arXiv subjects
Publications and source records attributed to Gerd Laures.
We describe a new link between the theory of topological modular forms and representations of vertex operator algebras obtained by certain lattices. The construction is motivated by the arithmetic Whitehead tower of the orthogonal groups. The tower discloses the role of codes in representation theory.
An IP-space is a pseudomanifold whose defining local properties imply that its middle perversity global intersection homology groups satisfy Poincaré duality integrally. We show that the symmetric signature induces a map of Quinn spectra from IP bordism to the symmetric $L$-spectrum of $\Z$, which is, up to weak equivalence, an $E_\infty$ ring map. Using this map, we construct a fundamental $L$-homology class for IP-spaces, and as a consequence we prove the stratified Novikov conjecture for IP-spaces.
We prove a relative version of the realization theorem for $h_\infty$-diagrams in case that the underlying diagram subcategory is factorization-closed.
We show that $K(2)$-locally, the smash product of the string bordism spectrum and the spectrum $T_2$ splits into copies of Morava $E$-theories. Here, $T_2$ is related to the Thom spectrum of the canonical bundle over $ΩSU(4)$.
We introduce cannibalistic classes for string bundles with values in $TMF$ with level structures. This allows us to compute the Morava $E$-homology of any map from the bordism spectrum $MString$ to $TMF$ with level structures.
We compute the completed $TMF_0(3)$ cohomology of the 7-connective cover $BString$ of $BO$. We use cubical structures on line bundles over elliptic curves to construct an explicit class which together with the Pontryagin classes freely generates the cohomology ring.
We introduce singularities to Quinn spectra. It enables us to talk about ads with prescribed singularities and to explicitly construct representatives for prominent spectra like Morava $K$-theories or for $L$-theory with singularities. We develop a spectral sequence for the computation of the associated bordism groups and investigate product structures in the presence of singularities.
Let $TMF_1(n)$ be the spectrum of topological modular forms equipped with a $Γ_1(n)$-structure. We compute the $K(2)$-local $TMF_1(3)$-cohomology of $B{\mathit String}$ and $B{\mathit Spin}$: both are power series rings freely generated by classes that we explicitly construct and which generalize the classical Pontryagin classes. As a first application of this computation, we show how to construct $TMF(3n)$-cohomology classes from stable positive energy representations of the loop groups $L{\mathit Spin}$.
We give a simple sufficient condition for Quinn's "bordism-type" spectra to be weakly equivalent to commutative symmetric ring spectra. We also show that the symmetric signature is (up to weak equivalence) a monoidal transformation between symmetric monoidal functors, which implies that the Sullivan-Ranicki orientation of topological bundles is represented by a ring map between commutative symmetric ring spectra. In the course of proving these statements we give a new description of symmetric L theory which may be of independent interest.
We give a simple sufficient condition for Quinn's "bordism-type spectra" to be weakly equivalent to strictly associative ring spectra. We also show that Poincare bordism and symmetric L-theory are naturally weakly equivalent to monoidal functors. Part of the proof of these statements involves showing that Quinn's functor from bordism-type theories to spectra lifts to the category of symmetric spectra. We also give a new account of the foundations.
This paper investigates the relation between Toda brackets and congruences of modular forms. It determines the $f$-invariant of Toda brackets and thereby generalizes the formulas of J.F.\ Adams for the classical $e$-invariant to the chromatic second filtration.
In previous work, the authors have each introduced methods for studying the 2-line of the p-local Adams-Novikov spectral sequence in terms of the arithmetic of modular forms. We give the precise relationship between the congruences of modular forms introduced by the first author with the Q-spectrum and the f-invariant of the second author. This relationship enables us to refine the target group of the f-invariant in a way which makes it more manageable for computations.