mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, specifically differential and algebraic

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

, 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 ...

^{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 Ne ...

of dimension

2n

(since

n

-connected

2n

-manifolds are

homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...

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 dim ...

between this boundary and a sphere, and showing this hypothetical cobordism invalidates certain properties of the

Hirzebruch signature theorem In differential topology, an area of mathematics, the Hirzebruch signature theorem (sometimes called the Hirzebruch index theorem) is Friedrich Hirzebruch's 1954 result expressing the signature of a smooth closed oriented manifold by a linear combi ...

References

{{reflist Differential topology Algebraic topology Topology

See also

References