide examples of spaces for which the groups differ. We also consider conditions on X such that e ∗ (X) = e ∗ (X) and obtain a class of spaces (including compact, oriented manifolds and H-spaces) for which this holds. Finally, we examine e#dim + r(X) for certain spaces X and completely determine the group when X = Sm × Sn and X = CPn ∨ S2n.

CHAPTER 23 – Co-H-Spaces

This chapter deals with Co-H-spaces. Co-H-spaces are important for at least two reasons. First of all, they are the duals in the sense of Eckmann and Hilton of H-spaces. The latter have played a significant and central role in topology for many years. Second, there is a large class of examples, namely the suspensions, which are co-H-spaces. It is the co-H-structure that allows addition of homotopy classes. It also presents maps defined on a suspension. This gives rise to the homotopy groups of a space. This chapter surveys the known results on co-H-spaces. It also provides sketches of the proofs of the major theorems and many illustrative examples. Although suspensions enter naturally into the exposition, the main focus is on co-H-spaces.

The Lefschetz-Hopf theorem and axioms for the Lefschetz number

The reduced Lefschetz number, that is, where denotes the Lefschetz number, is proved to be the unique integer-valued function on self-maps of compact polyhedra which is constant on homotopy classes such that (1) for and ; (2) if is a map of a cofiber sequence into itself, then ; (3) , where is a self-map of a wedge of circles, is the inclusion of a circle into the th summand, and is the projection onto the th summand. If is a self-map of a polyhedron and is the fixed point index of on all of , then we show that satisfies the above axioms. This gives a new proof of the normalization theorem: if is a self-map of a polyhedron, then equals the Lefschetz number of . This result is equivalent to the Lefschetz-Hopf theorem: if is a self-map of a finite simplicial complex with a finite number of fixed points, each lying in a maximal simplex, then the Lefschetz number of is the sum of the indices of all the fixed points of .

Noncommutativity of the group of self homotopy classes of Lie groups

Abstract Let G be a topological group and let [G,G] be the group of homotopy classes of maps from G into G. For a large class of simple Lie groups, we prove that the group [G,G] is non-abelian. For certain Lie groups we show that nil[G,G]⩾3.

Equivalence classes of homotopy-associative comultiplications of finite complexes

Let X be a finite, 1-connected CW-complex which admits a homotopy-associative comultiplication. Then X has the rational homology of a wedge of spheres, Sn1 + 1 V … V Snr + 1. Two comultiplications of X are equivalent if there is a self-homotopy equivalence of X which carries one to the other. Let ba(X), respectively bac(X), denote the set of equivalence classes of homotopy classes of homotopy-associative, respectively, homotopy-associative and homotopycommutative, comultiplications of X. We prove the following basic finiteness result: Theorem 6.1 (1) If for each i, (a) ni ≠ nj + nk for every j, k with j < k and (b) ni ≠ 2nj for every j with nj even, then ba(X) is finite. (2) bac(X) is always finite. The methods of proof are algebraic and consist of a detailed examination of comultiplications of the free Lie algebra π#(ΩX) ⊗ Q. These algebraic methods and results appear to be of interest in their own right. For example, they provide dual versions of well-known results about Hopf algebras. In an appendix we show the group of self-homotopy equivalences that induce the identity on all homology groups is finitely generated.

On the Nilpotency of Subgroups of Self-Homotopy Equivalences

If X is a topological space, we denote by e(X) the set of homotopy classes of self-homotopy equivalences of X. Then e(X) is a group with group operation given by composition of homotopy classes. The group e(X) is a natural object in homotopy theory and has been studied extensively—see [Ar] for a survey of known results and applications of e(X). In this paper we continue our investigation of e #(X), the subgroup of e(X) consisting of homotopy classes which induce the identity on homotopy groups, and, to a lesser extent, of e *#(X), the subgroup of e #(X) consisting of homotopy classes which also induce the identity on homology groups (see §2 for precise definitions), which was begun in [A-L]. These groups are nilpotent and we focus primarily on the nilpotency class of e #(X). The determination of this nilpotency class appears in the list of problems on e(X) in [Ka, Problem 10]. For rational spaces we obtain both general results on the nilpotency class and a complete determination of the nilpotency class in specific cases. This leads to a lower bound for the nilpotency class of the groups e #(X) for certain finite complexes X by using derationalization techniques.

Formal differential graded algebras and homomorphisms

Abstract Let B be a differential graded algebra over the rationals (DGA), M a minimal DGA, [ M , B ] the homotopy classes of DGA maps M → B , and I :[ M , B ]→Hom( H *( M ), H *( B )) the function which assigns the induced cohomology homomorphism to a homotopy class. Theorem. If M and B are formal, then I restricted to the homotopy classes of formal maps is a bijection. This theorem has several diverse consequences including results on the group of homotopy classes of homotopy equivalences of a formal DGA and results on the suspension Σ:[ X , Y ]→[Σ X ,Σ Y ] co-H when X and Y are formal spaces.

Homotopy classes that are trivial mod F

IfF is a collection of topological spaces, then a homotopy class in (X;Y ) is called F -trivial if =0:( A;X)! (A;Y ) for all A2F. In this paper we study the collection ZF (X;Y )o f all F - trivial homotopy classes in (X;Y )w hen F = S, the collection of spheres, F = M, the collection of Moore spaces, and F = , the collection of suspensions. Clearly Z(X;Y )ZM(X;Y )ZS(X;Y ); and we nd examples of nite complexes X and Y for which these inclu- sions are strict. We are also interested in ZF(X )= ZF (X;X), which under composition has the structure of a semigroup with zero. We show that if X is a nite dimensional complex and F =S , M or , then the semigroup ZF (X) is nilpotent. More precisely, the nilpotency of ZF (X) is bounded above by theF -killing length of X , a new numerical invariant which equals the number of steps it takes to make X contractible by successively attach- ing cones on wedges of spaces in F , and this in turn is bounded above by the F -cone length of X. We then calculate or estimate the nilpotency of ZF (X )w hen F = S , Mor for the following classes of spaces: (1) projective spaces (2) certain Lie groups such as SU(n )a ndSp(n). The paper concludes with several open problems. AMS Classication 55Q05; 55P65, 55P45, 55M30

Whitehead products as images of Pontrjagin products

A method is given for computing higher order Whitehead products in the homotopy groups of a space X. If X can be embedded in an H-space E such that the pair (E, X) has sufficiently high connectivity, then we prove that a higher order Whitehead product element in the homotopy of X is the homomorphic image of a Pontrjagin product in the homology of E. The two main applications determine a higher order Whitehead product element in (1) 7*(BUt), the homotopy groups of the classifying space of the unitary group U,, (2) the homotopy groups of a space with two nonvanishing homotopy groups.

Categories equivalent to the category of rational H-spaces.

AbstractThe category