The CDR pattern zero-surgery homology cobordism

The smooth disk tube uses [F5] directly. In the TOP branch the input is a single locally flat proper disk in the standard topological ball with its given end collar. Encode that standard ball, disk and finitely many pair charts as in [F7], apply the normal existence assertion of [F6] in $L[r]$, and export a compact tube with a wider domain and its actual boundary-collar certificates. Its zero-framing calculation and all subsequent Mayer–Vietoris and group pushout calculations are performed on the resulting actual objects in $V$. Thus no full-AC hypersurface supplier and no general homology-absoluteness assertion enters the no-choice Statement.

Write $P_0=P(U)$ and $R=\mathbb{Z}[1\mathrm{/}\mathrm{\mid }n\mathrm{\mid }]$. First take the supplied smooth disk $D\subset B$. Apply the finite smooth disk-tube construction [F5] to this actual disk and its boundary collar. They give $N=D^2\times D^2$, product-shaped at its boundary. The framing at $P_0$ is zero. Indeed the small parallel disk is disjoint from $D$; boundary linking equals their intersection over $R$ in the $R$-homology ball. To check the latter identity here, two choices of bounding two-chain differ by a two-cycle, which bounds since $H_2(B;R)=0$; the chain intersection boundary formula makes their intersections with a proper relative cycle equal. Compare with the pushed-in boundary Seifert chain to get boundary linking. Thus the integral boundary linking is zero in $R$, hence zero in $\mathbb{Z}$. Put $Z=\overline{B\setminus N}$. Its boundary is the actual preferred-longitude filling $M(P_0)$. In the excisive collared decomposition $B=Z\cup N$, the overlap is $D^2\times S^1$. Mayer–Vietoris, with its connecting map represented by the meridian disk, gives $H_0(Z;R)=R$, $H_1(Z;R)=R[{\mu }_0]$ and $H_j(Z;R)=0$ for $j\ge 2$. This uses only the stated $R$-homology of $B$. For a pseudo-slice disk the same construction applies smoothly with $R=\mathbb{Z}$. For a TOP disk this calculation is valid if its boundary-compatible zero-framed product normal neighborhood is actually supplied; this paragraph does not construct that neighborhood from local flatness.

Use the accepted actual standard infection cobordism Ordinary homology of standard infection and connected-sum cobordisms for $P_0\cup \eta$ and $K$, with $\mathrm{lk}(P_0,\eta )=n$, and call it $E$. Its construction glues the two surgery products along the axis tube and companion filling solid torus, exchanging ${\lambda }_{\eta }={\mu }_K$ and ${\mu }_{\eta }={\lambda }_K$. Its third boundary is exactly $M(P(K))$, with its zero slope and oriented exterior marking, as proved in that supplier's2.1. Glue $Z$ to the $M(P_0)$ end, with the reversed boundary orientation needed to form the oriented cobordism $V$. Product collars make $V$ smooth for smooth input and TOP for supplied TOP tube input. The two remaining ends are $M(K)$ and $M(P(K))$. In $H_1(E;R)$ the incoming meridians and outgoing meridian have coordinates $(1,n,1)$, respectively for $P_0,K,P(K)$. In $H_2(E;R)=R^2$ the incoming classes are the coordinate basis and the outgoing class is $(1,n)$. These are the actual inclusion maps of the accepted supplier, not inferred solely from abstract ranks.

Here is the full gluing homology calculation. The common surgery $M(P_0)$ has $H_j=R$ in degrees0,1,2,3 and zero otherwise. The product-union spine defining $E$ has the Mayer–Vietoris sequence for the common solid torus; it gives $H_3(E;R)=R^2$ with the two incoming oriented fundamental classes as basis, $H_4(E;R)=0$, and no homology above4. This is the same finite sequence as the accepted supplier's3.1, now in degree3: the solid torus has no degree2 or3 homology. In the sequence for $V=E{\cup }_{M(P_0)}Z$, the degree3 map sends $1$ to the first basis vector of $R^2$; the degree2 map sends $1$ to the first basis vector of $R^2$; the degree1 map is the diagonal inclusion $R\to R\oplus R$ with unit coordinates, since ${\mu }_0$ generates both groups. All three are injective and the degree0 difference map is injective. Exactness in every degree therefore gives $H_j(V;R)=R$ for $j=0,1,2,3$ and zero otherwise. The $M(K)$ inclusion is an isomorphism in degree3 by the second fundamental-class coordinate, degree2 by the second basis coordinate, and degree1 by multiplication by $n$, a unit in $R$. For the outgoing end, degree2 is multiplication by $n$ because $(1,n)$ becomes $n$ in the quotient by the first basis vector; degree1 is the unit1. In degree3 the oriented boundary of a relative fundamental chain of $E$ is the sum of its incoming components minus the outgoing component. Consequently its outgoing degree3 class is the sum of the incoming classes; after gluing $Z$, it becomes the second generator. Degree0 is an isomorphism for both connected ends. Thus BOTH boundary inclusions are homology isomorphisms in every degree. The exact marked degree1 relation is $[{\mu }_K]=n[{\mu }_{P(K)}]$, with the signs determined by the supplied peripheral orientations. It is equality up to the unit $n$, not literal equality for general $n$. This is the precise meaning of the cited CFHH 'rel meridians' convention.

Suppose now that the supplied disk lies in a simply connected ball (standard TOP or pseudo-smooth) and that $P$ is strong winding $\pm 1$. Van Kampen for $B=Z\cup N$ identifies ${\pi }_{1}B$ with ${\pi }_{1}Z\mathrm{/}⟨\text{\hspace{0.17em}⁣}⟨{\mu }_0⟩\text{\hspace{0.17em}⁣}⟩$, so ${\mu }_0$ normally generates ${\pi }_{1}Z$. Strong winding says that $\eta$ normally generates the closure knot-complement group; its image therefore normally contains ${\mu }_0$ in ${\pi }_{1}Z$, since the knot exterior maps into $Z$. Hence $\eta$ normally generates ${\pi }_{1}Z$ too. The actual product-union construction of $E$ and van Kampen give ${\pi }_{1}V=({\pi }_{1}Z\ast {\pi }_{1}M(K))\mathrm{/}⟨\text{\hspace{0.17em}⁣}⟨\eta {\mu }_K^{-1}⟩\text{\hspace{0.17em}⁣}⟩\mathrm{.}$ No injectivity of the gluing maps is assumed. Killing ${\mu }_K$ kills ${\pi }_{1}M(K)$ and then $\eta$, hence ${\pi }_{1}Z$; so ${\mu }_K$ normally generates ${\pi }_{1}V$. Killing ${\mu }_0$ instead kills ${\pi }_{1}Z$, hence $\eta ={\mu }_K$ and the other factor, so ${\mu }_0$ normally generates as well. The same ${\mu }_0$ is the outgoing satellite meridian in the actual infection boundary. Thus the image of either boundary group normally generates ${\pi }_{1}V$. The integer homology calculation is step3.1 with $n=\pm 1$. The smooth pseudo-slice branch and the conditional supplied-neighborhood TOP branch follow with their actual categories; no smooth identification of a homotopy ball is used.

The smooth localized and pseudo-slice calculations in steps1.1–4.1 are unchanged, including both boundary homology maps and the signed relation $[{\mu }_K]=n[{\mu }_{P(K)}]$. For the arbitrary locally flat disk in the standard topological ball, [F6] now supplies the actual zero-framed product neighborhood that those same steps require. Their collar gluing, ordinary-coefficient Mayer–Vietoris sequences and van Kampen pushouts apply to this TOP neighborhood, so they prove the stated integral homology and normal-generation conclusions there too. No identification of a smooth homotopy ball with the standard smooth ball is used. All original category and closure hypotheses are retained.