Cellular spaces, null spaces and homotopy localization.

*(English)*Zbl 0842.55001
Lecture Notes in Mathematics. 1622. Berlin: Springer-Verlag. xiv, 199 p. (1995).

The book gives a clear presentation of new methods in homotopy theory. It is well written and well organized. Each chapter starts with the statements of the results, followed by comments, examples, and explanations. The proofs are placed at the end of each chapter and are rather easy to follow.

“The book describes a certain framework for doing homotopy theory in which the function complexes play a central role. This approach emerged in the early 1990’s but has roots in earlier work of Bousfield about localization and in the big advances made by Mahowald, Ravenel, Devinatz, Hopkins and Smith towards deeper understanding of the role of periodicity in stable homotopy theory.” The key constructions are the homotopy idempotent localization functor \(L_f\) with respect to a map \(f\), and the augmented homotopy idempotent functor \(CW_A\) with respect to a space \(A\). The functor \(L_f\) assigns to a space \(X\) the “smallest” quotient \(X \to L_f X\), such that if \(\text{map}(f,Y)\) is a weak equivalence, then so is \(\text{map} (L_f X, Y) \to \text{map} (X,Y)\). For an appropriate \(f\), \(L_f\) is the localization functor with respect to a given homology theory. Therefore studying the functors \(L_f\) gives a general framework for looking at homological localizations. The functor \(CW_A\) assigns to a space \(X\) the biggest subobject \(CW_A X \to X\) such that \(\text{map}_* (A, CW_A X) \to \text{map}_*(A, X)\) is a weak equivalence. For example if \(A\) is the \(\mathbb{Z}/p\)-Moore space, the functor \(CW_A\) assigns to \(X\) the universal space \(CW_A X\) which has the same mod-\(p\) homotopy groups as \(X\). The author presents constructions of continuous versions of the functors \(L_f\) and \(CW_A\).

An important issue of the book is the behavior of \(L_f\) and \(CW_A\) with respect to fibration sequences and mapping spaces, thus in particular with respect to loop spaces. The author proves that applying \(L_f\) and \(CW_A\) to a loop space produces again a loop space. Even more: \(L_f \Omega X \simeq \Omega L_{\Sigma f} X\) and \(CW_A \Omega X \simeq \Omega CW_{\Sigma A} X\) (Theorems 3.A.1 and 3.A.2). Therefore in order to study to what extent \(L_f\) preserves fibration sequences, in particular to study the difference between \(L_f \Omega X\) and \(\Omega L_f X\), one has to investigate the map \(L_{\Sigma f} X \to L_f X\). Similarly the behaviour of \(CW_A\) with respect to fibration sequences is encoded in the map \(CW_{\Sigma A} X \to CW_A X\). The author proves that if \(f\) is a suspension, the homotopy fiber of \(L_{\Sigma f} X \to L_f X\) is a GEM i.e., a possibly infinite product of Eilenberg-Mac Lane spaces (Proposition 5.F.4). He also shows that the homological localizations with respect to complex mod-\(p\) \(K\)-theory or Morava \(K\)-theories preserve fibrations of the form \(\Omega^2(F \to E \to B)\) up to an error term, which is a product of only three Eilenberg-Mac Lane spaces (Theorem 6.A.1). If \(A\) is a suspension,then it is shown that the homotopy fiber of \(CW_{\Sigma A} X \to CW_A X\) is a polyGEM i.e., it is an element of the smallest class of spaces that contains GEMs and is closed under taking the total space of any fibration sequence for which the fiber and the base are polyGEMs (Theorem 5.E.3).

This brings us to another important issue of the book, which is preservations of GEMs and polyGEMs by \(L_f\) and \(CW_A\) (Theorems 4.B.3 and 4.B.4). In order to prove these theorems the symmetric product construction \(SP^\infty\) is being discussed very extensively. In particular the Bousfield Key Lemma is proven (Proposition 4.D.1). The study of the \(SP^\infty\) construction is used to prove two remarkable theorems. The first one is that \(L_f K(\pi,n)\) and \(CW_A K(\pi, n)\) are equivalent to a product of two Eilenberg-MacLane spaces (Corollary 4.B.4.1). The second theorem says that the homotopy fiber of the map \(P_{\Sigma W} (X^V) \to (P_{\Sigma W} X)^V\) has vanishing homotopy groups in dimensions bigger than \(\dim W\), where \(P_A = L_{A \to *}\) (Theorem 5.C.5). The consideration of mapping spaces becomes important, especially for studying localization with respect to the cofiber \(C\) of the Adams map \(v_1\). In particular it is proven that the map \(X \to P_{\Sigma C} X\) induces an isomorphism on \(v^{-1}_1 \pi_*\) (Proposition 8.A.9). This is applied to the \(K\)-theory localization (Chapter 8).

The functors \(P_A\) and \(CW_A\) are used to define two orders on spaces called cellular inequalities. A space \(B\) is said to be supported by \(A\), denoted by \(B > A\), if \(P_A B \simeq *\). A space \(B\) is said to be \(A\)-cellular, denoted by \(B \gg A\), if \(CW_A B \simeq B\). The equivalence relations defined by these orders are called respectively the nullity and the cellular types. Thus a space \(A\) has the same nullity type as \(B\) if \(A > B\) and \(B >A\). Similarly \(A\) has the same cellular type as \(B\) if \(A \gg B\) and \(B \gg A\). The book contains the classification of the nullity and cellular types of finite \(p\)-torsion suspension spaces (Theorems 7.B.2 and 7.C.2).

The relation \(\gg\) plays a special role in the book. The fact that \(SP^\infty X \gg X\) (Corollary 4.A.2.1) is one of the most crucial technical results which is used several times. The main theorem concerning the relation \(\gg\) is as follows. Let \(E \to B\) be a natural transformation between diagrams \(E\) and \(B\) indexed by a category \(I\). If for all \(\alpha \in I\) the homotopy fiber of the map \(E_\alpha \to B_\alpha\) is \(A\) cellular, then so is the homotopy fiber of \(\text{hocolim}_I E \to \text{hocolim}_I B\) (Theorem 9.A.8).

“The book describes a certain framework for doing homotopy theory in which the function complexes play a central role. This approach emerged in the early 1990’s but has roots in earlier work of Bousfield about localization and in the big advances made by Mahowald, Ravenel, Devinatz, Hopkins and Smith towards deeper understanding of the role of periodicity in stable homotopy theory.” The key constructions are the homotopy idempotent localization functor \(L_f\) with respect to a map \(f\), and the augmented homotopy idempotent functor \(CW_A\) with respect to a space \(A\). The functor \(L_f\) assigns to a space \(X\) the “smallest” quotient \(X \to L_f X\), such that if \(\text{map}(f,Y)\) is a weak equivalence, then so is \(\text{map} (L_f X, Y) \to \text{map} (X,Y)\). For an appropriate \(f\), \(L_f\) is the localization functor with respect to a given homology theory. Therefore studying the functors \(L_f\) gives a general framework for looking at homological localizations. The functor \(CW_A\) assigns to a space \(X\) the biggest subobject \(CW_A X \to X\) such that \(\text{map}_* (A, CW_A X) \to \text{map}_*(A, X)\) is a weak equivalence. For example if \(A\) is the \(\mathbb{Z}/p\)-Moore space, the functor \(CW_A\) assigns to \(X\) the universal space \(CW_A X\) which has the same mod-\(p\) homotopy groups as \(X\). The author presents constructions of continuous versions of the functors \(L_f\) and \(CW_A\).

An important issue of the book is the behavior of \(L_f\) and \(CW_A\) with respect to fibration sequences and mapping spaces, thus in particular with respect to loop spaces. The author proves that applying \(L_f\) and \(CW_A\) to a loop space produces again a loop space. Even more: \(L_f \Omega X \simeq \Omega L_{\Sigma f} X\) and \(CW_A \Omega X \simeq \Omega CW_{\Sigma A} X\) (Theorems 3.A.1 and 3.A.2). Therefore in order to study to what extent \(L_f\) preserves fibration sequences, in particular to study the difference between \(L_f \Omega X\) and \(\Omega L_f X\), one has to investigate the map \(L_{\Sigma f} X \to L_f X\). Similarly the behaviour of \(CW_A\) with respect to fibration sequences is encoded in the map \(CW_{\Sigma A} X \to CW_A X\). The author proves that if \(f\) is a suspension, the homotopy fiber of \(L_{\Sigma f} X \to L_f X\) is a GEM i.e., a possibly infinite product of Eilenberg-Mac Lane spaces (Proposition 5.F.4). He also shows that the homological localizations with respect to complex mod-\(p\) \(K\)-theory or Morava \(K\)-theories preserve fibrations of the form \(\Omega^2(F \to E \to B)\) up to an error term, which is a product of only three Eilenberg-Mac Lane spaces (Theorem 6.A.1). If \(A\) is a suspension,then it is shown that the homotopy fiber of \(CW_{\Sigma A} X \to CW_A X\) is a polyGEM i.e., it is an element of the smallest class of spaces that contains GEMs and is closed under taking the total space of any fibration sequence for which the fiber and the base are polyGEMs (Theorem 5.E.3).

This brings us to another important issue of the book, which is preservations of GEMs and polyGEMs by \(L_f\) and \(CW_A\) (Theorems 4.B.3 and 4.B.4). In order to prove these theorems the symmetric product construction \(SP^\infty\) is being discussed very extensively. In particular the Bousfield Key Lemma is proven (Proposition 4.D.1). The study of the \(SP^\infty\) construction is used to prove two remarkable theorems. The first one is that \(L_f K(\pi,n)\) and \(CW_A K(\pi, n)\) are equivalent to a product of two Eilenberg-MacLane spaces (Corollary 4.B.4.1). The second theorem says that the homotopy fiber of the map \(P_{\Sigma W} (X^V) \to (P_{\Sigma W} X)^V\) has vanishing homotopy groups in dimensions bigger than \(\dim W\), where \(P_A = L_{A \to *}\) (Theorem 5.C.5). The consideration of mapping spaces becomes important, especially for studying localization with respect to the cofiber \(C\) of the Adams map \(v_1\). In particular it is proven that the map \(X \to P_{\Sigma C} X\) induces an isomorphism on \(v^{-1}_1 \pi_*\) (Proposition 8.A.9). This is applied to the \(K\)-theory localization (Chapter 8).

The functors \(P_A\) and \(CW_A\) are used to define two orders on spaces called cellular inequalities. A space \(B\) is said to be supported by \(A\), denoted by \(B > A\), if \(P_A B \simeq *\). A space \(B\) is said to be \(A\)-cellular, denoted by \(B \gg A\), if \(CW_A B \simeq B\). The equivalence relations defined by these orders are called respectively the nullity and the cellular types. Thus a space \(A\) has the same nullity type as \(B\) if \(A > B\) and \(B >A\). Similarly \(A\) has the same cellular type as \(B\) if \(A \gg B\) and \(B \gg A\). The book contains the classification of the nullity and cellular types of finite \(p\)-torsion suspension spaces (Theorems 7.B.2 and 7.C.2).

The relation \(\gg\) plays a special role in the book. The fact that \(SP^\infty X \gg X\) (Corollary 4.A.2.1) is one of the most crucial technical results which is used several times. The main theorem concerning the relation \(\gg\) is as follows. Let \(E \to B\) be a natural transformation between diagrams \(E\) and \(B\) indexed by a category \(I\). If for all \(\alpha \in I\) the homotopy fiber of the map \(E_\alpha \to B_\alpha\) is \(A\) cellular, then so is the homotopy fiber of \(\text{hocolim}_I E \to \text{hocolim}_I B\) (Theorem 9.A.8).

Reviewer: W.Chacholski (Toronto)

##### MSC:

55-02 | Research exposition (monographs, survey articles) pertaining to algebraic topology |

55P60 | Localization and completion in homotopy theory |