15 problems

Let $K(n)$ denote the $2(p^n-1)$-periodic Morava $K$-theory. Its topological Hochschild homology, denoted $THH(K(n))$, may depend a priori on the chosen $A_\infty$ structure. Morav…

Let $G$ be a finite group, let $X$ be a based $G$-space, and let $H\leq G$. The trace map on fixed points is … Here $A_G(X)$ denotes genuine $G$-equivariant $A$-theory,…

Let $C_n$ be a cyclic group and let $R$ be a $C_n$-ring spectrum. The fixed-point map … is the map obtained by taking $C_n$-fixed points of an equivariant map from $K_{C_n}(R)$ to…

Let $\Theta$ be a tangential structure, let $R$ be its associated ring spectrum, and suppose $\operatorname{Flow}^{\Theta} \simeq \operatorname{Perf}({}_R\operatorname{mod}_R)$. Fo…

Let $T$ be a $(2n,2n)$-tangle, and let $\mathscr{X}(T)_{2n,2n}$ be the associated spectral $(\mathcal{H}_{2n},\mathcal{H}_{2n})$-bimodule. Let $\hat T\subset S^1\times S^2$ be the…

Let $\mathrm{THR}(\mathbb{Z})$ denote the real topological Hochschild homology of the integers, regarded as a $\mathbb{Z}/2$-spectrum. Write $\mathrm{H}\\,\underline{\mathbb{Z}}$ f…

Non-splitting conjecture. These sequences do not split.

Real equivariant conjectures. The six listed assertions in the source hold: the required $\mathbf{E}_{\rho}\rtimes\mathrm{U}(1)_\mathbf{R}$-algebra and splitting structures exist;…

Let $K^S(A)$ denote stable K-theory of an $mathbb S$-algebra $A$, and let $\mathrm{THH}(A)$ denote its topological Hochschild homology. Waldhausen's conjecture. Stable K-theory and…

Conjectural THH computation. There is an isomorphism

The cube decomposition conjecture. The spectrum $\mathrm{THH}(E(n))$ decomposes as a sum of $2^n$ factors: one copy of $E(n)$, together with $2^{n-i-1}$ suspended copies of…

The Becker–Gottlieb transfer is a transfer map associated to a suitable fibration, and the THH transfer is the transfer map constructed using topological Hochschild homology, denot…

Relative THH conjecture. The relative topological Hochschild homology spectrum should satisfy

Consider the $H\mathbb{F}_p$-THH-May spectral sequence, with differential $d_1$ and the operation $\sigma$ on classes. The $d_1$–$\sigma$ commutation conjecture. The differential…

Cyclotomic THH equivalence conjecture. There is an equivalence of cyclotomic spectra