Satellite operators preserve concordance

For the TOP annulus the ambient manifold is the standard compact product, and the given annulus and end collars have exactly the finite pair-code description in [F5]. Apply the external normal theorem [F3] inside $L[r]$ with those collars. Restrict the resulting tube to a compact disk bundle, still with a slightly wider domain, and export its forward/inverse, pair and collar certificates as in [F5]. This gives an actual relative tube in $V$ without ambient AC. For the smooth annuli in the other three categories, a finite chart partition of unity gives a metric and connection along the compact annulus; parallel transport and the compact inverse-function construction give the smooth tube. Only finitely many chart choices and least integer radii are needed. Thus the external TOP input used below is earned at this particular finite input; no global choice-free version of FQ is being asserted.

In the smooth product category, choose a tubular neighborhood of the smooth annulus A. Its oriented normal 2-plane bundle over A has relative Euler obstruction in H²(A,∂A;Z)≅Z. The boundary trivializations are the zero framings of K₀ and K₁; the relative obstruction is the difference of their self-linking numbers, hence zero. A relative framing therefore identifies a neighborhood of A with A×D² and agrees with the two prescribed endpoint zero-framed neighborhoods.

In that framed neighborhood insert $P$ in every disk fiber. Orient the inserted annulus by the interval direction followed by the pattern orientation. Its oriented boundary is then $-P(K_0)$ at the lower end and $P(K_1)$ at the upper end. The product collars of $A$ and the endpoint framings give product collars matching the ambient end collars, and the two endpoint patterns use the preferred zero framings. This proves concordance preservation for smooth annuli in a standard product.

In the exotic and localized cases the annulus is smooth, so the same smooth tubular construction is available in its given smooth ambient manifold. In the TOP case use the external input [F3] relative to the two actual smooth endpoint collars. In every case this gives an oriented rank-two vector normal bundle over $A$. Its obstruction to a framing extending the two preferred zero framings is its relative Euler number in $H^2(A,\mathrm{\partial }A;\mathbb{Z})$. Cap $A$ by boundary Seifert chains with opposite boundary orientations and push the caps into the product collars. Their union is a closed two-cycle. Its intersection with a relative small parallel copy of $A$ is the relative Euler number, because the two cap terms are precisely the endpoint zero-linking numbers. The cycle is zero over $R$ by [F4], so that Euler number is zero in $R$. The inclusion $\mathbb{Z}\hookrightarrow R$ is injective, hence it is zero integrally. Classification of oriented plane bundles with boundary framing now gives a framing of the actual normal bundle agreeing with both preferred endpoint framings. This proves the required relative framing, rather than inferring it from absolute triviality.

Insert the product annulus $P\times I$ into this framed neighborhood, as in step1.2. Its endpoint knots are the untwisted satellites because both endpoint framings are the preferred zero framings. In the TOP case the product model makes the new annulus locally flat; in the three smooth categories it is smooth. The ambient manifold is the original $W$, so its homeomorphism or $R$-homology conditions do not change. Thus the satellite assignment preserves concordance in all four categories and descends to the stated set maps.