Computations of relative topological coHochschild homology

Spanier–Whitehead duality for topological coHochschild homology

“…where |𝑡| = 0 and |𝑥 1 | = 1. This completes the result of [23] in which the 𝐸 2 -page of the relative coBökstedt spectral sequence from [4] computing the homotopy groups above was calculated. • Theorem 7.8: there is an 𝔸 ∞ -coalgebra structure on Ω𝐻𝔽 𝑝 as an 𝐻ℤ-module and its coTHH is given by:…”

“…Essentially, model categories currently can only capture coalgebras of the form Σ ∞ + 𝑋 induced by the diagonal on the space 𝑋. It is for this very reason that the work of [4,20], and [23] is limited to the computation of 𝖼𝗈𝖳𝖧𝖧(Σ ∞ + 𝑋). If a symmetric monoidal ∞-category  is endowed with an internal hom [−, −], we can define the notion of linear duality that generalizes the classical dual of a vector space over a field, or the Spanier-Whitehead dual of a spectrum.…”

“…\end{equation*}$$In [4, 2.3], the authors construct a coBökstedt spectral sequence for coTHH and show that this spectral sequence has a ‘$$\square$$‐Hopf algebra structure’, that is, it is endowed with a comultiplicative structure that is compatible with its multiplicative structure. The $$E_2$$‐page of this spectral sequence is computed for various interesting coalgebra spectra in [23]. In [6], Bohmann, Gerhardt and Shipley use the $$\square$$‐Hopf algebra structure on the coBökstedt spectral sequence and the equivalence above to obtain homology computations for various free loop spaces generalizing earlier computations of [25].…”

“…Essentially, model categories currently can only capture coalgebras of the form $$\Sigma ^\infty _+ X$$ induced by the diagonal on the space $$X$$. It is for this very reason that the work of [4, 20], and [23] is limited to the computation of $$\mathsf {coTHH}(\Sigma ^\infty _+ X)$$.…”

“… Theorem 7.2: there is an $${\mathbb {E}_\infty }$$‐coalgebra structure on $$H\mathbb {F}_p\wedge _{H\mathbb {Z}}H\mathbb {F}_p$$ as an $$H\mathbb {F}_p$$‐module, and its coHH is given by: $$$$\begin{equation*} \pi _*{\left(\mathsf {coTHH}^{H\mathbb {F}_p}(H\mathbb {F}_p\wedge _{H\mathbb {Z}}H\mathbb {F}_p)\right)}\cong \Lambda _{\mathbb {F}_p}(x_1) \otimes \mathbb {F}_p[[t$$]], \end{equation*}$$where $$\vert t \vert = 0$$ and $$\vert x_1 \vert = 1$$. This completes the result of [23] in which the $$E_2$$‐page of the relative coBökstedt spectral sequence from [4] computing the homotopy groups above was calculated. Theorem 7.8: there is an $${\mathbb {A}_\infty }$$‐coalgebra structure on $$\Omega H\mathbb {F}_p$$ as an …”