Michael Boardman

**John Michael Boardman** (1938-2021) was a British-born mathematician, specialised in algebraic topology and differential topology. He received his Ph.D. from the University of Cambridge in 1964. His thesis advisor was C. T. C. Wall. He was formerly at the University of Cambridge, England; later a full professor at Johns Hopkins University in Baltimore, USA.

Boardman introduced the stable homotopy category in 1969. His notes on this subject were never formally published, but Rainer Vogt gave a course on this subject in Aarhus in 1969:

He coauthored with Vogt the famous book

**Homotopy invariant algebraic structures on topological spaces**, Springer Lecture Notes in Math 347 (1973).

This book introduced the notion of weak Kan complex that was later popularized by André Joyal under the name quasi-category as a natural basis for the higher category theory of (∞,1)-categories.

See also: *Homotopy invariant algebraic structures*: *a conference in honor of J. Michael Boardman* edited by Jean-Pierre Meyer, Jack Morava, and W. Stephen Wilson. AMS, 1999, CONM/239.

Introducing the Boardman homomorphism:

- J. Michael Boardman,
*The eightfold way to BP-operations*, in:*Current trends in algebraic topology*, pp. 187–226, Canadian Mathematical Society Proceedings, 2, Part 1. Providence 1982 (ISBN:978-0-8218-6003-8)

On stable cohomology operations, but mostly reviewing the background of stable homotopy theory, Whitehead-generalized cohomology theory, ring spectra, multiplicative cohomology theory, complex oriented cohomology and the examples of KU, MU, BP:

- John Michael Boardman,
*Stable Operations in Generalized Cohomology*(pdf) in: Ioan Mackenzie James (ed.)*Handbook of Algebraic Topology*Oxford 1995 (doi:10.1016/B978-0-444-81779-2.X5000-7)

On unstable cohomology operations (hence: in non-abelian cohomology):

- John Michael Boardman, David Copeland Johnson, W. Stephen Wilson,
*Unstable Operations in Generalized Cohomology*(pdf), in: Ioan Mackenzie James (ed.)*Handbook of Algebraic Topology*Oxford 1995 (doi:10.1016/B978-0-444-81779-2.X5000-7)

Some of the above material is taken from Joyal's CatLab – Michael Boardman.

and Rainer Vogt’s article (with self explanatory title):

- Rainer Vogt,
*My time as Mike Boardman’s student and our work on infinite loop spaces*(pdf)

category: people