This book develops rigorous foundations for parametrized homotopy theory, which is the
algebraic topology of spaces and spectra that are continuously parametrized by the points of …

The author introduces the notion of a Galois extension of commutative $ S $-algebras ($
E_\infty $ ring spectra), often localized with respect to a fixed homology theory. There are …

We model the homotopy fiber $ E_m $ of the inclusion of the space of long knots in
dimension $ m $ into the corresponding space of immersions, through an operad structure …

We present two models for the space of knots which have endpoints at fixed boundary points
in a manifold with boundary, one model defined as an inverse limit of spaces of maps …

We establish that En-operads satisfy a rational intrinsic formality theorem for n≥ 3. We get
our results in the category of Hopf cooperads in cochain graded dg-modules which defines a …

The standard reduced bar complex B (A) of a differential graded algebra A inherits a natural
commutative algebra structure if A is a commutative algebra. We address an extension of …

We present two models for the space of knots which have endpoints at fixed boundary points
in a manifold with boundary, one model defined as an inverse limit of spaces of maps …

One-point compactification turns real vector spaces into spheres. In homotopy theory, this
transformation gets encoded in a map called the" real J-homomorphism". Here we define …

We compute the 2-primary Dyer-Lashof operations in the string topology of several families
of manifolds, specifically spheres and a variety of projective spaces. These operations, while …

The notion of Hochschild homology of a dg algebra admits a natural dualization, the
coHochschild homology of a dg coalgebra, introduced in [23] as a tool to study free loop …