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Finite winding-zero infection towers preserve the required derived quotients

Statement

Fix $n \geq 1$ and a finite marked smooth infection sequence $J_{j + 1} = P_{j} (J_{j})$ , $0 \leq j < n$ , with unknotted axes $η_{j}$ and winding number zero. Supply compatible product tubes, peripheral paths and finite disk-band Seifert-surface models with local ambient roundings for all knots $J_{j}$ and $P_{j} (U)$ . Let $E_{j}$ be their actual standard infection cobordisms. Let $W_{n}$ be any compact connected oriented topological four-manifold with a supplied collared boundary component $M (J_{n})$ , and form $W_{j} = W_{j + 1} \cup_{M (J_{j + 1})} E_{j} (j = n - 1, \dots, 0)$ using the marked collars. Other boundary components of $W_{n}$ are allowed. Base paths are supplied coherently through the finite gluings. Write $μ_{j}$ for the positive meridian of $J_{j}$ and $G_{j} = π_{1} (W_{j})$ .

For $0 \leq k < n$ , inclusion induces an epimorphism $q_{k} : G_{k + 1} \to G_{k}$ and a natural isomorphism $G_{k + 1} / G_{k + 1}^{(n - k + 1)} ≅ G_{k} / G_{k}^{(n - k + 1)} .$ Moreover $μ_{k} \in G_{k}^{(n - k)}$ . For any supplied finite commutative-ring list $R_{0}, \dots, R_{n}$ , define the mixed series by $P^{0} G = G, P^{i + 1} G = \ker ⁣ (P^{i} G ⟶ (P^{i} G / [P^{i} G, P^{i} G]) \otimes_{Z} R_{i}) .$ The same $q_{k}$ identifies $G_{k + 1} / P^{i} G_{k + 1}$ with $G_{k} / P^{i} G_{k}$ for $i \leq n - k + 1$ , and identifies $P^{i} G_{k + 1} / P^{i + 1} G_{k + 1}$ with $P^{i} G_{k} / P^{i + 1} G_{k}$ for $0 \leq i < n - k + 1$ . These are assertions for the actual inclusion maps. No nontriviality or order assertion about a meridian in a mixed quotient is included; such an assertion requires the separate Blanchfield argument. No choice principle or smoothing of $W_{n}$ is assumed.

Facts & Assumptions

Given: The finite marked knot models, zero-winding axes, actual product cobordisms, collared $W_{n}$ , paths and finite coefficient-ring list.

[F1]

Adjoining the actual standard infection cobordism along its outgoing surgery induces precisely the quotient by the normal closure of the companion longitude, and its companion meridian is the marked pattern-axis core. Standard infection adjunction kills precisely the companion longitude

[F2]

For the supplied finite knot models, exterior and zero-surgery groups are normally generated by their meridian, and the preferred exterior longitude lies in the exterior group's second derived subgroup. Preferred longitudes of finite disk-band knots lie in the second derived subgroup

[F3]

Killing a normal subgroup contained in $G^{(r)}$ preserves the ordinary derived quotient at index $r$ and all mixed-series quotients and factors through index $r$ , for the actual quotient map. Deep derived relations preserve finite mixed-coefficient quotients

[F4]

The standard product-spine markings identify the outgoing satellite meridian with the pattern meridian. The peripheral calculation gives $H_{1} (M (P_{j} (U)); Z) = Z [μ_{P_{j}}]$ and the axis class $[η_{j}] = lk (P_{j} (U), η_{j}) [μ_{P_{j}}] = 0$ : this is the longitude calculation in the two-component exterior, before filling back the axis tube. Ordinary homology of standard infection and connected-sum cobordisms

[F5]

Zero first-homology class is equivalent to membership in the commutator subgroup, without choice. The first Hurewicz map is abelianization

Proof

1.1F1F2F4F5givenconstructalgebra

Fix $k < n$ and work first in $G = G_{k + 1}$ . For $k + 1 \leq j < n$ , the piece $E_{j}$ is already present in $W_{k + 1}$ . Let $A_{j}$ be the image of the group of its pattern boundary $M (P_{j} (U))$ in $G$ . Its meridian is the image of $μ_{j + 1}$ by [F4], so [F2] implies $A_{j} \subseteq ⟨ ⁣ ⟨ μ_{j + 1} ⟩ ⁣ ⟩_{G}$ . By [F4] and [F5], its axis lies in the commutator subgroup of that pattern-boundary group. Its image is $μ_{j}$ by [F1]. Hence $μ_{j} \in [A_{j}, A_{j}] \subseteq [⟨ ⁣ ⟨ μ_{j + 1} ⟩ ⁣ ⟩_{G}, ⟨ ⁣ ⟨ μ_{j + 1} ⟩ ⁣ ⟩_{G}] .$ Only this inclusion is needed; no equality between a single axis normal closure and an entire commutator subgroup is asserted.

2.1F2step 1.1constructalgebra

Starting with $μ_{n} \in G^{(0)}$ , descend through the finite range $j = n - 1, \dots, k + 1$ . If $μ_{j + 1} \in G^{(n - j - 1)}$ , its normal closure is contained in that derived subgroup because derived subgroups are normal: conjugation preserves every finite commutator word. Step 1.1 then gives $μ_{j} \in G^{(n - j)}$ . Consequently $μ_{k + 1} \in G^{(n - k - 1)}$ ; when $k = n - 1$ this just says $μ_{n} \in G$ . The companion exterior $X_{J_{k}}$ is an actual subspace of the outgoing boundary $M (J_{k + 1})$ . By normal generation of that surgery group in [F2], the entire image of its group, and hence the image of the companion-exterior group, lies in $⟨ ⁣ ⟨ μ_{k + 1} ⟩ ⁣ ⟩_{G} \subseteq G^{(n - k - 1)}$ . Its longitude lies in its second derived subgroup by [F2]. Expanding the finite commutator products under the inclusion homomorphism therefore puts the longitude image $λ_{k}$ in $G^{(n - k + 1)}$ .

3.1F1F2F3F4F5step 1.1step 2.1constructalgebra

By [F1], the actual inclusion gives $G_{k} = G / ⟨ ⁣ ⟨ λ_{k} ⟩ ⁣ ⟩_{G}$ . Derived subgroups are normal, so step 2.1 puts the entire kernel in $G^{(n - k + 1)}$ . Apply [F3] with $r = n - k + 1$ to obtain exactly the ordinary and mixed-quotient isomorphisms in the Statement, for the actual inclusion map and every supplied finite ring prefix. To check the remaining meridian depth, observe in the new piece $E_{k}$ that the image $A_{k}$ of its pattern-boundary group lies in the normal closure of $q_{k} (μ_{k + 1})$ , and that $μ_{k} \in [A_{k}, A_{k}]$ , by the same argument as step 1.1. The image of $μ_{k + 1}$ has depth at least $n - k - 1$ by step 2.1, because a homomorphism sends each finite commutator word to the corresponding word of its images. Thus $μ_{k} \in G_{k}^{(n - k)}$ .

4.1F1F2F3F4F5step 1.1step 2.1step 3.1algebra∎

The topological adjunctions used only supplied three-dimensional boundary collars and the explicit smooth product pieces. Finite paths fix all based inclusions; changing any such path conjugates its relevant image and does not affect normal closures or derived-depth statements. The calculation never asserted that a derived meridian is nonzero, of infinite order, or of prime order: those require additional homological and Blanchfield inputs. Likewise positive-characteristic mixed factors are not called PTFA merely because the rational prefixes are. Every commutator, representative and path used above belongs to a fixed finite calculation, and [F3] proves the kernel comparisons without an infinite representative selection. Thus the full stated finite-tower naturality and depth assertions hold in the supplied TOP/smooth scope without a choice axiom.

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Sources

Cha, Amenable L2-theoretic methods and knot concordance (standard reference, not scraped)