Norman Steenrod
Encyclopedia
Norman Earl Steenrod was a preeminent mathematician
most widely known for his contributions to the field of algebraic topology
.
, 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.
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
. José Ádem
studied the relations between the Steenrod operations, and discovered secondary cohomology operations. Using these secondary operations, Frank Adams
obtained the definitive answer to the problem of counting the number of linearly independent vector fields on a sphere
.
His book The Topology of Fiber Bundles is a standard reference. In collaboration with Samuel Eilenberg
, he was a founder of the axiomatic approach to homology theory
. See Eilenberg–Steenrod axioms.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
most widely known for his contributions to the field of algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which 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.Although algebraic topology...
.
Life
He was born in Dayton, OhioDayton, Ohio
Dayton is the 6th largest city in the U.S. state of Ohio and the county seat of Montgomery County, the fifth most populous county in the state. The population was 141,527 at the 2010 census. The Dayton Metropolitan Statistical Area had a population of 841,502 in the 2010 census...
, and educated at Miami University
Miami University
Miami University is a coeducational public research university located in Oxford, Ohio, United States. Founded in 1809, it is the 10th oldest public university in the United States and the second oldest university in Ohio, founded four years after Ohio University. In its 2012 edition, U.S...
and University of Michigan
University of Michigan
The University of Michigan is a public research university located in Ann Arbor, Michigan in the United States. It is the state's oldest university and the flagship campus of the University of Michigan...
(A.B. 1932). After receiving a master's degree from Harvard University
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
in 1934, he enrolled at Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
. He completed his Ph.D. under the direction of Solomon Lefschetz
Solomon Lefschetz
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.-Life:...
, with a thesis titled Universal homology groups. He held positions at the University of Chicago
University of Chicago
The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
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 productCup product
In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. This defines an associative graded commutative product operation in cohomology, turning the cohomology of a space X into a...
structure of cohomology
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
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
Steenrod algebra
In algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod p cohomology.For a given prime number p, the Steenrod algebra Ap is the graded Hopf algebra over the field Fp of order p, consisting of all stable cohomology operations for mod p...
. José Ádem
José Ádem
José Ádem Chaín was a Mexican mathematician who worked inalgebraic topology, and proved the Ádem relations between Steenrod squares....
studied the relations between the Steenrod operations, and discovered secondary cohomology operations. Using these secondary operations, Frank Adams
Frank Adams
John Frank Adams FRS was a British mathematician, one of the founders of homotopy theory.-Life:He was born in Woolwich, a suburb in south-east London. He began research as a student of Abram Besicovitch, but soon switched to algebraic topology. He received his Ph.D. from the University of...
obtained the definitive answer to the problem of counting the number of linearly independent vector fields on a sphere
Vector fields on spheres
In mathematics, the discussion of vector fields on spheres was a classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification of division algebras....
.
His book The Topology of Fiber Bundles is a standard reference. In collaboration with Samuel Eilenberg
Samuel Eilenberg
Samuel Eilenberg was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.He earned his Ph.D. from University of Warsaw in 1936. His thesis advisor...
, he was a founder of the axiomatic approach to homology theory
Homology theory
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...
. See Eilenberg–Steenrod axioms.
See also
- Abstract nonsenseAbstract nonsenseIn mathematics, abstract nonsense, general abstract nonsense, and general nonsense are terms used facetiously by some mathematicians to describe certain kinds of arguments and methods related to category theory. roughly speaking, category theory is the study of the general form of mathematical...
- Eilenberg-Steenrod axiomsEilenberg-Steenrod axiomsIn mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common...
- Steenrod algebraSteenrod algebraIn algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod p cohomology.For a given prime number p, the Steenrod algebra Ap is the graded Hopf algebra over the field Fp of order p, consisting of all stable cohomology operations for mod p...
- Steenrod operations
- Steenrod problemSteenrod problemIn mathematics, and particularly homology theory, Steenrod's Problem is a problem concerning the realisation of homology classes by singular manifolds.-Formulation:...