Thom isomorphism theorem

Thom isomorphism theorem
Name	Thom isomorphism theorem
Field	Algebraic topology
Introduced by	René Thom
Year	1952
Related	Cobordism theory, K-theory, cohomology, characteristic classes

Thom isomorphism theorem The Thom isomorphism theorem is a fundamental result in algebraic topology connecting the cohomology of a base space to the cohomology of a vector bundle's Thom space. It plays a central role in the development of cobordism theory, characteristic classes, and topological K-theory, and it has influenced work by Élie Cartan, Henri Poincaré, Jean Leray, and John Milnor. The theorem underpins constructions by René Thom and subsequent advances by Michel Atiyah, Raoul Bott, and Michael Atiyah and Friedrich Hirzebruch.

Statement

For a rank-n real vector bundle E over a CW complex or smooth manifold M, the theorem asserts an isomorphism between the reduced cohomology of the Thom space Th(E) and the cohomology of M shifted by degree n. In the classical singular cohomology with coefficients in a ring R this isomorphism is realized by cup product with a Thom class u_E in H^n(Th(E); R). The formulation and hypotheses were clarified in the work of Jean-Pierre Serre, Samuel Eilenberg, Norman Steenrod, and Henri Cartan, and later formalized in the contexts used by John Milnor and James D. Stasheff. Variants appear in complex K-theory developed by Michael Atiyah and Friedrich Hirzebruch and in cobordism developed by René Thom and Lev Pontryagin.

Thom class and orientation

The Thom class u_E is a cohomology class in H^n(Th(E); R) whose restriction to each fiber generates the cohomology of the one-point compactified fiber. Existence and uniqueness of u_E depend on orientability of E with respect to R; orientability conditions were studied by Élie Cartan and Évariste Galois in other contexts and clarified by René Thom and Lev Pontryagin in cobordism. For R = Z the existence of an integral Thom class corresponds to an orientation in the sense used by John Milnor and James Stasheff; for R = Z/2Z orientation issues relate to work of Henri Poincaré and Emmy Noether. The Thom isomorphism ties to characteristic classes such as the Stiefel–Whitney classes, Chern classes, and Pontryagin classes introduced by Hassler Whitney, Shiing-Shen Chern, and Lev Pontryagin, which detect obstructions to existence of the Thom class. The notion of orientation used here interacts with orientability concepts in the theories of Jean Leray, Georges de Rham, and André Weil.

Proofs and constructions

Proofs of the Thom isomorphism theorem use cellular approximation and spectral sequence techniques developed by Jean Leray, Jean-Pierre Serre, and Samuel Eilenberg, or use transversality and tubular neighborhood arguments associated with Hassler Whitney and René Thom. Constructions via Thom spaces invoke one-point compactification and pairings exploited in the work of John Milnor, James D. Stasheff, and Jacob Lurie in modern homotopy theoretic settings. Alternative proofs appear in the context of topological K-theory by Michael Atiyah and Friedrich Hirzebruch using Bott periodicity discovered by Raoul Bott, and in generalized homology theories via the Brown representability theorem studied by Edgar H. Brown Jr. Some treatments leverage sheaf-theoretic methods related to Alexander Grothendieck, Jean-Pierre Serre, and Henri Cartan, while others use surgery theory and cobordism techniques advanced by William Browder, Andrew Casson, and Dennis Sullivan.

Applications

The Thom isomorphism theorem underlies the Pontryagin–Thom construction central to René Thom's solution of cobordism problems and to work by Lev Pontryagin, John Milnor, and René Thom on manifold classification. It is instrumental in defining Gysin homomorphisms and pushforward maps in cohomology used by Raoul Bott, Michael Atiyah, and Isadore Singer in index theory, and by Alexander Grothendieck in the context of Riemann–Roch theorems. In topological K-theory, the Thom isomorphism yields the Thom class in K-theory used by Michael Atiyah and Frank Adams in Adams operations and the solution of the Hopf invariant one problem by J. F. Adams. The theorem also feeds into characteristic class calculations for fiber bundles in the work of Shiing-Shen Chern, Hassler Whitney, and René Thom, and into equivariant topology studied by Glen Bredon and Graeme Segal.

Generalizations and variants

Generalized cohomology theories such as complex K-theory, cobordism MU, and extraordinary theories studied by J. F. Adams and Douglas C. Ravenel admit Thom isomorphisms under appropriate orientability conditions; these generalizations appear in the work of Michael Atiyah, Friedrich Hirzebruch, and Dennis Sullivan. Equivariant Thom isomorphisms for group actions relate to work by Graeme Segal and Glen Bredon. Spectral and stable homotopy formulations are central to the stable homotopy category developed by J. P. May, Peter May, and Jacob Lurie, and to modular representations investigated by Daniel Quillen. Twisted Thom isomorphisms connect to the theory of gerbes and to the work of Jean-Luc Brylinski and André Henriques, while parametrized and fiberwise versions appear in the research of Ralph Cohen and Robert D. Edwards.

Examples and computations

Classic computations include the Thom isomorphism for the tangent bundle of the n-sphere S^n used in the proof of the generalized Poincaré conjecture approaches by René Thom and later work by Stephen Smale and Michael Freedman, and for complex line bundles such as the tautological bundle over complex projective space CP^n studied by Shiing-Shen Chern and Alexander Grothendieck. Computations of Thom classes for real vector bundles over real projective spaces RP^n relate to Stiefel–Whitney classes introduced by Hassler Whitney. Applications to Bott periodicity yield explicit K-theory Thom isomorphisms used by Raoul Bott and Michael Atiyah in calculations on unitary groups U(n) and orthogonal groups O(n). Examples in cobordism theory feature prominently in René Thom's classification results and in later computations by Milnor, Novikov, and Daniel Quillen.

Category:Algebraic topology