# Norman Steenrod

**Norman Earl Steenrod** (April 22, 1910 – October 14, 1971) was an American mathematician most widely known for his contributions to the field of algebraic topology.[1]

Norman Steenrod | |
---|---|

Born | |

Died | 14 October 1971 61) | (aged

Nationality | American |

Alma mater | University of Michigan Harvard University Princeton University |

Known for | Eilenberg–Steenrod axioms Steenrod squares |

Scientific career | |

Fields | Mathematics |

Institutions | University of Chicago University of Michigan Princeton University |

Doctoral advisor | Solomon Lefschetz |

Doctoral students | José Ádem Peter J. Freyd Samuel Gitler Wu-Chung Hsiang Jerome Levine William S. Massey Paul A. Schweitzer Edwin Spanier George W. Whitehead |

## Life

He was born in Dayton, Ohio, and educated at Miami University and University of Michigan (A.B. 1932). After receiving a master's degree from Harvard University in 1934, he enrolled at Princeton University. He completed his Ph.D. under the direction of Solomon Lefschetz, with a thesis titled *Universal homology groups*. He held positions at the University of Chicago from 1939 to 1942, and the University of Michigan from 1942 to 1947. He moved to Princeton University in 1947, and remained on the Faculty there for the rest of his career. He died in Princeton.

## Work

Thanks to Lefschetz and others, the cup product structure of cohomology was understood by the early 1940s. Steenrod was able to define operations from one cohomology group to another (the so-called Steenrod squares) that generalized the cup product. The additional structure made cohomology a finer invariant. The Steenrod cohomology operations form a (non-commutative) algebra under composition, known as the Steenrod algebra.

His book *The Topology of Fibre Bundles*[2] is a standard reference. In collaboration with Samuel Eilenberg, he was a founder of the axiomatic approach to homology theory. See Eilenberg–Steenrod axioms.

## See also

## Publications

- Steenrod, Norman E. (1943). "Homology with local coefficients".
*Annals of Mathematics*.**44**(4): 610–627. doi:10.2307/1969099. MR 0009114. - Eilenberg, Samuel; Steenrod, Norman E. (1945). "Axiomatic approach to homology theory".
*Proceedings of the National Academy of Sciences of the United States of America*.**31**(4): 117–120. doi:10.1073/pnas.31.4.117. MR 0012228. PMC 1078770. PMID 16578143. - Steenrod, Norman E. (1947). "Products of cocycles and extensions of mappings".
*Annals of Mathematics*.**48**(2): 290–320. doi:10.2307/1969172. MR 0022071. - Steenrod, Norman E. (1962), Epstein, David B. A. (ed.),
*Cohomology operations*, Annals of Mathematics Studies,**50**, Princeton University Press, ISBN 978-0-691-07924-0, MR 0145525[3]

## References

- Steenrod, Norman, et al.
*First Concepts of Topology*. The Mathematical Association of America New Mathematical Library. Miami: 1966. - Milnor, John (1958). "Review: Norman Steenrod,
*The topology of fibre bundles*".*Bulletin of the American Mathematical Society*.**64**(4): 202–203. doi:10.1090/s0002-9904-1958-10211-6. - Szczarba, Robert H. (1964). "Review:
*Cohomology operations*. Lectures by N. E. Steenrod. Written and revised by D. B. A. Epstein".*Bulletin of the American Mathematical Society*.**70**(4): 482–483. doi:10.1090/s0002-9904-1964-11157-5.

## External links

- O'Connor, John J.; Robertson, Edmund F., "Norman Steenrod",
*MacTutor History of Mathematics archive*, University of St Andrews. - Norman Steenrod at the Mathematics Genealogy Project
- Michael Hoffman (2013) Norman Steenrod from US Naval Academy