The Groove of Math

or in other words, the Devil is in the Details

Random Exercise #2: Problem 11-C in Milnor’s Characteristic Classes

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Todays’ random exercise will be the pretty famous exercise 11-C  one can find in Milnor’s Book. Let me recall the statement:

Let M = M^n and A = A^p be compact oriented manifolds with smooth embedding i : M \rightarrow A. Let k = p-n. Show that the Poincare duality isomorphism  \frown \mu_A : H^k(A) \rightarrow H_n(A) maps the cohomology class u^{'}|A dual to M to the homology class (-1)^{nk} i_{*} (\mu_M). Assume moreover that the normal bundle \nu^k is oriented so that \tau_M \oplus \nu^k is orientation preserving isomorphic to \tau_A|M.

Hint: consider the following commutative diagram:


where N is a tubular neighbourhood of M in A.


Before going all in with the proof, let me recall two things:

Theorem [Corollary 11.2 page 117  in Milnor’s Characteristic Classes] If M is embedded as  a closed subset of A, the cohomology ring H^*(E,E_0; \Lambda) associated with the normal bundle \nu^k\colon E \to M in A is canonically isomorphic to the cohomology ring H^*(A,A\setminus M; \Lambda). Here \Lambda can be any coefficient ring.

Definition: Let \nu^k be oriented, let u \in H^k(E,E_0\setminus M; Z) be its Thom Class (denoted  occasionally also Th(\nu^k))  Define u'  \in  H^k(A,A\setminus M; Z) to be the image of the Thom Class under the above iso. 

Theorem [Thm 11.3 page 119]: If M is embedded as a closed subset of A, then the composition of the two restriction homomorphisms

H^k(A,A\setminus M) \to H^k(A)\to H^k(M)

with \mathbb{Z}_2 coefficients, maps the fundamental class u' to the top Stifle-Whitney class w_k(\nu^k) of the normal bundle. Similarly, if \nu^k is oriented, then the corresponding composition with integer coefficients maps the integral fundamental class u' to the Euler Class e(\nu^k).

Definition: The image of u' \in H^k(A) is called the dual cohomology class to the submanifold M of codimension k. The terminology will be explained in the exercise

Proof of exercise:  Denote with \mu_A \in H_p(A) the fundamental class of A. Start picking in the upper left angle the element u' \otimes \mu_A \in H^k(A,A\setminus M)\otimes H_p(A). We need to chase this element a little bit. The left “path” (\downarrow and then \rightarrow) is straightforward. We end up with the element u'|A\frown \mu_A.

For the “right” path (\rightarrow and then \downarrow and then \leftarrow ) we can use the so called coherency of the fundamental class \mu_A and Corollary 11.2 to prove that \mu_A is sent to \mu_{A,A\setminus M} which is the generator of H_p(A, A\setminus M) ( See Bredon’s Topology and Geometry Theorem 7.8 page 344). Following the next arrow, which is an iso, we end up in H^k(N,N\setminus M)\otimes H_p(N,N\setminus M) with Th(\nu)\otimes \mu_{N,N\setminus M} . For the curious reader (yes please!) interested in this kind of stuff, I worked with compactly supported sections of the orientation bundle of A, which we know being trivial since A is orientable, for more details see Bredon’s Topology and Geometry chapter VI.7.

Since the vertical right arrow (after choosing the Thom class of the normal bundle as the cohomology class in H^k(N,N\setminus M)\cong H^k(E,E_0)) is the so called homological Thom isomorphism (tom Dieck’s Algebraic Topology Theorem 18.1.2 page 439), it’s an isomorphism so it maps \mu_{N,N\setminus M} to \mu_M (up to a sign and using H_n(N)\cong H_n(M)). The conclusion follows at once.

(inspired by my answer here in M.Se)


Author: RicPed

Young mathematician :)

One thought on “Random Exercise #2: Problem 11-C in Milnor’s Characteristic Classes

  1. Pingback: Filling the Details #2: Representing 2-homology classes of a 4-manifold. | The Groove of Math

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