In mathematics, specifically differential and algebraic

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

, 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 Un ...

^{pg 14} was trying to understand the structure of

(n-1)

-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 ...

of dimension

2n

(since

n

-connected

2n

-manifolds are homeomorphic 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

V \to S^n

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

S^

, 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 manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same ...

between this boundary and a sphere, and showing this hypothetical cobordism invalidates certain properties of the Hirzebruch signature theorem.

References

{{reflist Differential topology Algebraic topology Topology

See also

References