On the Pontryagin-Thom Construction

Rasheed, Safarul (2025) On the Pontryagin-Thom Construction. Masters thesis, Indian Institute of Science Education and Research Kolkata.

Text (MS Dissertation of Safarul Rasheed (20MS178)) 20MS178_Thesis_file.pdf - Submitted Version Restricted to Repository staff only Download (732kB)

Abstract

We follow Differential Forms in Algebraic Topology by Raoul Bott and Loring Tu to define the de Rham cohomology H∗ dR(M) and cohomology with compact supports H∗ c (M) of a smooth manifold M. We define vector bundles π : E −→ M over a smooth manifold M and discuss their orientability. We use differential forms on E with compact support when restricted to the inverse images of compact subsets of M to define the compact vertical cohomology H∗ cv(E). Integration along the fiber π∗ : Ω∗cv(E) −→ Ω∗−n(M) is defined and shown to be a cochain map. This leads to the Thom Isomorphism theorem in de Rham cohomology. We define the Thom class Φ(E) ∈ Hn cv(E) of a vector bundle E −→ M and show that the Thom class and the Poincaré dual of the zero section of E can be represented by the same form. We also construct the Euler class e(E) and show that it is the pullback of the Thom class by the zero section. We proceed to follow Bundles, Homotopy, and Manifolds by Ralph Cohen to define characteristic classes for principal G-bundles, where G is a topological group. To identify the rings of characteristic classes, CharU(n)(Z) and CharO(n)(Z2), H∗(D(γn), S(γn)) must be studied, where γn is the universal U(n)- or O(n)-bundle. This leads to the Thom space Th(E) of a vector bundle E −→ X. We give E a Riemannian metric and consider the unit disk bundle D(E) and the unit sphere bundle S(E) and define Th(E) := D(E)/S(E). This leads to the Thom Isomorphism theorem in singular cohomology. We proceed to study George Whitehead’s construction of the J-homomorphism J : πm(SO(n)) −→ πm+n(Sn) generalizing Hopf’s construction for the case m = n. It is shown that J is an isomorphism for m = 1, n = 2. We use J to give a description of the Thom space Th(E), where E −→ Sn is an oriented rank k vector bundle, in terms of a CW complex structure. In particular, this describes Th(TSn). The particular case of Th(TS2) is described in detail.

Actions (login required)

View Item