In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, specifically
differential and
algebraic topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, during the mid 1950's
John Milnor
John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Uni ...
was trying to understand the structure of
-connected manifolds
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
of dimension
(since
-connected
-manifolds are
homeomorphic
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
to spheres, this is the first non-trivial case after) and found an example of a space which is homotopy equivalent to a sphere, but was not explicitly diffeomorphic. He did this through looking at real vector bundles
over a sphere and studied the properties of the associated disk bundle. It turns out, the boundary of this bundle is homotopically equivalent to a sphere
, but in certain cases it is not diffeomorphic. This lack of diffeomorphism comes from studying a hypothetical
cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact space, compact manifolds of the same dimension, set up using the concept of the boundary (topology), boundary (French ''wikt:bord#French, bord'', giving ''cob ...
between this boundary and a sphere, and showing this hypothetical cobordism invalidates certain properties of the
Hirzebruch signature theorem.
See also
*
Exotic sphere
In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold ''M'' that is homeomorphic but not diffeomorphic to the standard Euclidean ''n''-sphere. That is, ''M'' is a sphere from the point of view of ...
*
Gromoll–Meyer sphere, special Milnor sphere
*
Oriented cobordism
References
{{reflist
Differential topology
Algebraic topology
Topology