Hood’s Reading Guide for the Thom Paper
Siddharth’s talk will cover chapter 2, with a focus on proving the main results in sections II.3 and II.6. My
talk will cover chapter 4, focusing on sections IV.4–7. I suggest not reading chapters 1 or 3, sections II.4
and II.7–9, or section IV.3. If you know what the cohomology of BO looks like, you could also skip section
II.5. Do read II.10.
A light skim of chapter 1 should suffice if you’re familiar with some basic differential geometry. It introduces
the notions of regular values, transversality (called t-regularity in the paper), and tubular neighborhoods.
An automorphism of a tubular neighborhood T is defined a self-diffeomorphism A of M that restricts to
the identity away from T and that preserves the fibers of T . The key point is as follows: Suppose that
f : V → M is smooth, N is a submanifold of M , and T is a tubular neighborhood of N in M . Then there
is an automorphism A of T so that A ◦ f is transverse to N . Also, if A and A0 are two automorphisms of T
such that A ◦ f and A0 ◦ f are both transverse to N then (A ◦ f )−1 (N ) and (A0 ◦ f )−1 (N ) are isotopic, so
if we only care about preimages up to isotopy (or some coarser equivalence relation), then it doesn’t matter
how we pick A. The last point is that such a map f induces a fiberwise isomorphism ν(f −1 (N )) → ν(N )
between normal bundles, so that it expresses ν(f −1 (N )) = f ∗ (ν(N )). This implies in particular that there
is a map between the Thom spaces.
Chapter 2 discusses G-representable (co)homology classes of a compact manifold, where G is a closed subgroup of O(k). Let ξG(k) be the universal k-bundle of structure group G, with total space EG(k) and base
space BG(k). Sections II.1 and II.2 contain the definitions of BG(k), the Thom space of a vector bundle,
the Thom isomorphism, and M G in particular.
Section II.3 is about G-representability. In the case of homology, if V n is a manifold, a class x ∈ Hk (V ) is said
to be represented by a submanifold W k ⊂ V if i∗ ([W ]) = x. The class x is G-representable if there is such
a submanifold W whose normal bundle has structure group G. Representable means O(k)-representable. In
the case of cohomology, suppose again that W k is a submanifold of V whose normal bundle has structure
G. Then the normal bundle ν(W ) is classified by a map ν : W → BG(n − k), which gives a map of bundles
ν(W ) → ξG(n − k), and so a map of Thom spaces T (ν) : T (ν(W )) → T (ξG(n − k)) = M G(n − k). Pick a
tubular neighborhood of W in V , say t : E(ν(W )) → V . Then we get a map t+ : V → T (ν(W )) given by
(
t−1 (v) v ∈ t(E(ν(W )))
v 7→
∞
otherwise
The map t+ is called the Pontryagin-Thom collapse map. From this we get a map T (ν)◦t+ : V → M G(n−k).
We that W represents the cohomology class (T (ν) ◦ t+ )∗ (U ) where U ∈ H n−k (M G(n − k)) is the Thom class.
Theorem II.1 says that these two notions of G-representability are Poincare dual – that is, if W represents
the homology class x and the cohomology class u, then u and x are Poincare dual classes. Siddharth will
prove theorem II.1 in his talk.
Theorem II.1 is interesting because it implies that a homology class x ∈ Hk (V ; A) is SO(k)-representable
if and only if the map u : V → K(A, n − k) classifying the Poincare dual of x factors through the map
U : M SO(n − k) → K(A, n − k) classifying the Thom class in H n−k (M SO(n − k)). Similarly, a homology
class x ∈ Hk (V ; Z/2) is representable if and only if the map u : V → HZ/2(n − k) factors through U :
M O(n − k) → HZ/2(n − k). It turns out that the stable version of this problem is given by representability
by singular manifolds. That is, a class x ∈ Hk (V ; Z/2) is representable by a singular manifold if there is a
manifold W k and a continuous map f : W → V such that f∗ ([W ]) = x. This is true if and only if the map
u : V → HZ/2 factors through the map U : M O → HZ/2
In subsection II.4, the paper discusses G-representability when G = e ⊆ O(k). You could safely skip this
subsection.
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In section II.5, the paper determines the cohomology of M O(k) and M SO(k). The paper does this calculation
in a fair amount of detail, but the answer is determined by the Thom isomorphism theorem, which says that
e ∗ (M O(k)) should be a free module over H ∗ (BO(k)) on a single generator in degree k, and similarly
H
for M SO(k) and BSO(k). Since the cohomology of BO(k) and BSO(k) is where characteristic classes
live, these rings are well known. At the end of section II.5, the paper describes the suspension maps
ΣM G(k) → M G(k + 1) which give rise to the spectrum M G. This is given by applying the Thom space
functor to the map ξG(k)⊕R → ξG(k +1) classifying the sum of the universal k-bundle with a trivial bundle.
You can view this sum as the external product (ξGk ) × (R ↓ {pt}), and since T (η × ν) = T (η) ∧ T (ν) and
T (R) = S 1 , we see that we have a map ΣM G(k) → M G(k + 1). Thom uses this to prove that πk+i (M G(k))
doesn’t depend on k for 0 ≤ i < k.
Section II.6 is the most important section in chapter 2. It contains the proof that M O splits as a wedge
of HZ/2’s, which Siddharth will give in his talk. The splitting of M O implies in particular that there is a
map s : HZ/2 → M O such that s∗ (U ) = ι. This immediately implies that all HZ/2 homology classes are
representable by a singular manifold. The map s also appears as a map s : HZ/2(n − k) → M O(n − k)
only defined in the stable range – that is, it is only defined on the 2(n − k) skeleton of HZ/2(n − k). We
deduce that every Z/2 homology class in the stable range is representable by an embedded submanifold:
given u : X → HZ/2(n − k), we get a map s ◦ u : X → M O(n − k) representing the homology class Poincare
dual to u as long as the image of u lies in the 2(n − k) skeleton of HZ/2(n − k). In particular, this holds
whenever X = X 2(n−k) is a 2(n − k)-dimensional CW complex.
Section II.7 you can skip. It examines M O(k) for k ≤ 3 to get some results about representability of
cohomology classes in degrees less than or equal to 3.
Section II.8 you can skip. It is the M SO version of section II.6, except that Thom is unable to prove
the corresponding result about M SO, which is significantly harder since M SO doesn’t split into Eilenberg
Maclane spectra. Instead Thom explicitly constructs a space K that looks like M SO through dimension 8.
This gives him a condition for SO(k) representability of integral cohomology classes in certain cases, but not
in general. There is a problem in dimension 9 that makes it impossible to continue this construction over Z.
After localizing M SO at a prime, it’s possible to continue the construction to get what’s called the BrownPeterson spectrum, which would give necessary and sufficient conditions for classes in H k (X, Z/p) to be
SO(k)-representable. If you want to read this, note that the k-invariant of K is a map K(Z, k) → K(Z, k +5)
which Thom denotes as St53 . This map takes a cohomology class, reduces mod 3, applies P 1 , and then applies
the integral mod-3 Bockstein.
Section II.9 you can skip. It is the MSO version of section II.7. If you want to read it, you should read
section II.8 carefully first.
Section II.10 you should read. It has a proof
W that any integral class has a multiple that is representable, and
deduces that M SO splits rationally into k HZ[4k] into an HZ cell in each dimension divisible by 4. At the
end of section II.10 is a complete list of results from chapter II.
Chapter III is short, and consists primarily of an unnecessarily complicated (but nicely geometric) proof that
all Z/2 homology classes are representable by a singular manifold – using modern language, this immediately
follows because M O splits.
Chapter IV is about the oriented and unoriented cobordism rings, denoted Ωk and Rk respectively. Section IV.1 defines cobordism as an equivalence relation and section IV.2 quotes prior results that StiefelWhitney numbers are unoriented cobordism invariants (and hence also oriented cobordism invariants) and
that Pointryagin numbers are oriented cobordism invariants.
Section IV.3 extends the results from chapter 1 to manifolds with boundary. I don’t recommend reading it,
at least not very carefully.
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Section IV.4 defines Lk (V ), the set L-equivalence classes of k-dimensional submanifolds of V . In effect, this
is cobordism over V . It also proves theorem IV.6, which says that that Lk (V n ) ∼
= [V, M SO(n − k)] as long
as k and n are chosen in the stable range. The Thom collapse makes another appearance in the proof of
this theorem, as well as all of the chapter 1 stuff about transversality. When one side of this isomorphism
is a group, the other is also a group, and this isomorphism respects the group structure. In general, M SOcohomology of a space X is given by singular manifolds in X up to cobordism over X. The corresponding
results also hold for unoriented L-equivalence and M O.
Section IV.5 proves lemma IV.7, which says that Lk (S n ) ∼
= Ωk as long as n is large enough that k-dimensional
n
manifolds embed into R in a unique way up to isotopy. From theorem IV.6 and lemma IV.7, theorem IV.8
follows immediately: Ωk ∼
= πk (M SO) and likewise Rk ∼
= πk (M O).
Section IV.6 proves that if all Stiefel-Whitney numbers of a manifold vanish, then that manifold is unoriented
null-cobordant. This is a pretty simple deduction from the fact that MO splits.
Section IV.7 proves that R ∼
= Z/2[Vn ]n6=2i −1 , where Vn is a certain n-dimensional manifold. This is the last
thing I’ll cover in my talk. In the case that n is even, it shows that Vn = [RP2n ] modulo decomposables.
Section IV.8 gives the first few oriented cobordism groups, which were calculated in section II.9. It deduces
from II.10 that Ω ⊗ Q = Q[V4k ]k∈N (it incorrectly cites II.5 for this). These classes V4k are represented
modulo decomposables by CP2k . At the very end, Thom demonstrates the smallest two cases of the signature
theorem.
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