Joseph Roitberg

Homotopy-epimorphisms, homotopy-monomorphisms and homotopy-equivalences

Abstract Two variants of a classical theorem of J.H.C. Whitehead are established. The first is used to infer that a pointed map ƒ:X → Y of pointed, path-connected CW-spaces which is both a homotopy-epimorphism and a homotopy-monomorphism is a homotopy-equivalence and the second characterizes the plus construction in homotopy theory.

Phantom maps and spaces of the same n-type for all n

Abstract This paper develops the connection between the set of phantom maps from X to Y and the set of homotopy types W having the same n -type as Z , for all n , where Z = X × ΩY or Y ∨ σX . Using recent work making calculations of the space of phantom maps possible, we can give explicit constructions of various W s.

Note on sequences of Mayer-Vietoris type

AMsmAcr. In this largely expository note, we reexamine the construction of the homotopical Mayer-Vietoris sequence associated to a homotopy pullback. We show that in this situation, the Mayer-Vietoris sequence may be realized simply as the homotopy sequence of a suitable fibration. The usual approaches to constructing the Mayer-Vietoris sequence involve some auxiliay algebraic result, such as the Barratt-Whitehead lemma; the present approach avoids any such considerations. An additional beneficial feature of our approach is the attention paid to the bottom end of the Mayer-Vietoris sequence. Thus we are led to a cleaner proof of Proposition II.7.11 of [HMRJ; moreover, we show that the converse of this latter result is also true. The homological Mayer-Vietoris sequence associated to a homotopy pushout may be established in a very similar manner, as we point out at the end of the paper.

The Lusternik–Schnirelmann category of certain infinite CW-complexes

Abstract Examples are constructed to illustrate: (i) The LS category of a 1-connected, finite type CW-complex X which is the homotopy colimit of a sequence X1→X2→X3→⋯ of 1-connected, finite CW-complexes may exceed the LS category of each Xi; and (ii) LS category is not an invariant of the localization genus of a 1-connected, finite type CW-complex.

Phantom maps and torsion

Abstract For 1-connected, finite type CW-spaces X and Y with Y a loop space, the group Ph( X , Y ) of pointed homotopy classes of phantom maps from X to Y is divisible abelian. We obtain sharpened and generalized versions of examples demonstrating that the group Ph( X , Y ) may possess torsion. Moreover, we give a complete and explicit description of those divisible abelian groups that may be realized as Ph( X , Y ) for suitable X and Y .

PHANTOM MAPS, COGROUPS AND THE SUSPENSION MAP

ABSTRACT The set Ph(X, Y) of pointed homotopy classes of phantom maps from X to Y admits a natural group structure if either Y is a grouplike space or X is a cogroup. In the present paper, the group structure on Ph(X,Y) is examined in the second case. (The first case was examined in an earlier paper.) The results in the two cases are similar—for instance, the group structure turns out to be abelian, divisible and independent of the grouplike structure on Y or the cogroup structure on X—but the techniques used to establish the results differ substantially in the two cases. In addition, a study of the map g*: Ph(X,Y1) → Ph(X,Y2) induced by a map g: Y1 → Y2 of grouplike spaces is initiated. A particularly interesting special case of this situation is the suspension map Ph(X, Y) → Ph(X, ΩσY) ≅ Ph(σX, σY) with Y a grouplike space.

The homotopy fiber of profinite completion

Abstract We show that the basepoint-component of the homotopy fiber of Sullivans profinite completion map c :Y→ Y is always an H-space, provided Y is nilpotent of finite type. This observation allows for a reworking of the completion/rationalization approach to phantom map theory and leads to more direct, transparent proofs of various results in the recent literature on phantom maps.

Rationally equivalent nilpotent groups and spaces

Examples are constructed of rationally isomorphic, finitely generated nilpotent groups L and M such that there are no homomorphisms L → M,M → L inducing rational isomorphisms. Similar examples are constructed of nilpotent, or even simply-connected, finite CW-complexes.

Automorphism groups of nilpotent groups and spaces

Abstract A pair of finitely generated, torsion-free nilpotent groups G 1 , G 2 is constructed with the properties that G 1 and G 2 are p -isomorphic for all primes p , yet Aut( G 1 ) and Aut( G 2 ) are not isomorphic. The example constructed is compared to an analogous example in the homotopy category of simply connected, finite CW -complexes.

The Mislin genus, phantom maps and classifying spaces

Abstract For a finite type, nilpotent space X , we prove that the cardinality of the set Ph( X , Y ), where Ph(−, −) denotes homotopy classes of phantom maps, depends only on the Mislin genus of X , at least if Y has countable higher homotopy groups. In the special case where X = BG , the classifying space of a 1-connected Lie group G , and Y is the iterated loop space of a 1-connected, finite CW-complex, we prove the stronger result that the isomorphism class of the group Ph( X , Y ) depends only on the Mislin genus of X . The latter strengthening depends on two results of independent interest 1. (i) Under a fairly mild connectivity condition on X , the torsionization of X , that is the homotopy fiber of a rationalization map X → X (0) , is a Mislin genus invariant; 2. (ii) the torsionization of BG , localized away from a prime p , is homotopy equivalent to the plus construction applied to a space of the form BΛ , where λ is a suitable locally finite, perfect (discrete) group.