Topological spaces

Vector bundles

Group cohomology

Homology groups

The group of all vector bundles over X

The group of all homotopy classes of maps from X to a point

The group of all homology groups of X

The group of all cohomology groups of X

The K-theory of a sphere is isomorphic to the integers

The K-theory of a torus is isomorphic to the integers

The K-theory of a Klein bottle is isomorphic to the integers

The K-theory of a projective space is isomorphic to the integers

It relates the index of an elliptic operator to the topological invariants of a manifold

It relates the homology groups of a manifold to its cohomology groups

It relates the K-theory of a manifold to its cohomology groups

It relates the homotopy groups of a manifold to its homology groups

Classifying vector bundles

Studying the topology of manifolds

Calculating the index of elliptic operators

All of the above

K-Theory is a generalization of cohomology theory

Cohomology theory is a generalization of K-Theory

K-Theory and cohomology theory are unrelated

K-Theory and cohomology theory are equivalent

It is a map from the K-group of a space to its cohomology ring

It is a map from the cohomology ring of a space to its K-group

It is a map from the homology group of a space to its K-group

It is a map from the K-group of a space to its homology group

It provides a framework for defining and studying the index of elliptic operators

It provides a framework for defining and studying the homology groups of a manifold

It provides a framework for defining and studying the cohomology groups of a manifold

It provides a framework for defining and studying the homotopy groups of a manifold

K-Theory can be used to study the algebraic geometry of varieties

Algebraic geometry can be used to study the K-Theory of varieties

K-Theory and algebraic geometry are unrelated

K-Theory and algebraic geometry are equivalent

Michael Atiyah

Isadore Singer

Alain Connes

All of the above

The Baum-Connes conjecture

The Novikov conjecture

The Milnor conjecture

All of the above

The development of topological K-Theory

The introduction of bivariant K-Theory

The application of K-Theory to string theory

All of the above

It provides a framework for classifying vector bundles over manifolds

It allows for the calculation of topological invariants such as the signature and Euler characteristic

It helps in understanding the relationship between the homology and cohomology groups of a manifold

All of the above

It establishes a connection between the K-Theory of a space and the K-Theory of its suspension

It provides a way to calculate the K-Theory of a space using its homology groups

It relates the K-Theory of a space to its cohomology groups

It allows for the classification of vector bundles over a space