We describe the topological Hochschild homology of the periodic complex K-theory spectrum, T HH(KU ), as a commutative KU -algebra: it is equivalent to KU [K(Z, 3)] and to F (ΣKU Q ), where F is the free commutative KU -algebra functor on a KU -module. Moreover,In order to prove these results, we first establish that topological Hochschild homology commutes, as an algebra, with localization at an element.Then, we prove that T HH n (KU ), the n-fold iteration of T HH(KUwhere G is a certain product of integral Eilenberg-Mac Lane spaces, and to a free commutative KU -algebra on a rational KU -module. We prove that S n ⊗KU is equivalent to KU [K(Z, n + 2)] and to F (Σ n KU Q ). We describe the topological André-Quillen homology of KU as KU Q .

We describe the topological Hochschild homology of the periodic complex K-theory spectrum, T HH(KU), as a commutative KU-algebra: it is equivalent to KU [K(Z, 3)] and to F (ΣKU Q), where F is the free commutative KU-algebra functor on a KU-module. Moreover, F (ΣKU Q) KU ∨ ΣKU Q , a square-zero extension. In order to prove these results, we first establish that topological Hochschild homology commutes, as an algebra, with localization at an element. Then, we prove that T HH n (KU), the n-fold iteration of T HH(KU), i.e. T n ⊗ KU , is equivalent to KU [G] where G is a certain product of integral Eilenberg-Mac Lane spaces, and to a free commutative KU-algebra on a rational KU-module. We prove that S n ⊗KU is equivalent to KU [K(Z, n + 2)] and to F (Σ n KU Q). We describe the topological André-Quillen homology of KU as KU Q .

We define a bar construction endofunctor on the category of commutative augmented monoids A of a symmetric monoidal category V endowed with a left adjoint monoidal functor F : sSet → V. To do this, we need to carefully examine the monoidal properties of the well-known (reduced) simplicial bar construction B • (1, A, 1). We define a geometric realization |−| with respect to the image under F of the canonical cosimplicial simplicial set. This guarantees good monoidal properties of | − |: it is monoidal, and given a left adjoint monoidal functor G : V → W, there is a monoidal transformation |G − | ⇒ G| − |. We can then consider BA = |B • A| and the iterations B n A. We establish the existence of a graded multiplication on these objects, provided the category V is cartesian and A is a ring object. The examples studied include simplicial sets and modules, topological spaces, chain complexes and spectra.

The cotangent complex of a map of commutative rings is a central object in deformation theory. Since the 1990s, it has been generalized to the homotopical setting of $$E_\infty $$ E ∞ -ring spectra in various ways. In this work we first establish, in the context of $$\infty $$ ∞ -categories and using Goodwillie’s calculus of functors, that various definitions of the cotangent complex of a map of $$E_\infty $$ E ∞ -ring spectra that exist in the literature are equivalent. We then turn our attention to a specific example. Let R be an $$E_\infty $$ E ∞ -ring spectrum and $$\mathrm {Pic}(R)$$ Pic ( R ) denote its Picard $$E_\infty $$ E ∞ -group. Let Mf denote the Thom $$E_\infty $$ E ∞ -R-algebra of a map of $$E_\infty $$ E ∞ -groups $$f:G\rightarrow \mathrm {Pic}(R)$$ f : G → Pic ( R ) ; examples of Mf are given by various flavors of cobordism spectra. We prove that the cotangent complex of $$R\rightarrow Mf$$ R → M f is equivalent to the smash product of Mf and the connective spectrum associated to G.