Poincaré duality and manifold topology
Poincaré spaces and pairs, Spivak normal fibrations, surgery, finiteness, and stable thickenings.
Professor of Mathematics · Wayne State University
John R. Klein
Research in algebraic and geometric topology, with a focus on Poincaré duality, embedding problems, homotopy theory, transfers, and algebraic K-theory.
- Office
- 1213 Faculty/Administration Building
- ae9462@wayne.edu
Current directions
Poincaré spaces embeddings transfers Hopf invariants
01 · Research
Algebraic topology across spaces, embeddings, and invariants
My work develops homotopy-theoretic tools for geometric problems and follows their connections to K-theory, dynamics, and mathematical physics.
Embeddings and functor calculus
Embedding and compression problems, disjunction theorems, relative Hopf invariants, and calculus of homotopy functors.
Transfers, A-theory, and torsion
Transfer maps, algebraic K-theory of spaces, parametrized topology, Euler characteristics, and zeta functions.
Topology in mathematical physics
Topological methods for stochastic dynamics, fluctuating currents, graph trajectories, and invariants of quantum states.
02 · Recent work
New and ongoing research
2026Algebraic topology
A relative Euclidean thickening
Constructs a Poincaré triad realizing a finite CW pair while retaining a trivial Spivak normal fibration.
arXiv:2607.296312025Surgery theory
On Poincaré Surgery
Gives a homotopy-theoretic proof of the Fundamental Theorem of Poincaré surgery in the simply connected case and derives the Poincaré transversality exact sequence.
arXiv:2503.200522026Algebraic topology
On the stable Hopf invariant
Develops the stable Hopf invariant and examines the limits of an elementary axiomatization.
Cambridge First View · 20262026Geometric topology
Embedding, compression, and the relative Hopf invariant
Relates compression questions in embedding theory to a relative form of the Hopf invariant.
arXiv:2602.1926603 · Teaching
From calculus to modern homotopy theory
I teach undergraduate and graduate mathematics at Wayne State, including calculus, algebraic topology, and advanced topics in geometry and topology.
04 · Contact
Mathematics at Wayne State
Department of Mathematics, College of Liberal Arts and Sciences, Wayne State University.
Email
ae9462@wayne.edu
Office
1213 Faculty/Administration Building
Detroit, Michigan 48202