Friedhelm Waldhausen
Encyclopedia
Friedhelm Waldhausen is a German
mathematician
known for his work in algebraic topology
.
After visits to Princeton University
, the University of Illinois and the University of Michigan
he moved in 1968 to Kiel
, where he habilitated (qualified to assume a professorship).
In 1969, he was professor at the Ruhr-Universität before in 1970 becoming a professor of mathematics at the Universität Bielefeld, an appointment he held until his retirement in 2004.
s. He dealt mainly with Haken manifold
s and Heegaard splitting
. Among other things, he proved that, roughly speaking, any homotopy equivalence of Haken manifolds is homotopic to a homeomorphism, i.e. that closed Haken manifolds are topologically rigid. He put forward the Waldhausen conjecture about Heegaard splitting.
In the mid-seventies, he extended the connection between geometric topology and algebraic K-theory
by introducing a kind of algebraic K-theory for topological spaces, closed. This led to new foundations for algebraic K-theory (using what are now called Waldhausen categories
) and also gave new impetus to the study of highly structured ring spectra
. Articles: Algebraic K-Theory of Topological Spaces I (1976) and Algebraic K-theory of spaces (1983).
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
mathematician
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....
known for his work in 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...
.
Academic life
He studied mathematics in Göttingen, Munich and Bonn, where he obtained his PhD in 1966 with "Eine Klasse von 3-dimensionalen Mannigfaltigkeiten" (A class of 3-dimensional manifolds) from the Rheinische Friedrich-Wilhelms-Universität. His advisor was Friedrich HirzebruchFriedrich Hirzebruch
Friedrich Ernst Peter Hirzebruch is a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation.-Life:He was born in Hamm, Westphalia...
After visits to 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....
, the University of Illinois and the 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...
he moved in 1968 to Kiel
Kiel
Kiel is the capital and most populous city in the northern German state of Schleswig-Holstein, with a population of 238,049 .Kiel is approximately north of Hamburg. Due to its geographic location in the north of Germany, the southeast of the Jutland peninsula, and the southwestern shore of the...
, where he habilitated (qualified to assume a professorship).
In 1969, he was professor at the Ruhr-Universität before in 1970 becoming a professor of mathematics at the Universität Bielefeld, an appointment he held until his retirement in 2004.
Publications
His early work was mainly on the theory of 3-manifold3-manifold
In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
s. He dealt mainly with Haken manifold
Haken manifold
In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface...
s and Heegaard splitting
Heegaard splitting
In the mathematical field of geometric topology, a Heegaard splitting is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.-Definitions:...
. Among other things, he proved that, roughly speaking, any homotopy equivalence of Haken manifolds is homotopic to a homeomorphism, i.e. that closed Haken manifolds are topologically rigid. He put forward the Waldhausen conjecture about Heegaard splitting.
In the mid-seventies, he extended the connection between geometric topology and algebraic K-theory
Algebraic K-theory
In mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequenceof functors from rings to abelian groups, for all integers n....
by introducing a kind of algebraic K-theory for topological spaces, closed. This led to new foundations for algebraic K-theory (using what are now called Waldhausen categories
Waldhausen category
In mathematics a Waldhausen category is a category C with a zero object equipped with cofibrations co and weak equivalences we, both containing all isomorphisms, both compatible with pushout, and co containing the unique morphisms\scriptstyle 0\,\rightarrowtail\, Afrom the zero-object to any...
) and also gave new impetus to the study of highly structured ring spectra
Ring spectrum (homotopy theory)
In stable homotopy theory, a ring spectrum is a spectrum E together with a multiplication mapand a unit mapwhere S is the sphere spectrum. These maps have to satisfy associativity and unitality conditions up to homotopy much in the same way as the multiplication of a ring is associative and unital...
. Articles: Algebraic K-Theory of Topological Spaces I (1976) and Algebraic K-theory of spaces (1983).
Honors
Among others, he was awarded the von Staudt Prize in 2004 along with Guenter Harder, and an honorary doctorate from the Universität Osnabrück.See also
- Graph manifoldGraph manifoldIn topology, a graph manifold is a 3-manifold which is obtained by gluing some circle bundles. They were invented and classified by the German topologist Friedhelm Waldhausen in 1967...
- Loop theoremLoop theoremIn mathematics, in the topology of 3-manifolds, the loop theorem is a generalization of Dehn's lemma. The loop theorem was first proven by Christos Papakyriakopoulos in 1956, along with Dehn's lemma and the Sphere theorem....
- Smith conjectureSmith conjectureIn mathematics, the Smith conjecture states that if f is a diffeomorphism of the 3-sphere, of finite order then the fixed point set of f cannot be a nontrivial knot....
- Surface subgroup conjectureSurface subgroup conjectureIn mathematics, the surface subgroup conjecture of Friedhelm Waldhausen states that the fundamental group of every closed, irreducible 3-manifold with infinite fundamental group has a surface subgroup. By "surface subgroup" we mean the fundamental group of a closed surface not the 2-sphere...
- Virtually Haken conjectureVirtually Haken conjectureIn topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken...
- History of knot theoryHistory of knot theoryFor thousands of years, knots have been used for basic purposes such as recording information, fastening and tying objects together. Over time people realized that different knots were better at different tasks, such as climbing or sailing. Knots were also regarded as having spiritual and...