Eells–Kuiper invariant

Eells–Kuiper invariant
Name	Eells–Kuiper invariant
Field	Differential topology; Spin geometry; Algebraic topology
Introduced	1962
Introduced by	James Eells; Noboru Kuiper
Related	Rokhlin invariant; Â-genus; Hirzebruch signature; Pontryagin class; Dirac operator

Eells–Kuiper invariant

The Eells–Kuiper invariant is an invariant of certain smooth closed manifolds introduced by James Eells and Noboru Kuiper in 1962. It assigns to a closed spin manifold of specific dimension a rational value mod 1 that refines information carried by Pontryagin classes and the Â-genus, and interacts with index theory for the Dirac operator, Rochlin-type results, and exotic sphere classification. The invariant plays a role in the study of differentiable structures on spheres, relations among characteristic classes, and obstructions in cobordism and surgery theory.

Definition and basic properties

For a closed smooth spin manifold M of dimension 4k+3 satisfying appropriate cohomological hypotheses, the Eells–Kuiper invariant is defined using a bounding spin manifold W of dimension 4k+4 and characteristic numbers built from Pontryagin classes and the signature. The construction uses the Hirzebruch signature theorem, the Â-genus defined by Friedrich Hirzebruch and Atiyah–Singer index ideas, and the Rokhlin-type congruences appearing in work of Rokhlin and Milnor. Its value is well-defined in Q/Z (often represented mod 1), independent of the choice of W by Gauß–Bonnet type cancellation with the signature and the index of the Dirac operator studied by Atiyah, Patodi, and Singer. Basic properties include additivity under connected sum (relating to work of Kervaire and Milnor on homotopy spheres), behavior under orientation reversal (related to Pontryagin classes transformations studied by Novikov), and constraints from stable framing results of Browder and Adams.

Construction and examples

The classical construction begins with a closed spin 7-manifold M whose second Stiefel–Whitney class vanishes and whose third integral cohomology satisfies torsion conditions studied by Bott and Samelson. Choose a compact spin 8-manifold W with ∂W = M; form the characteristic number combining p1(W)∪p1(W) and the signature σ(W) as in formulas influenced by work of Hirzebruch and Thom. The resulting rational number is taken modulo 1 to yield the invariant. Key examples appear among exotic 7-spheres classified by Milnor and Kervaire; for these, computation uses plumbing constructions introduced by Brieskorn and techniques of Wall. Other examples include total spaces of certain sphere bundles over four-manifolds studied by Smale, and circle bundles over spin 6-manifolds appearing in constructions by Crowley and Goette.

Computation and formulas

Explicit formulas express the invariant in terms of Pontryagin numbers p_i(W), the signature σ(W), and the index of the Dirac operator ind(D_W) as in Atiyah–Singer formulas. For dimension 7 the invariant μ(M) can be written as a combination (1/28)(p1(W)^2 − 2σ(W)) mod 1, echoing congruences related to Rokhlin and the signature theorem of Hirzebruch. Computations for homotopy spheres use the plumbing description of Milnor and the Eells–Kuiper values computed by Crowley and Escher for families of manifolds built via surgery theory of Wall and Kreck. Techniques employ spectral flow computations of families of Dirac operators developed by Atiyah, Patodi, Singer, and later analytic refinements by Bismut and Cheeger.

Relations to other invariants and topology

The Eells–Kuiper invariant refines and interacts with the Rokhlin invariant (for spin 3- and 7-manifolds) studied by Rokhlin and Kirby–Melvin, and with the Â-genus central to work of Hirzebruch and Atiyah–Singer. It links to the Kervaire–Milnor classification of exotic spheres and to the group of homotopy spheres Θ_n investigated by Kervaire and Milnor. Relations with quadratic forms and linking forms on torsion cohomology invoke the algebraic topology developed by Wall and Levine, while connections to analytic torsion and eta invariants exploit the analytic apparatus introduced by Atiyah and Patodi and extended by Dai and Freed. In high-dimensional surgery theory the invariant provides an obstruction appearing in works of Kreck, Sullivan, and Ranicki.

Applications in differential topology and spin geometry

Applications include distinguishing smooth structures on spheres and other manifolds, classification results for 7-manifolds by Crowley and Escher, and constraints on possible metrics with special holonomy studied by Joyce and Salamon. The invariant enters existence questions for metrics of positive scalar curvature through index-theoretic obstructions from Lichnerowicz and Rosenberg, and it informs classifications of almost contact and Sasakian structures on odd-dimensional manifolds in research by Boyer and Galicki. In gauge theory contexts the Eells–Kuiper invariant complements invariants from Donaldson and Seiberg–Witten theory when analyzing smooth structures on 4- and 7-dimensional manifolds.

History and development

The invariant was introduced by James Eells and Noboru Kuiper in 1962 in their study of manifolds with special spin and framing properties, building on prior results by Rokhlin, Hirzebruch, Kervaire, and Milnor. Subsequent developments linked the invariant to index theory through the Atiyah–Singer index theorem and to analytic eta invariants by Atiyah, Patodi, and Singer. Later classification work for 7-manifolds by Crowley, Escher, Kreck, and Eells himself expanded computational techniques and applications to exotic sphere theory and special holonomy, while analytic refinements by Bismut, Cheeger, and Dai clarified spectral contributions. Contemporary research continues in the study of spin cobordism, exotic smooth structures, and interactions with geometric analysis pursued by groups around Kreck, Crowley, Goette, and Hambleton.

Category:Differential topology