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Let $G$ be a compact Lie group, $\Pi$ be a normal subgroup of $G$, $\mathcal G=G/\Pi$, $X$ be a $\mathcal G$-space and $Y$ be a $G$-space. There are a number of results in the literature giving a direct sum decomposition of the group $[\Sigma^\infty X,\Sigma^\infty Y]_G$ of equivariant stable homotopy classes of maps from $X$ to $Y$. Here, these results are extended to a decomposition of the group $[B,C]_G$ of equivariant stable homotopy classes of maps from an arbitrary finite $\mathcal G$-CW sptrum $B$ to any $G$-spectrum $C$ carrying a geometric splitting (a new type of structure introduced here). Any naive $G$-spectrum, and any spectrum derived from such by a change of universe functor, carries a geometric splitting. Our decomposition of $[B,C]_G$ is a consequence of the fact that, if $C$ is geometrically split and $(\mathfrak F',\mathfrak F)$ is any reasonable pair of families of subgroups of $G$, then there is a splitting of the cofibre sequence $(E\mathfrak F_+ \wedge C)^\Pi \longrightarrow (E\mathfrak F'_+ \wedge C)^\Pi \longrightarrow (E(\mathfrak F', \mathfrak F) \wedge C)^\Pi$ constructed from the universal spaces for the families. Both the decomposition of the group $[B,C]_G$ and the splitting of the cofibre sequence are proven here not just for complete $G$-universes, but for arbitrary $G$-universes. Various technical results about incomplete $G$-universes that should be of independent interest are also included in this paper. These include versions of the Adams and Wirthmüller isomorphisms for incomplete universes. Also included is a vanishing theorem for the fixed-point spectrum $(E(\mathfrak F',\mathfrak F) \wedge C)^\Pi$ which gives computational force to the intuition that what really matters about a $G$-universe $U$ is which orbits $G/H$ embed as $G$-spaces in $U$.

Abstract. The main goal of this paper is to prove the following conjecture of Baues and Lemaire: the differential graded Lie algebra associated with the Sullivan model of a space is homotopy equivalent to its Quillen model. In addition we show the same for the cellular Lie algebra model which we build from the simplicial analog of the classical Adams - Hilton model. It turns out that this cellular Lie algebra model is one link in a chain of models connecting the models of Quillen and Sullivan. The key result which makes all this possible is Anick's correspondence between differential graded Lie algebras and Hopf algebras up to homotopy. In addition we show that the Quillen model is a rational homotopical equivalence, and we conclude the same for the other models using our main result. The construction of the three models is given in detail. The background from homotopy theory, differential algebra, and algebra is presented in great generality.

The memoir presents a systematic study of rational $S^1$-equivariant cohomology theories, and a complete algebraic model for them. It provides a classification of such cohomology theories in simple algebraic terms and a practical means of calculation. The power of the model is illustrated by analysis of the Segal conjecture, the behaviour of the Atiyah-Hirzebruch spectral sequence, the structure of $S^1$-equivariant $K$-theory, and the rational behaviour of cyclotomic spectra and the topological cyclic homology construction.

Let $p$ be a fixed prime number. Let $G$ denote a finite $p$-perfect group. This book looks at the homotopy type of the $p$-completed classifying space $BG_p$, where $G$ is a finite $p$-perfect group. The author constructs an algebraic analog of the Quillen's “plus” construction for differential graded coalgebras. This construction is used to show that given a finite $p$-perfect group $G$, the loop spaces $BG_p$ admits integral homology exponents. Levi gives examples to show that in some cases our bound is best possible. It is shown that in general $B\ast _p$ admits infinitely many non-trivial $k$-invariants. The author presents examples where homotopy exponents exist. Classical constructions in stable homotopy theory are used to show that the stable homotopy groups of these loop spaces also have exponents.

The need to address the appropriate three-dimensional generalization of category (tricategory) has been felt in homotopy theory, low-dimensional topology, cohomology theory, category theory, and quantum field theory. Benabou's bicategories provide the two-dimensional notion into which examples naturally fit. In developing the theory of bicategories it is very reassuring to know the coherence theorem: They can be replaced by the stricter 2-categories which are merely categories enriched in the category of categories. In this book, the authors provide… the unique source of the full definition of tricategory a thorough and complete proof of the coherence theorem for tricategories a wholly modern source of material on Gray's tensor product of 2-categories

In algebraic topology, obstruction theory provides a way to study homotopy classes of continuous maps in terms of cohomology groups; a similar theory exists for certain spaces with group actions and maps that are compatible (that is, equivariant) with respect to the group actions. This work provides a corresponding setting for certain spaces with group actions and maps that are compatible in a stronger sense, called isovariant. The basic idea is to establish an equivalence between isovariant homotopy and equivariant homotopy for certain categories of diagrams. Consequences include isovariant versions of the usual Whitehead theorems for recognizing homotopy equivalences, an obstruction theory for deforming equivariant maps to isovariant maps, rational computations for the homotopy groups of certain spaces of isovariant functions, and applications to constructions and classification problems for differentiable group actions.

This book gives an axiomatic presentation of stable homotopy theory. It starts with axioms defining a “stable homotopy category”; using these axioms, one can make various constructions—cellular towers, Bousfield localization, and Brown representability, to name a few. Much of the book is devoted to these constructions and to the study of the global structure of stable homotopy categories. Next, a number of examples of such categories are presented. Some of these arise in topology (the ordinary stable homotopy category of spectra, categories of equivariant spectra, and Bousfield localizations of these), and others in algebra (coming from the representation theory of groups or of Lie algebras, as well as the derived category of a commutative ring). Hence one can apply many of the tools of stable homotopy theory to these algebraic situations. Features: Provides a reference for standard results and constructions in stable homotopy theory. Discusses applications of those results to algebraic settings, such as group theory and commutative algebra. Provides a unified treatment of several different situations in stable homotopy, including equivariant stable homotopy and localizations of the stable homotopy category. Provides a context for nilpotence and thick subcategory theorems, such as the nilpotence theorem of Devinatz-Hopkins-Smith and the thick subcategory theorem of Hopkins-Smith in stable homotopy theory, and the thick subcategory theorem of Benson-Carlson-Rickard in representation theory. This book presents stable homotopy theory as a branch of mathematics in its own right with applications in other fields of mathematics. It is a first step toward making stable homotopy theory a tool useful in many disciplines of mathematics.