Perspectives of higher signatures: codimension 2 homotopy invariance and L2-acyclicity

• Applicant: Professor Dr. Thomas Schick
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 556897787

Manifolds are fundamental objects of study in geometric topology. Surgery theory has been developed to classify and construct them. For this purpose, higher signatures with values in L-theory groups have to be understood. These are invariant under oriented homotopy equivalence. The Novikov conjecture asserts that easier to compute numerical higher signatures also are homotopy invariant. To this day, the Novikov conjecture is not known in general. The first part of the project studies a new candidate of geometrically defined higher signature, which should be a homotopy invariant: the L-theoretic signature of submanifolds of codimension 2 (with small additional conditions). To do this, the L-theory groups have to be understood better and suitable transfer homomorphisms have to be developed and computed. For the second part of the project we observe that the ordinary signatures can be computed using L2-homology. Interestingly, these invariants can completely vanish. We call such manifolds L2-bettiless. The second part of the project deals with the bordism classification of this class of manifolds: when is a given manifold bordism equivalent to an L2-bettiless one? Which secondary invariants exist to classify L2-bettiless manifolds? What is the corresponding bordism theory? To answer these questions, we will further develop and use tools from a variety of mathematical areas, from functional analysis via the theory of quadratic forms and in particular to surgery theory.

• DFG Programme: Research Grants
• International Connection: USA
• Cooperation Partner: Professor James Frederic Davis, Ph.D.