# The odd-primary Thom module: an explicit reduction

<a id="DG-CHAR-13F-thom.proof"></a>

This proof identifies the actual odd-primary Thom module, using the full operation-algebra computation in [the one-group companion](odd-primary-one-group-cohomology.md), Sections 8–9. The [exterior resolution](odd-primary-exterior-resolution.md) then computes its possible Ext degrees; [finite detection](finite-adams-tower-detection.md) supplies the homotopy implication.

All graded vector spaces below are nonnegatively graded over \(k=\mathbb F_p\), where \(p\) is an arbitrary odd prime. Tensor products use the graded sign convention. Every element is a finite sum. The graded dual of a vector space finite dimensional in each degree is the direct sum of its degreewise duals.

## 1. The exact operation-algebra input

The input to the topological application is the following statement, denoted **O**.

The full algebra \(A\) of stable mod-\(p\) cohomology operations, including Bocksteins, is a connected graded Hopf algebra finite dimensional in every degree. Its graded dual has the presentation

\[
H=A^\#=k[\xi_1,\xi_2,\ldots]\otimes\Lambda(\tau_0,\tau_1,\ldots),
\quad |\xi_n|=2(p^n-1),\quad |\tau_n|=2p^n-1.
\tag{1}
\]

Set \(\xi_0=1\). With the right-coaction convention on cohomology, the coproduct is

\[
\begin{aligned}[c]
\Delta\xi_n&=\sum_{i=0}^n\xi_{n-i}^{p^i}\otimes\xi_i,\\
\Delta\tau_n&=\tau_n\otimes1+
                    \sum_{i=0}^n\xi_{n-i}^{p^i}\otimes\tau_i.
\end{aligned}
\tag{2}
\]

The positive generators have counit zero. For the degree-two first Chern class \(x\) of the universal complex line, its right coaction is

\[
\psi(x)=\sum_{j\geq0}x^{p^j}\otimes\xi_j.
\tag{3}
\]

Equation (3) is interpreted operation-degree by operation-degree: only finitely many terms contribute to an operation of a specified degree. It is not an assertion that the entire displayed series is an element of the uncompleted algebraic tensor product. Operations obey naturality, stability and the external Cartan formula.

The [one-group companion](odd-primary-one-group-cohomology.md), Section 9, together with the [coordinate proof](odd-primary-operation-coordinates.md), Sections 2–5, proves **O** for the full stable-operation algebra. Constructing reduced powers alone would not establish (1)–(3).

## 2. Right exterior freeness, with a specified change of variables

First work algebraically: let \(H\) be the Hopf algebra (1)-(2), and let \(A=H^\#\). The ideal generated by the positive \(\xi_n\) is a Hopf ideal. Indeed (2) sends every such generator to the sum of its two endpoint terms and terms with a positive \(\xi\) in both factors. Thus its quotient is

\[
T=\Lambda(\bar\tau_0,\bar\tau_1,\ldots),
\qquad \Delta\bar\tau_n=\bar\tau_n\otimes1+1\otimes\bar\tau_n.
\]

Write \(q\colon H\twoheadrightarrow T\) for the quotient and \(E=T^\#\hookrightarrow A\) for its graded dual. Dualizing the displayed primitive coproduct, or expanding it on an exterior monomial, identifies

\[
E=\Lambda(Q_0,Q_1,\ldots),\qquad |Q_n|=2p^n-1.
\tag{4}
\]

In particular \(Q_n^2=0\), and interchanging two distinct generators changes sign. A consistent graded pairing fixes the signs of the ordered exterior basis; no divided-power factorial occurs in this identification.

Let \(P=k[\xi_1,\xi_2,\ldots]\). Give \(H\) its right \(T\)-coaction

\[
\rho=(1\otimes q)\Delta:H\longrightarrow H\otimes T.
\]

It satisfies

\[
\rho(\xi_n)=\xi_n\otimes1,\qquad
\rho(\tau_n)=\tau_n\otimes1+
               \sum_{i=0}^n\xi_{n-i}^{p^i}\otimes\bar\tau_i.
\tag{5}
\]

Define odd elements of \(H\) recursively by

\[
\lambda_0=\tau_0,\qquad
\lambda_n=\tau_n-\sum_{i=0}^{n-1}\xi_{n-i}^{p^i}\lambda_i.
\tag{6}
\]

Every summand has degree \(2p^n-1\). For each finite initial set of indices this is a triangular change of exterior generators over the even polynomial ring \(P\), with diagonal entries one. Its inverse is the explicit equation
\(\tau_n=\lambda_n+\sum_{i<n}\xi_{n-i}^{p^i}\lambda_i\).
Consequently the monomials \(\xi^R\lambda_I\), with \(I\) a finite ordered set, are a vector-space basis of \(H\). Passing through the finite initial sets covers every element of the infinite algebra.

Induction in (5)-(6) gives

\[
\rho(\lambda_n)=\lambda_n\otimes1+1\otimes\bar\tau_n.
\tag{7}
\]

In detail, \(\rho\) fixes each polynomial coefficient. The terms
\(\xi_{n-i}^{p^i}\otimes\bar\tau_i\) for \(i<n\) cancel against the second coaction terms of the subtracted \(\lambda_i\); the remaining first terms combine to \(\lambda_n\otimes1\), and the last term is \(1\otimes\bar\tau_n\). This proves the assertion without treating the original \(\tau_n\) as primitive relative to \(T\).

There is therefore an isomorphism of right \(T\)-comodules

\[
\begin{aligned}[c]
\Phi:P\otimes T&\longrightarrow H,\\
f\otimes\bar\tau_I&\longmapsto f\lambda_I,
\end{aligned}
\tag{8}
\]

where the coaction on the source is \(1_P\otimes\Delta_T\). Multiplication and the graded signs extend (7) to every exterior monomial. Each graded piece is finite dimensional, so dualizing (8) yields

\[
A\cong P^\#\otimes E
\quad\text{as graded right }E\text{-modules}.
\tag{9}
\]

The side in (9) is fixed by the definition of \(\rho\): pairing \(\rho h\) with \(a\otimes e\) is pairing \(h\) with the product \(ae\). Thus the second dual factor has the right regular \(E\)-action. The tensor decomposition is degreewise finite, not a replacement of a direct sum by a completion. Taking the monomial dual basis of \(P\) in (9) proves right freeness, not only flatness or an equality of dimensions.

The coinvariants of \(P\otimes T\) are exactly \(P\otimes1\). To verify this, write an element using linearly independent polynomial coefficients; for each coefficient its \(T\) factor must obey \(\Delta t=t\otimes1\), and application of \(\epsilon\otimes1\) then gives \(t=\epsilon(t)1\). By (8),

\[
\{h\in H:\rho h=h\otimes1\}=P.
\tag{10}
\]

Let \(E_+\) be the augmentation ideal. A functional \(h\in H=A^\#\) annihilates the left ideal \(AE_+\) exactly when all pairings of \(\rho h\) with \(a\otimes e\), \(e\in E_+\), vanish. By the counit identity this is equivalent to the condition in (10). Degreewise finite duality now gives

\[
(A/AE_+)^\#=P.
\tag{11}
\]

Equation (2) makes \(P\) a Hopf subalgebra of \(H\): its coproduct, counit and product restrict, and the connected graded antipode restricts by its degree recursion. Its annihilator in \(A\) is therefore a two-sided Hopf ideal. By (11) this annihilator is precisely \(AE_+\). In particular, the notation

\[
B=A/AE_+=A\otimes_E k
\tag{12}
\]

is justified both as a quotient algebra and as an induced left \(A\)-module. We have not assumed that a left ideal is automatically two-sided.

Since \(B^\#=P\), its basis \(b_R\) dual to the monomials \(\xi^R\) has degree

\[
|b_R|=\sum_{j\geq1}2r_j(p^j-1).
\tag{13}
\]

Every such degree is divisible by four. The sequence \(R\) is finitely supported, including \(R=0\) for the identity.

## 3. The actual stable Thom coefficient coalgebra

Assume **O** from Section 1, and let

\[
C=H^*(MSO;k).
\]

Here the stable grading places the Thom class in degree zero. More explicitly, represent degree \(q\) by
\(\widetilde H^{2m+q}(\operatorname{Th}(\gamma_{2m});k)\)
for sufficiently large \(m\), using the Thom isomorphism and normal stabilization. Theorem 5.1 of *Pontryagin classes and oriented universal cohomology* gives, for \(2m>q\), precisely the polynomials in the stable Pontryagin classes of total degree \(q\). The extra rank-\(2m\) Euler generator has degree \(2m\), so it is absent in this range. The stabilization maps preserve every Pontryagin class and the fibre-normalized Thom class and are isomorphisms in this range.

Here is why taking the stable object introduces no unrecorded class. Put
\(Y_m=\Sigma^{-2m}\Sigma^\infty\operatorname{Th}(\gamma_{2m})\), with the maps induced by adjoining an oriented trivial two-plane. The telescope model of \(MSO\) is the cofiber of \(1-\mathrm{shift}:\bigvee_mY_m\to\bigvee_mY_m\). Applying cohomology gives the exact sequence

\[
0\longrightarrow\operatorname{coker}D_{q-1}
\longrightarrow H^q(MSO;k)
\longrightarrow\ker D_q\longrightarrow0,
\quad
D_q:\prod_m H^q(Y_m;k)\longrightarrow\prod_m H^q(Y_m;k),
\quad (z_m)\longmapsto(z_m-f_m^*z_{m+1}).
\]

In each fixed degree the transition maps are eventually isomorphisms. To solve \(D_qz=w\), choose \(z_M=0\) after this point, solve
\(z_{m+1}=(f_m^*)^{-1}(z_m-w_m)\) successively for \(m\geq M\), and then solve
\(z_m=w_m+f_m^*z_{m+1}\) backwards for \(m<M\). Thus every \(D_q\) is surjective, its kernel is the stable value, and the cohomology of the telescope is exactly that value. This proves the required stability comparison, rather than presuming an inverse-limit identification.

Thus the Thom isomorphisms identify the coefficient vector space as

\[
C=k[p_1,p_2,\ldots]u,\qquad |p_j|=4j,\quad |u|=0.
\tag{14}
\]

This is a coefficient description, **not** an assertion that ordinary cup product in a finite Thom space makes its Thom class a multiplicative identity. In particular, (14) shows that \(C_q=0\) unless \(4\mid q\), and that every component is finite dimensional.

Whitney sum gives a coalgebra on (14). At ranks \(2m,2n\), the typed maps are the bundle-sum classifying map

\[
BSO(2m)\times BSO(2n)\longrightarrow BSO(2m+2n)
\]

and the associated Thom map

\[
\operatorname{Th}(\gamma_{2m})\wedge
\operatorname{Th}(\gamma_{2n})\longrightarrow
\operatorname{Th}(\gamma_{2m+2n}).
\]

Pullback sends the total Thom class to the external product of the two Thom classes. This follows from its value one on each oriented fibre, which characterizes it under the Thom isomorphism. The field Kunneth isomorphism and stabilization give

\[
\Delta_C(p_j u)=\sum_{a+b=j}(p_a u)\otimes(p_b u),
\quad p_0=1,\quad \Delta_Cu=u\otimes u,
\tag{15}
\]

extended to coefficient polynomials by the Whitney formula. The counit is restriction to a trivial fibre, followed by desuspension. Associativity and the unit of bundle sum give coassociativity and the counit laws; they may also be checked on the coefficient generators in (15).

Every \(Q_i\) has odd degree, so (14) implies \(Q_i c=0\) for every \(c\in C\). By Section 2 the \(A\)-action therefore factors through the algebra \(B\). Naturality and external Cartan for the typed Thom maps give

\[
\Delta_C(bc)=\sum(-1)^{|b''||c'|}b'c'\otimes b''c'',
\tag{16}
\]

for \(\Delta_B b=\sum b'\otimes b''\) and \(\Delta_Cc=\sum c'\otimes c''\). All degrees here are multiples of four, so the displayed signs are one. Restriction to the trivial fibre makes the counit a \(B\)-module map. Thus \(C\), with its actual stable-operation action, is a connected \(B\)-module coalgebra of exactly the kind required by the existing general-field free-module criterion.

## 4. Faithfulness of the orbit of the Thom class

It remains to prove that \(i:B\to C\), \(b\mapsto bu\), is injective. No dimension-series argument suffices for this step.

For \(m\) universal complex line bundles, the underlying real bundle has rank \(2m\) and its complex orientation. Write

\[
X_m=(\mathbb{CP}^{\infty})^m,\qquad
H^*(X_m;k)=k[x_1,\ldots,x_m],\quad |x_i|=2.
\]

Its classifying map \(f_m:X_m\to BSO(2m)\) sends \(p_j\) to the elementary symmetric polynomial \(e_j(y_1,\ldots,y_m)\), where \(y_i=x_i^2\). Indeed the complexification of a complex line considered as a real plane has Chern roots \(x_i,-x_i\); the definition \(p_j=(-1)^j c_{2j}\) gives total Pontryagin class \(\prod_i(1+x_i^2)\). The zero section of its Thom space pulls the Thom class back to its Euler class

\[
e=x_1\cdots x_m.
\tag{17}
\]

For a stable coefficient \(cu\), denote substitution \(p_j\mapsto e_j(y)\) in \(c\) by \(r_m(cu)\). Zero-section pullback is then \(e\,r_m(cu)\). This identity applies to any fixed stable degree by choosing the Thom rank sufficiently large.

Odd operations vanish on \(H^*(X_m;k)\), which is concentrated in even degrees, so its action also factors through \(B\). From (3) and Cartan,

\[
\psi(e)=\prod_{i=1}^m\left(\sum_{j\geq0}x_i^{p^j}\otimes\xi_j\right).
\tag{18}
\]

For a sequence \(R=(r_1,r_2,\ldots)\), let
\(\ell(R)=\sum_jr_j\) and \(a_j=(p^j-1)/2\).
The coefficient of \(\xi^R\) in (18) is

\[
b_R(e)=e\,m_{\lambda(R)}(y_1,\ldots,y_m),
\tag{19}
\]

where \(\lambda(R)\) has \(r_j\) parts equal to \(a_j\), and \(m_\lambda\) is the sum of the **distinct** monomials whose exponent list is the partition \(\lambda\), padded with zeros to length \(m\). It is zero if \(\ell(R)>m\).

To check coefficients, select the term with index \(j\) from exactly \(r_j\) of the \(m\) distinct factors in (18). Every assignment of indices to factors produces one monomial \(\prod_i x_i^{p^{j_i}}\), with coefficient one. Since the values \(p^j\) are distinct, two different assignments produce different exponent lists. After dividing by \(e\), these are exactly the distinct monomials of \(m_{\lambda(R)}(y)\). There is no factorial associated with repeated parts. For example, at \(p=3\) with \(R=(3,0,\ldots)\) and \(m=3\), the result after division is \(y_1y_2y_3\), not \(3!y_1y_2y_3=0\).

Apply naturality to the zero section and to \(f_m\). For \(m\) large enough for the degree of \(b_R\),

\[
e\,r_m(b_Ru)=b_R(e).
\]

The polynomial ring \(k[x_1,\ldots,x_m]\) is an integral domain and \(e\ne0\). Cancelling \(e\) and using (19) gives

\[
r_m(b_Ru)=m_{\lambda(R)}(y_1,\ldots,y_m).
\tag{20}
\]

Suppose \(\sum_R c_R b_Ru=0\), with finitely many nonzero coefficients. Choose \(m\) large enough for every involved stable degree and every \(\ell(R)\). The positive integers \(a_j\) are strictly increasing, so distinct sequences \(R\) give distinct partitions \(\lambda(R)\). Their monomial symmetric polynomials have disjoint monomial supports and nonzero coefficients one. Substituting (20) forces every \(c_R=0\). This proves injectivity of \(i\) over every odd prime field, including primes smaller than \(m\). In particular no division by \(m!\), averaging over a symmetric group, or characteristic-zero argument has been used.

## 5. The decomposition and its precise consequence

Let \(V=C/B_+C\), choose a graded linear section \(s:V\to C\), and define

\[
\theta:B\otimes V\longrightarrow C,\qquad b\otimes v\longmapsto b\,s(v).
\tag{21}
\]

The general-field Theorem D.1 in *Stiefel-Whitney numbers and unoriented bordism* applies by (14)-(16) and the injectivity proved in Section 4. For clarity, its two exact mechanisms are these. Degree induction proves surjectivity because \(c-s\pi c\in B_+C\) is a sum of positive-degree operations applied to lower-degree elements. For injectivity, apply \((1\otimes\pi)\Delta_C\) to (21). Its leading component in the second-factor degree is \(bu\otimes v\); all terms with a positive-degree operation in the second factor vanish under \(\pi\). The leading map is \(i\otimes1\), which is injective. Each element has finite support, so a largest nonzero second degree detects it. This proves (21) is an isomorphism, not just a surjection.

By (14), \(V\) is concentrated in degrees divisible by four. Choosing a homogeneous basis \((v_\lambda)\) therefore gives, with \(m_\lambda=|v_\lambda|\),

\[
H^*(MSO;k)\cong
\bigoplus_\lambda\Sigma^{m_\lambda}(A\otimes_E k),
\qquad 4\mid m_\lambda.
\tag{22}
\]

This is an isomorphism of left \(A\)-modules, since both actions factor through \(B\), and (12) identifies that quotient with the induced module. Equation (9) supplies the **right** \(E\)-freeness needed to induce the existing exterior resolution. Together they imply

\[
\operatorname{Ext}^{s,t}_A(H^*(MSO;k),k)=0
\quad\text{if }t-s\not\equiv0\pmod4.
\tag{23}
\]

As a consequence of the already proved module isomorphism, one may also compute the dimensions of the generator space. With a variable \(z\) recording one quarter of the degree,

\[
\sum_{d\geq0}\dim_k V_{4d}\,z^d
=\frac{\prod_{j\geq1}(1-z^{(p^j-1)/2})}
        {\prod_{n\geq1}(1-z^n)}.
\tag{24}
\]

Every coefficient is a finite calculation. The numerator cancels precisely the denominator factors indexed by \((p^j-1)/2\). Thus the dimension is the number of partitions with none of these parts. Equation (24) does not assert that arbitrary particular characteristic polynomials indexed by these partitions form a free basis; (21) constructs a free basis from a section of the actual quotient.

## 6. From the module to bordism

The [exterior resolution](odd-primary-exterior-resolution.md) proves the Ext-degree implication in (23). The [finite detection theorem](finite-adams-tower-detection.md) turns an empty Ext stem into divisibility by each prime power, without an infinite-convergence assumption. Combined with the exact finite-odd-order theorem, it gives the [seven-dimensional bounding theorem](oriented-seven-manifold-bounding.md).

René Thom, [*Travaux de Milnor sur le cobordisme*](https://www.numdam.org/item/SB_1958-1960__5__169_0.pdf), pp. 171–173, gives the operation-on-roots route to this module calculation.