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

**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] }

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*.

Steenrod 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 was editor of the Annals of Mathematics and a member of the National Academy of Sciences. He died in Princeton, survived by his wife, the former Carolyn Witter, and two children.^{ [2] }

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*^{ [3] } is a standard reference. In collaboration with Samuel Eilenberg, he was a founder of the axiomatic approach to homology theory. See Eilenberg–Steenrod axioms.

- Steenrod, Norman E. (1943). "Homology with local coefficients".
*Annals of Mathematics*.**44**(4): 610–627. doi:10.2307/1969099. JSTOR 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. JSTOR 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^{ [4] }

**Algebraic topology** is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.

In mathematics, specifically in homology theory and algebraic topology, **cohomology** is a general term for a sequence of abelian groups associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.

**Witold Hurewicz** was a Polish mathematician.

**Samuel Eilenberg** was a Polish-American mathematician who co-founded category theory and homological algebra.

**Henri Paul Cartan** was a French mathematician who made substantial contributions to algebraic topology. He was the son of the French mathematician Élie Cartan and the brother of composer Jean Cartan.

**Armand Borel** was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in algebraic topology, in the theory of Lie groups, and was one of the creators of the contemporary theory of linear algebraic groups.

In mathematics, particularly algebraic topology and homology theory, the **Mayer–Vietoris sequence** is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups. The result is due to two Austrian mathematicians, Walther Mayer and Leopold Vietoris. The method consists of splitting a space into subspaces, for which the homology or cohomology groups may be easier to compute. The sequence relates the (co)homology groups of the space to the (co)homology groups of the subspaces. It is a natural long exact sequence, whose entries are the (co)homology groups of the whole space, the direct sum of the (co)homology groups of the subspaces, and the (co)homology groups of the intersection of the subspaces.

**Solomon Lefschetz** was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations.

**Sir William Vallance Douglas Hodge** was a British mathematician, specifically a geometer.

**Daniel Gray** "**Dan**" **Quillen** was an American mathematician. He is known for being the "prime architect" of higher algebraic *K*-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 1978.

**Donald Clayton Spencer** was an American mathematician, known for work on deformation theory of structures arising in differential geometry, and on several complex variables from the point of view of partial differential equations. He was born in Boulder, Colorado, and educated at the University of Colorado and MIT.

In mathematics, the **cohomology operation** concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if *F* is a functor defining a cohomology theory, then a cohomology operation should be a natural transformation from *F* to itself. Throughout there have been two basic points:

- the operations can be studied by combinatorial means; and
- the effect of the operations is to yield an interesting bicommutant theory.

**Ralph Hartzler Fox** 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.

In algebraic topology, a **Steenrod algebra** was defined by Henri Cartan (1955) to be the algebra of stable cohomology operations for mod cohomology.

In mathematics, specifically in algebraic topology, the **Eilenberg–Steenrod axioms** are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular homology, developed by Samuel Eilenberg and Norman Steenrod.

**William Schumacher Massey** was an American mathematician, known for his work in algebraic topology. The Massey product is named for him. He worked also on the formulation of spectral sequences by means of exact couples, and wrote several textbooks, including *A Basic Course in Algebraic Topology* (ISBN 0-387-97430-X).

**Andrew Hugh Wallace** was a Scottish-American mathematician.

**Sze-Tsen Hu**, also known as **Steve Hu**, was a Chinese-American mathematician, specializing in homotopy theory.

**Ivan Smith** is a British mathematician who deals with symplectic manifolds and their interaction with algebraic geometry, low-dimensional topology, and dynamics. He is a professor at the University of Cambridge.

- ↑ Steenrod, Norman, et al.
*First Concepts of Topology*. The Mathematical Association of America New Mathematical Library. Miami: 1966. - ↑ "Norman Steenrod, Expert in Topology".
*The New York Times*. October 16, 1971. p. 34. Retrieved March 29, 2020. - ↑ 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 .

- 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

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