The Milnor lambda candidate from a supplied filling

Assume the Axiom of Choice as inherited from the duality and characteristic-class suppliers. Let $M$ be a closed oriented smooth seven-manifold with $H^3(M;\mathbb{Z})=H^4(M;\mathbb{Z})=0,$ and let $W$ be a supplied compact oriented smooth eight-manifold with $\mathrm{\partial }W=M$. AC implies countable choice (AC implies DC implies countable choice), so $W$ has a finite CW homotopy model (Compact smooth manifolds have finite CW models under countable choice); each connected component is therefore in the path-connected CW-type domain of Pontryagin classes by complexification. More explicitly, the components $W_{\alpha }$ are open and path connected by local path connectivity, and compactness makes their number finite and each component compact. Every singular simplex lies in one component, so restriction gives a canonical isomorphism $H^4(W;\mathbb{Z})\cong {\prod }_{\alpha }{H}^4(W_{\alpha };\mathbb{Z})$. For a possibly disconnected filling, define $p_1(TW)$ to be the unique class whose restriction to each $W_{\alpha }$ is $p_1(T{W}_{\alpha })$ from that supplier. This agrees with its definition for connected $W$ and commutes with restriction and diffeomorphism pullback componentwise. For empty $W$ it is the zero class. The pair sequence $H^3(M;\mathbb{Z})\to H^4(W,M;\mathbb{Z})\stackrel{j}{\to }{H}^4(W;\mathbb{Z})\to H^4(M;\mathbb{Z})$ of Long exact sequence of a pair in singular cohomology shows that $j$ is an isomorphism: $H^3(M;\mathbb{Z})=0$ gives injectivity (uniqueness of a relative lift) and $H^4(M;\mathbb{Z})=0$ gives surjectivity (existence), the latter because the target group $H^4(M;\mathbb{Z})$ is zero. Define ${\bar{p}}_1(W):=j^{-1}{p}_1(TW)\in H^4(W,M;\mathbb{Z}),$ with $p_1$ the first Pontryagin class Pontryagin classes by complexification, and $q(W):=⟨{\bar{p}}_1(W)⌣{\bar{p}}_1(W),[W,M]⟩,$ the relative square evaluated on the relative fundamental class as in Relative cup product for an excisive triad; the identity $⟨{\bar{p}}_1⌣{\bar{p}}_1,[W,M]⟩=⟨{\bar{p}}_1⌣j({\bar{p}}_1),[W,M]⟩$ of The relative square equals the mixed evaluation makes the mixed form of the evaluation available. Finally define the filling-level candidate ${\lambda }_W(M):=2q(W)-\sigma (W)(\mathrm{m}\mathrm{o}\mathrm{d}7),$ where $\sigma (W)$ is the boundary signature of Boundary middle form and boundary signature and the congruence class is taken in $\mathbb{Z}\mathrm{/}7$.

This definition is conditional on the supplied filling $W$: it asserts no general existence of a compact oriented filling for a manifold satisfying the cohomology vanishing, and it asserts no independence of the choice of $W$. Well-definedness modulo seven under a change of filling, and invariance under orientation-preserving boundary diffeomorphisms, are the content of the following theorem The Milnor lambda invariant is well defined modulo seven ↗. Orientation reversal of a filling negates both $q$ and $\sigma$ and therefore negates ${\lambda }_W$; the class $p_1$ itself is orientation-independent. For the concrete Milnor bundles the filling $W=D({\xi }_{h,j})$ is explicitly available, so no universal bounding theorem is needed for the exotic-sphere examples of this page.