Home > J. W. Alexander

He was a pioneer in algebraic topology, setting the foundations for Henri Poincaré's ideas on homology theory and furthering it by founding cohomology theory, which developed gradually in the decade after he gave a definition of cochain ). For this, in 1928 he was awarded the Bôcher Memorial Prize. He also contributed to the beginnings of knot theory by inventing the Alexander invariant of a knot, which in modern terms is a graded module obtained from the homology of a cyclic covering of the knot complement . From this invariant, he obtained the first polynomial knot invariant.

With Garland Briggs , he also gave a combinatorial description of knot invariance based on certain moves, now (against the history) called the Reidemeister movesReidemeister moves In 1927, working with this diagrammatic form of knots, Kurt Reidemeister demonstrated that all the allowable moves on a knot could be reduced to three kinds of move on the diagram, shown left. These operations, now called the Reidemeist; and also a means of computing homological invariants from the knot diagram .

Alexander was also a noted mountaineer, having succeeded in many major ascents. When in Princeton, he liked to climb the university buildings, and always left his office window on the top floor of Fine Hall open so that he could enter by climbing the building.

1 See also

• Alexander horned sphereAlexander's Horned Sphere is one of the most famous pathological examples in mathematics. A closely related object is Antoine's horned sphere. The Horned Sphere was introduced by James Waddell Alexander in 1924 as a counterexample to his previous claim of
• Alexander polynomial
• Alexander cochain
• Alexander-Spanier cohomologyIn mathematics, particularly in algebraic topology Alexander-Spanier cohomology is a cohomology theory arising from differential forms with compact support on a manifold. It is similar to and in some sense dual to de Rham cohomology. It is named for J.
• Alexander duality

2 External link

• MacTutor biography

3 References

• James, I. M., Portrait of Alexander (1888--1971), Bull. Amer. Math. Soc. (N.S.) 38 (2001), no. 2, 123-129.
• Cohen, Leon W., James Waddell Alexander (1888-1971), Bull. Amer. Math. Soc. 79 (1973), 900--903.