We define natural $A_\inf$-transformations and construct $A_\inf$-category of $A_\inf$-functors. The notion of non-strict units in an $A_\inf$-category is introduced. The 2-category of (unital) $A_\inf$-categories, (unital) functors and transformations is described.

The theorem of the title is deduced from the equivalence between crossed complexes and cubical $\omega$-groupoids with connections proved by the authors in 1981. In fact we prove the equivalence of five categories defined internally to an additive category with kernels.

We use extensions to study the semi-simple quotient of the group ring $\mathbf{F}_pAut(P)$ of a finite $p$-group $P$. For an extension $E: N \to P \to Q$, our results involve relations between $Aut(N)$, $Aut(P)$, $Aut(Q)$ and the extension class $[E]\in H^2(Q, ZN)$. One novel feature is the use of the intersection orbit group $\Omega([E])$, defined as the intersection of the orbits $Aut(N)\cdot[E]$ and $Aut(Q)\cdot [E]$ in $H^2(Q,ZN)$. This group is useful in computing $|Aut(P)|$. In case $N$, $Q$ are elementary Abelian $2$-groups our results involve the theory of quadratic forms and the Arf invariant.

Motivated by ideas from stable homotopy theory we study the space of strongly homotopy associative multiplications on a two-cell chain complex. In the simplest case this moduli space is isomorphic to the set of orbits of a group of invertible power series acting on a certain space. The Hochschild cohomology rings of resulting $A_\infty$-algebras have an interpretation as totally ramified extensions of discrete valuation rings. All $A_\infty$-algebras are supposed to be unital and we give a detailed analysis of unital structures which is of independent interest.

Il est bien connu que la seule donnée des groupes d'homologie et des groupes d'homotopie ne permet pas en général de déterminer le type d'homotopie d'un CW-complexe. J.H.C. Whitehead a introduit un invariant plus fin: la "certaine" suite exacte associée à un CW-complexe et en tire la classification des CW-complexes simplement connexes de dimension $% 4.$ Dans ce travail, en exploitant les propriétés de la suite exacte de Whitehead, nous montrons comment cette suite peut-être utiliser afin de classifier le type d'homotopie d'un CW-complexe simplement connexe de dimension quelconque.

An equivariant Thom isomorphism theorem in operator $K$-theory is formulated and proven for infinite rank Euclidean vector bundles over finite dimensional Riemannian manifolds. The main ingredient in the argument is the construction of a non-commutative $C^*$-algebra associated to a bundle $\mathfrak{E} \to M$, equipped with a compatible connection $\nabla$, which plays the role of the algebra of functions on the infinite dimensional total space $\mathfrak{E}$. If the base $M$ is a point, we obtain the Bott periodicity isomorphism theorem of Higson-Kasparov-Trout [19] for infinite dimensional Euclidean spaces. The construction applied to an even finite rank $\rm{spin}^c$-bundle over an even-dimensional proper $\rm{spin}^c$-manifold reduces to the classical Thom isomorphism in topological $K$-theory. The techniques involve non-commutative geometric functional analysis.

May's $J$-theory diagram is generalized to an equivariant setting. To do this, equivariant orientation theory for equivariant periodic ring spectra (such as $KO_G$) is developed, and classifying spaces are constructed for this theory, thus extending the work of Waner. Moreover, $Spin$ bundles of dimension divisible by 8 are shown to have canonical $KO_G$-orientations, thus generalizing work of Atiyah, Bott, and Shapiro. Fiberwise completions for equivariant spherical fibrations are constructed, also on the level of classifying spaces. When $G$ is an odd order $p$-group, this allows for a classifying space formulation of the equivariant Adams conjecture. It is also shown that the classifying space for stable fibrations with fibers being sphere representations completed at $p$ is a delooping of the 1-component of $Q_G(S^0)\hat{_p}$. The "Adams-May square," relating generalized characteristic classes and Adams operations, is constructed and shown to be a pull-back after completing at $p$ and restricting to $G$-connected covers. As a corollary, the canonical map from the $p$-completion of $J_G^k$ to the $G$-connected cover of $Q_G(S^0)\hat{_p}$ is shown to split after restricting to $G$-connected covers.

Generalizations of Borsuk's characterization theorems for AR and ANR spaces (for metrizable spaces) are provided. Moreover, a simple proof of Morita's theorem is given, in generalization of Borsuk's homotopy extension theorem.

We introduce the notion of (Weak) Corestriction Principle and prove some relations between the validity of this principle for various connecting maps in non-abelian Galois cohomology over fields of characteristic 0. We also prove the validity of Weak Corestriction Principle for images of coboundary maps $\rm{H}^1(k,G) \to \rm{H}^2(k,T)$, where $T$ is a finite commutative $k$-group of multiplicative type, $G$ is adjoint, semisimple and contains only almost simple factors of certain inner types.

Atiyah's work [1] describes the relationship between multiplication in a central extension of the mapping class group of a surface of genus $n$ and the signatures of $4$-dimensional manifolds. This work studies a subgroup of the central extension, which comes from the image of a representation of the pure framed braid group on $n$-strands found in [5], and the signatures of corresponding $4$-manifolds via a split exact sequence. We construct a splitting map to prove the sequence is split exact, and we use the splitting to give a topological description of homology classes in $4$-dimensional manifolds with non-zero intersection. We conclude with a description of multiplication in the subgroup.

We define a cotriple (co)homology of crossed modules with coefficients in a $\pi_1$-module. We prove its general properties, including the connection with the existing cotriple theories on crossed modules. We establish the relationship with the (co)homology of the classifying space of a crossed module and with the cohomology of groups with operators. An example and an application are given.

Bousfield has shown how the 2-primary $v_1$-periodic homotopy groups of certain compact Lie groups can be obtained from their representation ring with its decomposition into types and its exterior power operations. He has formulated a Technical Condition which must be satisfied in order that he can prove that his description is valid.

We prove that a simply-connected compact simple Lie group satisfies his Technical Condition if and only if it is not $E_6$ or Spin$(4k+2)$ with $k$ not a 2-power. We then use his description to give an explicit determination of the 2-primary $v_1$-periodic homotopy groups of $E_7$ and $E_8$. This completes a program, suggested to the author by Mimura in 1989, of computing the $v_1$-periodic homotopy groups of all compact simple Lie groups at all primes.

Given an orientable complete hyperbolic $3$-manifold of finite volume $M$ we construct a canonical class $\alpha(M)$ in $H_3(B(SL_2(\mathbb{C}),\mathfrak{T}))$ with $B(SL_2(\mathbb{C}),\mathfrak{T})$ the $SL_2(\mathbb{C})$-orbit space of the classifying space for a certain family of isotropy subgroups. We prove that $\alpha(M)$ coincides with the Bloch invariant $\beta(M)$ of $M$ defined by Neumann and Yang in [13], giving with this a simpler proof that the Bloch invariant is independent of an ideal triangulation of $M$. We also give a new proof of the fact that the Bloch invariant lies in the Bloch group $B(\mathbb{C})$.

We introduce a fibre homotopy relation for maps in a category of cofibrant objects equipped with a choice of cylinder objects. Weak fibrations are defined to be those morphisms having the weak right lifting property with respect to weak equivalences. We prove a version of Dold's fibre homotopy equivalence theorem and give a number of examples of weak fibrations. If the category of cofibrant objects comes from a model category, we compare fibrations and weak fibrations, and we compare our fibre homotopy relation, which is defined in terms of left homotopies and cylinders, with the fibre homotopy relation defined in terms of right homotopies and path objects. We also dualize our notion of weak fibration in a category of cofibrant objects to a notion of weak cofibration in a category of fibrant objects, and give examples of these weak cofibrations. A section is devoted to the case of chain complexes in an abelian category.

Let $\Lambda$ be a semiring with 1. By a Takahashi extension of a $\Lambda$-semimodule $X$ by a $\Lambda$-semimodule $Y$ we mean an extension of $X$ by $Y$ in the sense of M. Takahashi [10]. Let $A$ be an arbitrary $\Lambda$-semimodule and $C$ a $\Lambda$-semimodule which is normal in Takahashi's sense, that is, there exist a projective $\Lambda$-semimodule $P$ and a surjective $\Lambda$-homomorphism $\varepsilon : P \longrightarrow C$ such that $\varepsilon$ is a cokernel of the inclusion $\mu:\rm{Ker}(\varepsilon)\hookrightarrow P$. In [11], following the construction of the usual satellite functors, M. Takahashi defined $\rm{Ext}_{{}_\Lambda}(C,A)$ by

$$ \rm{Ext}_{{}_\Lambda}(C,A)=\rm{Coker}(\rm{Hom}_{{}_\Lambda}(\mu,A))$$

and used it to characterize Takahashi extensions of normal $\Lambda$-semimodules by $\Lambda$-modules.

In this paper we relate $\rm{Ext}_{{}_\Lambda}(C,A)$ with other known satellite functors of the functor $\rm{Hom}_{{}_\Lambda}(-,A)$.

We study $A_\infty$-structures extending the natural algebra structure on the cohomology of $\oplus_{n\in\mathbb{Z}} L^n$, where $L$ is a very ample line bundle on a projective $d$-dimensional variety $X$ such that $H^i(X,L^n)=0$ for 0 > i > d and all $ n \in \mathbb{Z}$. We prove that there exists a unique such nontrivial$A_{\infty}$-structure up to a strict $A_{\infty}$-isomorphism (i.e., an $A_{\infty}$-isomorphism with the identity as the first structure map) and rescaling.

In the case when $X$ is a curve we also compute the group of strict $A_{\infty}$-automorphisms of this $A_{\infty}$-structure.

For a given commutative graded algebra $A^*$, we study the set ${\cal M}_{A^*} =$ $\{\mbox{rational homotopy type of }X \ $ $| \ H^*(X;Q)\cong A^*\}$. For example, we see that if $A^*$ is isomorphic to $H^*(S^3\vee S^5\vee S^{16};Q)$, then ${\cal M}_{A^*}$ corresponds bijectively to the orbit space $P^3(Q)/Q^*\coprod \{*\}$, where $P^3(Q)$ is the rational projective space of dimension 3 and the point $\{*\}$ indicates the formal space.

In this work, we study the 2-category ${\bf Ext}(\underline{\cal K},\underline{\cal G})$ of extensions of a $gr$-category $\underline{\cal K}$ by a $gr$-category $\underline{\cal G}$. Such an extension consists of a $gr$-category $\underline{\cal H}$, an essentially surjective homomorphism $p : \underline{\cal H} \longrightarrow \underline{\cal K}$ and a monoidal equivalence $q : \underline{\cal G} \longrightarrow N(p)$ where $N(p)$ is the {\it homotopy kernel} of the homomorphism $p$. The main result is a classification theorem which constructs a biequivalence between the 2-category ${\bf Ext}(\underline{\cal K},\underline{\cal G})^{op}$ and the bicategory ${\bf Bimon}(\underline{\cal K}, \underline{\bf Bieq}(\underline{\cal G}))$ of monoidal bicategory homomorphisms between $\underline{\cal K}$ and $\underline{\bf Bieq}(\underline{\cal G})$, where $\underline{\bf Bieq}(\underline{\cal G})$ is the monoidal bicategory of biequivalences of $\underline{\cal G}$ with itself.

We construct a cofibrantly generated model structure on the category of flows such that any flow is fibrant and such that two cofibrant flows are homotopy equivalent for this model structure if and only if they are S-homotopy equivalent. This result provides an interpretation of the notion of S-homotopy equivalence in the framework of model categories.

Using the chain rule, we give a homotopy theoretic approach to identifying the derivative of the functor $X \mapsto Q_+(X^K)$.