We start by clarifying and extending the multibraces notation, which economically describes substitutions of multilinear maps and tensor products of vectors. We give definitions and examples of weakly homotopy algebras, homotopy Gerstenhaber and Gerstenhaber bracket algebras, and homotopy Batalin-Vilkovisky algebras. We show that a homotopy algebra structure on a vector space can be lifted to its Hochschild complex, and also suggest an induction method to generate some of the explicit (weakly) homotopy Gerstenhaber algebra maps on a topological vertex operator algebra (TVOA), their existence having been indicated by Kimura, Voronov, and Zuckerman in 1996 (later amended by Voronov). The contention that this is the fundamental structure on a TVOA is substantiated by providing an annotated dictionary of weakly homotopy BV algebra maps and identities found by Lian and Zuckerman in 1993.

Let $A \otimes _tC$ be a twisted tensor product of an algebra $A$ and a coalgebra $C$, along a twisting cochain $t:C \rightarrow A$. By means of what is called the tensor trick and under some nice conditions, Gugenheim, Lambe and Stasheff proved in the early 90s that $A \otimes C$ is homology equivalent to the objects $M \otimes _{t^\prime}C$ and $A \otimes _{t''}N$, where $M$ and $N$ are strong deformation retracts of $A$ and $C$, respectively. In this paper, we attack this problem from the point of view of contractions. We find explicit contractions from $A\otimes _t C$ to $M \otimes_{t'}C$ and $A\otimes_{t''}N$. Applications to the comparison of resolutions which split off of the bar resolution, as well as to some homological models for central extensions are given.

This article explores $\mathbb{Z}_2$-graded $L_\infty$ algebra structures on a $2|1$-dimensional vector space. The reader should note that our convention on the parities is the opposite of the usual one, because we define our structures on the symmetric coalgebra of the parity reversion of a space, so our $2|1$-dimensional \linf\ algebras correspond to the usual $1|2$-dimensional algebras.

We give a complete classification of all structures with a nonzero degree 1 term. We also classify all degree 2 codifferentials, which is the same as a classification of all $1|2$-dimensional $\mathbb{Z}_2$-graded Lie algebras. For each of these algebra structures, we calculate the cohomology and a miniversal deformation.

We construct an example of a finite dimensional $L_{\infty}$ algebra which is generated by a Lie algebra together with a non-Lie action on another vector space. We then show how this example fits into the gauge transformation theory of Berends, Burgers and Van Dam.

We reinterpret the classical theory of cocyclic operations in terms of permutations and homotopy equivalences of explicit chains. The essential tools we use are Homological Perturbation Theory and Eilenberg-Zilber Theorem. The main objective of this technique is the final identification of cohomology operations at cochain level.

We show that the cobar construction of a DG-bialgebra is a homotopy G-algebra. This implies that the bar construction of this cobar is a DG-bialgebra as well.

An explicit description of free strongly homotopy associative and free strongly homotopy Lie algebras is given. A variant of the Poincaré-Birkhoff-Witt theorem for the universal enveloping A($m$)-algebra of a strongly homotopy Lie algebra is formulated.

The classical problem of algebraic models for homotopy types is precisely stated here in terms of our ability to compute with the models. Two different natural statements for this problem are produced, the simplest one being entirely solved by the notion of $\mathcal{SS}_{EH}$-structure, due to the authors. Other tentative solutions, Postnikov towers and $E_\infty$-chain complexes, are considered and compared with the $\mathcal{SS}_{EH}$-structures. In particular, an imprecision in the usual definition of the k-"invariants" is explained, which implies we seem far from a solution for the ideal statement of our problem. On the positive side, the problem stated below in the framed quotation is solved.

An $A_{\infty }$-bialgebra is a DGM $H$ equipped with structurally compatible operations $\left\{ \omega ^{j,i}:H^{\otimes i}\rightarrow H^{\otimes j}\right\} $ such that $\left( H,\omega ^{1,i}\right) $ is an $% A_{\infty }$-algebra and $\left( H,\omega ^{j,1}\right) $ is an $A_{\infty }$% -coalgebra. Structural compatibility is controlled by the biderivative operator $Bd$, defined in terms of two kinds of cup products on certain cochain algebras of pemutahedra over the universal PROP $U=End\left(TH\right)$.

Homotopy theories over operads are defined. The corresponding spectral sequences for the homotopy groups are constructed. The calculations of the spectral sequences of the homotopy groups over the "$n$-dimensional little cubes" operads are produced.