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Register a new userThe topological period-index conjecture
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We prove the topological analogue of the period-index conjecture in each dimension away from a small set of primes.
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We establish upper bounds of the indices of topological Brauer classes over a closed orientable 8-manifolds. In particular, we verify the Topological Period-Index Conjecture (TPIC) for topological Brauer classes over closed orientable 8-manifolds of order not congruent to 2 mod 4. In addition, we provide a counter-example which shows that the TPIC fails in general for closed orientable 8-manifolds.
We give a new proof of a result of Lazarev, that the dual of the circle $S^1_+$ in the category of spectra is equivalent to a strictly square-zero extension as an associative ring spectrum. As an application, we calculate the topological cyclic homology of $DS^1$ and rule out a Koszul-dual reformulation of the Novikov conjecture.
Let $p$ be a prime, $n geq 1$, $K(n)$ the $n$th Morava $K$-theory spectrum, $mathbb{G}_n$ the extended Morava stabilizer group, and $K(A)$ the algebraic $K$-theory spectrum of a commutative $S$-algebra $A$. For a type $n+1$ complex $V_n$, Ausoni and Rognes conjectured that (a) the unit map $i_n: L_{K(n)}(S^0) to E_n$ from the $K(n)$-local sphere to the Lubin-Tate spectrum induces a map [K(L_{K(n)}(S^0)) wedge v_{n+1}^{-1}V_n to (K(E_n))^{hmathbb{G}_n} wedge v_{n+1}^{-1}V_n] that is a weak equivalence, where (b) since $mathbb{G}_n$ is profinite, $(K(E_n))^{hmathbb{G}_n}$ denotes a continuous homotopy fixed point spectrum, and (c) $pi_ast(-)$ of the target of the above map is the abutment of a homotopy fixed point spectral sequence. For $n = 1$, $p geq 5$, and $V_1 = V(1)$, we give a way to realize the above map and (c), by proving that $i_1$ induces a map [K(L_{K(1)}(S^0)) wedge v_{2}^{-1}V_1 to (K(E_1) wedge v_{2}^{-1}V_1)^{hmathbb{G}_1},] where the target of this map is a continuous homotopy fixed point spectrum, with an associated homotopy fixed point spectral sequence. Also, we prove that there is an equivalence [(K(E_1) wedge v_{2}^{-1}V_1)^{hmathbb{G}_1} simeq (K(E_1))^{widetilde{h}mathbb{G}_1} wedge v_2^{-1}V_1,] where $(K(E_1))^{widetilde{h}mathbb{G}_1}$ is the homotopy fixed points with $mathbb{G}_1$ regarded as a discrete group.
In recent work, Hess and Shipley defined a theory of topological coHochschild homology (coTHH) for coalgebras. In this paper we develop computational tools to study this new theory. In particular, we prove a Hochschild-Kostant-Rosenberg type theorem in the cofree case for differential graded coalgebras. We also develop a coBokstedt spectral sequence to compute the homology of coTHH for coalgebra spectra. We use a coalgebra structure on this spectral sequence to produce several computations.
We calculate the integral homotopy groups of THH(l) at any prime and of THH(ko) at p=2, where l is the Adams summand of the connective complex p-local K-theory spectrum and ko is the connective real K-theory spectrum.
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