An embedded sphere with nontrivial normal bundle is not valid framed surgery data

Given: AC and the manifold $M=S^2\times S^2$ with the product orientation, the diagonal $\mathrm{\Delta }=\{(x,x):x\in S^2\}$, and the framing lemma of this page.

The normal bundle of the diagonal is canonically the tangent bundle: for a smooth boundaryless manifold $M$, the difference map $T(M\times M){\mathrm{\mid }}_{{\mathrm{\Delta }}_M}\to TM$, $(v,w)\mapsto w-v$ has kernel $T{\mathrm{\Delta }}_M$ and induces a canonical isomorphism of smooth vector bundles ${\nu }_{{\mathrm{\Delta }}_M}\to TM$; under the stated orientation conventions it is orientation-preserving.

The diagonal self-intersection is the Euler number of the tangent bundle: for a closed oriented smooth $n$-manifold $M$ with the product orientation on $M\times M$, the diagonal is a closed oriented embedded $n$-submanifold with $2\dim {\mathrm{\Delta }}_M=\dim (M\times M)$ and ${\mathrm{\Delta }}_M\cdot {\mathrm{\Delta }}_M=⟨e(TM),[M]⟩$, where $e(TM)$ is the Euler class and the self-intersection number is that of The self-intersection number of a complementary-dimensional oriented submanifold.

The self-intersection number is the Euler number of the normal bundle: for a closed oriented embedded submanifold $A$ with $2\dim A=\dim M$, the self-intersection number is well defined and satisfies $A\cdot A=⟨e({\nu }_A),[A]⟩$; the value is independent of the tubular embedding and of the transverse push-off. The Euler class here is that of Euler class by zero-section pullback of the Thom class.

A nowhere-zero section forces the Euler data to vanish: every trivial bundle ${\epsilon }_B^n$ of positive rank $n\ge 1$, with its standard product orientation, has $e({\epsilon }_B^n)=0$.

The framing obstruction lives in the normal bundle of the surgery sphere: an embedded $p$-sphere $S\subseteq \mathrm{int}M$ occurs as the underlying sphere of a framed embedded surgery sphere if and only if its normal bundle is trivial.

The tangent field $X(p)=e_3-zp$ on $S^2$ has zeros only at the poles. In the projection charts $(x,y)\mapsto (x,y,\pm \sqrt{1-x^2-y^2})$ its components are $(-zx,-zy)$, whose derivatives are $-I_2$ and $I_2$ at the two poles, both with determinant $+1$. The zero signs above and the Euler-number formula for a rank-two bundle on a closed oriented surface in The self-intersection number is the Euler number of the normal bundle give $⟨e(T{S}^2),[S^2]⟩=2$. A nowhere-zero section would force this Euler class to vanish, contradicting that evaluation; hence $T{S}^2$ has no such section and is not trivial (Normal push-off zeros are the self-intersection points, A nowhere-zero section forces the Euler data to vanish).

Framed embedded surgery sphere: a framed embedded surgery sphere is an embedding $S^p\times D^q\hookrightarrow M$ whose restriction to the disk factor exhibits a trivialization of the normal bundle of its underlying sphere.