March 24, 1913|
|Died||December 23, 1973
|Doctoral advisor||Solomon Lefschetz|
Ralph Hartzler Fox (March 24, 1913 – December 23, 1973) was an American mathematician. As a professor at Princeton University, he taught and advised many of the contributors to the Golden Age of differential topology, and he played an important role in the modernization and main-streaming of knot theory.
Ralph Fox attended Swarthmore College for two years, while studying piano at the Leefson Conservatory of Music in Philadelphia. He earned a master's degree from Johns Hopkins University, and a Ph.D. degree from Princeton University in 1939. His doctoral dissertation, On the Lusternick-Schnirelmann Category, was directed by Solomon Lefschetz. (In later years he disclaimed all knowledge of Lusternik–Schnirelmann category, and certainly never published on the subject again.) He directed 21 doctoral dissertations, including those of John Milnor, John Stallings, Francisco González-Acuña, Guillermo Torres-Diaz and Barry Mazur.
He was an Invited Speaker in 1950 at Cambridge, Massachusetts. His mathematical contributions include Fox n-coloring of knots, the Fox-Artin arc, and the free differential calculus. He also identified the compact-open topology on function spaces as being particularly appropriate for homotopy theory.
Aside from his strictly mathematical contributions, he was responsible for introducing several basic phrases to knot theory: the phrases slice knot, ribbon knot, and Seifert circle all appear in print for the first time under his name, and he also popularized (if he did not introduce) the phrase Seifert surface.
- Introduction to Knot Theory, Richard H. Crowell and Ralph H. Fox, Reprint of the 1963 original, Graduate Texts in Mathematics, No. 57, Springer-Verlag, New York-Heidelberg, 1977. ISBN 0-387-90272-4
- "A quick trip through knot theory", in: M. K. Fort (Ed.), Topology of 3-Manifolds and Related Topics, Prentice-Hall, New Jersey, 1961, pp. 120–167. MR0140099
- Metacyclic invariants of knots and links, Canadian Journal of Mathematics 22 (1970) 193–201. MR0261584
- "Rolling". Bull. Amer. Math. Soc. 72, Part 1: 162–164. 1966. doi:10.1090/s0002-9904-1966-11467-2. MR 0184221.
- with N. Smythe: "An ideal class invariant of knots". Bull. Amer. Math. Soc. 15: 707–709. 1964. doi:10.1090/s0002-9939-1964-0165516-2. MR 0165516.
- "On topologies for function spaces". Bulletin of the American Mathematical Society. 51: 429–432. 1945. doi:10.1090/s0002-9904-1945-08370-0. MR0012224
- with W. A. Blankinship: "Remarks on certain pathological open subsets of 3-space and their fundamental groups". Proc. Amer. Math. Soc. 1: 618–624. 1950. doi:10.1090/s0002-9939-1950-0042120-8. MR 0042120.
- "Torus Homotopy Groups". Proceedings of the National Academy of Sciences of the United States of America. 31 (2): 71–74. February 1945. doi:10.1073/pnas.31.2.71. PMC . PMID 16588687.
- "On fibre spaces. I". Bulletin of the American Mathematical Society. 49: 555–557. 1943. doi:10.1090/s0002-9904-1943-07969-4. MR 0008702.
- "On fibre spaces. II". Bulletin of the American Mathematical Society. 49: 733–735. 1943. doi:10.1090/s0002-9904-1943-08015-9. MR 0009109.
- "A characterization of absolute neighborhood retracts". Bulletin of the American Mathematical Society. 48: 271–275. 1942. doi:10.1090/s0002-9904-1942-07652-x. MR 0006508.
- "On Homotopy and Extension of Mappings". Proceedings of the National Academy of Sciences of the United States of America. 26 (1): 26–28. 15 January 1940. doi:10.1073/pnas.26.1.26. PMC . PMID 16577957.
- Fox, Ralph; Neuwirth, Lee (1962). "The braid groups". Mathematica Scandinavica. 10: 119–126. doi:10.7146/math.scand.a-10518. MR 0150755.
- Fox, R. H. (1950). "Recent developments of knot theory at Princeton" (PDF). In: Proceedings of the International Congress of Mathematicians, Cambridge, Massachusetts, U.S.A., August 30–September 6, 1950. vol. 2. pp. 453–458.
- Neuwirth, Lee P. (1964). "Review: Introduction to knot theory by R. H. Crowell and R. H. Fox" (PDF). Bull. Amer. Math. Soc. 70 (2): 235–238. doi:10.1090/s0002-9904-1964-11096-x.