In this paper we compute the 3-primary v1-periodic homotopy groups of the exceptional Lie group E7. The p-primary v1-periodic homotopy groups of a space X, denoted v −1 1 π∗(X; p) or just v 1 π∗(X), were defined in [21]. They are a localization of the actual homotopy groups, telling roughly the portion which is detected by K-theory and its operations. If X is a compact Lie group, each v 1 πi(X;...

In [5], we computed the cyclic homology of finite extensions of Dedekind domains. The crucial result was a formula for the cyclic homology, relative to a ground ring R, of A = R[x]/(P (x)). This paper presents two extensions of the methods and results of this paper. Recall that Hochschild homology, though generally defined in terms of a bar complex, is almost never, in practice, computed from i...

4.3. (i) Since HomA(A (I),M) ∼= M (I) for a set I and ⊕ is exact, it is clear that HomA(A (I),−) is exact, i.e. A(I) is projective. (ii) “Only if” part: Let A(P ) = ⊕ p∈P Ap be the free A-module indexed by P and ψ : A (P ) → p be the A-homomorphism such that ψ(1p) = p. Then ψ is surjective. If P is projective, then ψ has a section, i.e. an A-homomorphism φ : P → A(P ) such that ψ ◦ φ = idP . Th...

̂ M of the connection ∇M to the universal cover ̂ M of M is a locally flat connection. The affine structure of ̂ M is defined by a local diffeomorphism DM : ̂ M → R called the developing map. The developing map gives rise to a representation hM : π1 M → Aff R called the holonomy. The linear part L hM of hM is the linear holonomy. The affine manifold M,∇M is complete if and only if DM is a diffeomor...

In [28], Lawrence Roberts, extending the work of Ozsváth and Szabó in [23], showed how to associate to a link, L, in the complement of a fixed unknot, B ⊂ S, a spectral sequence whose E term is the Khovanov homology of a link in a thickened annulus defined in [2], and whose E term is the knot Floer homology of the preimage of B inside the double-branched cover of L. In [6], we extended [23] in ...

Let $V\subseteq A$ be a conformal inclusion of vertex operator algebras and let $\mathcal{C}$ category grading-restricted generalized $V$-modules that admits the algebraic braided tensor structure Huang-Lepowsky-Zhang. We give conditions under which inherits semisimplicity from $A$-modules in $\mathcal{C}$, vice versa. The most important condition is $A$ rigid $V$-module with non-zero categoric...

The p-primary v1-periodic homotopy groups of a space X, denoted v −1 1 π∗(X; p) or just v 1 π∗(X), were defined in [15]. They are a localization of the actual homotopy groups, telling roughly the portion which is detected by K-theory and its operations. If X is a compact Lie group or spherically resolved space, each v 1 πi(X; p) is a direct summand of some actual homotopy group of X. By use of ...

We prove that the Farrell-Jones assembly map for connective algebraic K-theory is rationally injective, under mild homological finiteness conditions on the group and assuming that a weak version of the Leopoldt-Schneider conjecture holds for cyclotomic fields. This generalizes a result of Bökstedt, Hsiang, and Madsen, and leads to a concrete description of a large direct summand of Kn(Z[G])⊗Z Q...