The Hilbert-Smith Conjecture for Four-Manifolds: A Proof via the GM2 Module

THE HILBERT-SMITH CONJECTURE FOR FOUR-MANIFOLDS: A PROOF VIA THE GM2 MODULE SHIQIAO LUO Abstract. We prove the Hilbert–Smith conjecture in dimension four: a compact connected 4–manifold admits no faithful continuous action of Zp. The proof verifies the GM2 compatibility module using a low– degree, Leray–type equivariant cover: in degrees ≤3, the Borel equivariant cohomology groups Hd Gk(Nk;Fp)are finite–dimensional and satisfy a Mittag–Leffler stabilization property along the tower. Crucially, this control is required only in a truncated range and on intersections of bounded order, and is obtained without any smooth or PL assumptions. Alexander–Čech duality then produces nontrivial homology on the free side, contradicting an algebraic quotient–level inverse–limit vanishing engine. The constructions of the natural boundary map δG(independent of neighborhood choices and functorial in k) and the tower compatibility of Smith detector classes are given in detail in the appendices. All arguments are carried out in the purely continuous category. Contents 1. Introduction 2 2. Motivation and Conceptual Framework 3 2.1. Topological Good Blocks and the Fixed–Point Tower 3 2.2. Low–Degree Equivariant Coverage 3 2.3. Reconstruction of the Boundary Map 4 2.4. From Low–Degree Stabilization to GM2 4 2.5. Conceptual Perspective 5 3. Set-up, towers, and the GM2 module 5 4. Algebraic inverse–limit vanishing on the free side 6 5. Lemma A: low–degree equivariant coverage in the continuous 4–dimensional category 6 6. From low–degree stabilization to GM2 7 7. Hilbert–Smith Conjecture in dimension 4 8 Conclusion and Remarks 8 References 8 Appendix A. A schematic of towers and dualities 9 Appendix B. Proof of Proposition 4.1 9 B.0. Setup: the free–side tower and the detectable subsystem 9 B.1. Compact windows and slice covers 10 B.2. Nerves and finite–dimensional targets 11 Date: December 21, 2025. 1 2 SHIQIAO LUO B.3. Transfer generated subspaces and coinvariants 12 B.4. Final module: put the detectable class into the subspace that is killed by the detection quotient. 14 Appendix C. Supplement for Lemma A 19 C.1. Lemma A: Low–degree equivariant Leray cover (ELC–weak) 19 C.2. Cofinal systems and Borel cochains 23 Appendix D. Compatibility of Smith Detector Classes 26 1. Introduction The Hilbert–Smith conjecture (HS) asserts that a compact connected manifold admits no faithful continuous action of the p–adic integers Zp. We prove the 4–dimensional Hilbert–Smith conjecture via a modular argument centered on the GM2 compatibility module: the existence of a nontrivial compatible family of Čech cohomology classes along the fixed–point tower. Construction. Let Pk=pkZp⊴Zpact continuously on a compact connected 4–manifold M, and set Gk=Zp/Pk∼ =Z/pk. Denote Fk= Fix(Pk), choose nested Gk–invariant neighborhoods Nkof the fixed sets Fk, and set Vk:= M\Nk. Such a Gk–invariant neighborhood system exists by standard equivariant topology, and its construction is given in the appendix. Each Nkserves as a topological good block surrounding the fixed points of the Pk–action, replacing the smooth tubular neighborhoods that would normally appear under local linearity assumptions by purely topological constructions. Note that Nkbeing Gk–invariant does not mean it is pointwise fixed: it contains the fixed set Fkas a subset while generally including non–fixed points, so the construction is not circular. The goal is to produce a nonzero compatible family 0={αk}, αk∈ˇ Hr(Fk;Fp), r ∈ {1,2}, so that the inverse limit on the fixed side is nontrivial. Alexander–Čech duality then yields nontrivial homology on the free side, contradicting an algebraic quotient–level inverse–limit vanishing statement for the tower (Vk). The route to GM2. We establish GM2 from a low–degree stabilization principle. Using a low–degree, Leray–type equivariant cover (Lemma A, ELC–weak), we show that for degrees d≤3the Borel equivariant cohomology groups Hd Gk(Nk;Fp)are finite–dimensional and that the images ImHd Gj(Nj;Fp)→Hd Gk(Nk;Fp) stabilize for j≫k. Crucially, this finiteness and stabilization are required only in a truncated range of degrees and on intersections of bounded order, and are sufficient for the equivariant spectral sequence arguments used in HILBERT–SMITH IN DIMENSION FOUR VIA GM2 3 the proof. This finite–dimensional stabilization replaces the usual Noetherian module–theoretic input and yields a compatible family of Smith detector classes on the fixed–point tower. We then derive GM2 and complete the argument using Alexander–Čech duality and an algebraic quotient–level inverse–limit vanishing engine. Additionally, in what follows, by “vanishing on the free side” we mean vanishing after applying a natural detection quotient: the eventual images in the inverse system become zero in the detected groups (see Proposition 4.1). 2. Motivation and Conceptual Framework The guiding principle of this work is that the obstruction underlying the Hilbert–Smith conjecture in dimension four is not fundamentally algebraic or representation–theoretic, but structural and topological. Continuous p–adic actions destroy the local linear and smooth features on which classical slice and tubular–neighborhood arguments depend. Accordingly, the strategy pursued here is to reconstruct the GM2 compatibility module entirely within a purely topological and homotopical framework, avoiding all smooth, PL, or locally linear assumptions. The proof proceeds by isolating the minimal topological input required to force a contradiction between the fixed–point tower and the free side, and by showing that this input persists in the continuous category through low–degree cohomological control. 2.1. Topological Good Blocks and the Fixed–Point Tower. In dimension four, the fixed–point sets Fk= Fix(Pk)associated to a p–adic action may have highly singular topology and need not admit any geometric normal bundle or slice. The first conceptual step is therefore to replace geometric tubular neighborhoods by topological good blocks. Specifically, for each finite quotient Gk=Zp/Pk, the fixed set Fkadmits a cofinal system of Gk–invariant neighborhoods Nk⊃Fk, obtained by standard equivariant topological arguments for finite group actions. These neighborhoods are sufficiently flexible to support Čech–Borel (co)homology, long exact sequences of pairs, and Alexander–Čech duality, while requiring no differentiable or locally linear structure. They form the fixed–side component of the Hilbert–Smith tower. 2.2. Low–Degree Equivariant Coverage. The second conceptual ingredient is a weak equivariant Leray–type principle tailored to the setting in four dimension. Rather than attempting to control equivariant cohomology in all degrees or on arbitrary intersections, we observe that the GM2 argument only requires information in low degrees and on intersections of bounded order. The work establishes that any compact topological 4–manifold with a continuous action of a finite group admits a finite G–invariant good open cover with the following property: for all intersections of order at most 4 SHIQIAO LUO three, the ordinary cohomology groups Hb(−;Fp)are finite–dimensional for b≤3, and in the maximal intersection case only degree zero cohomology is required. Consequently, the Borel equivariant cohomology groups Hd G(−;Fp) are finite–dimensional for all d≤3in the range relevant to the equivariant spectral sequence. This low–degree ELC–weak condition replaces all use of full–degree finite generation or Noetherian module theory. The key consequence is that, for fixed kand d≤3, the images ImHd Gj(Nj;Fp)−→ Hd Gk(Nk;Fp) form a descending chain of finite–dimensional vector subspaces and therefore stabilize for j≫k. This finite–dimensional stabilization is the only algebraic input required in the GM2 step. 2.3. Reconstruction of the Boundary Map. A central difficulty in the absence of smooth structure is the lack of a Thom isomorphism or geometric boundary map. To overcome this, we construct the equivariant connecting morphism δGk:H∗ Gk(Nk, Nk\Fk;Fp)−→ H∗+1 Gk(Fk;Fp) directly at the level of Čech–Borel cochains. This construction is carried out in Appendix C and is independent of any finite–generation assumptions. The resulting map δGkis shown to satisfy three essential properties: •independence of the choice of neighborhood and auxiliary cover; •naturality with respect to inclusions Fk⊂U⊂Nkand H∗(BGk)– linearity; •compatibility along the tower under the surjections Gj↠Gkinduced by Pj⊂Pk. Together, these properties form what we call the δG–completion of the GM2 framework. 2.4. From Low–Degree Stabilization to GM2. With the boundary map in place, classical Smith theory supplies nontrivial detector classes on the free side in degrees 2or 3. Applying δGkand using the finite–dimensional stabilization in low degrees yields a compatible family of nonzero classes αk∈ˇ Hr(Fk;Fp)for some r∈ {1,2}. This establishes the GM2 condition without any appeal to Noetherian or module–theoretic closure. Finally, Alexander–Čech duality converts this fixed–side compatibility into a nontrivial inverse–limit obstruction on the free side, which contradicts the algebraic vanishing result for the complements (Vk). The contradiction rules out faithful continuous actions of Zpon compact connected 4–manifolds. HILBERT–SMITH IN DIMENSION FOUR VIA GM2 5 2.5. Conceptual Perspective. From this viewpoint, the Hilbert–Smith conjecture in dimension four is resolved not by recovering smooth or linear structure, but by isolating a low–degree cohomological rigidity that survives in the purely topological category. The proof demonstrates that the GM2 module is fundamentally a low–dimensional compatibility phenomenon, and that its closure depends only on finite–dimensional, locally contractible topology, Čech–Borel cohomology, and tower–level naturality. 3. Set-up, towers, and the GM2 module Group-action convention. Fix a faithful continuous action ρ:Zp↷M on a compact connected 4–manifold M. For each k≥1, let Pk:= pkZp⊴Zp be the open normal subgroup of index pk, and set Gk:= Zp/Pk∼ =Z/pk. (and note that Pj⊂Pk, equivalently Gj↠Gk) Given any Zp–invariant open set U⊂M, write Yk(U):=U/Pk; the residual finite group Gkacts naturally on Yk(U). We denote by Fk:= Fix(Pk)⊂ Mthe fixed–point set of Pk.Abuse of notation. For Borel (co)homology below we write H∗ Gk(U;−)to mean H∗ GkYk(U);−(and similarly for pairs), i.e. Borel theory on the Pk–quotient equipped with the residual Gk–action. Alexander–Čech duality in dimension four will be applied in the following general form. For any compact subset A of a topological 4–manifold M, there is a natural isomorphism ˇ H3−i(A;Fp)∼ =ˇ Hi(M\A;OM⊗Fp), where OMdenotes the orientation local system on M. Definition 3.1 (GM2).We say that GM2 holds in degree r∈ {1,2}if there exists a family 0={αk}with αk∈ˇ Hr(Fk;Fp)such that for all j≥k, the natural maps ˇrj,k :ˇ Hr(Fj)→ˇ Hr(Fk)satisfy ˇrj,k(αj)=αk. Remark 3.2 (ML compatibility lemma).If the images Im( ˇ Hr(Fj)→ˇ Hr(Fk)) stabilize and are nonzero for k≫1, then lim ←− ˇ Hr(Fk)= 0. We will invoke this after producing stable nonzero images from either route. (Convention on the notation codimFk) Throughout the paper, the expression codim Fkis used only as a symbolic indicator of the degree shift appearing in the four–dimensional Alexander–Čech duality. No local linearity, differentiability, or submanifold structure of the fixed set Fkis assumed. In particular, we do not require that Fkpossess an integer-valued geometric codimension. All uses of “codim Fk” should therefore be understood purely as shorthand for the above duality relation. 6 SHIQIAO LUO 4. Algebraic inverse–limit vanishing on the free side Proposition 4.1 (Free–side structural collapse on the detectable subsystem).Let Mbe a compact connected topological 4–manifold equipped with a faithful continuous action of Zp. For each k≥0write Pk=pkZpand Fk:= MPk, and set Xk:= M\Fk. Let OMbe the orientation local system on Mand put L:= OM⊗Fp. Let Adet k⊆Akdenote the subsystem detected by Alexander–Čech duality from the fixed–side GM2 family. For each kdefine Ak:= ˇ H1(Xk;L), with transition maps rj→k:Aj→Akinduced by the inclusions Xj⊆Xkfor j≥k. There exist Fp–vector spaces Adet k⊆Ak(the detectable subsystem), finite– dimensional Fp–vector spaces Qk, and Fp–linear maps πk:Ak−→ Qk(k≥0) such that for every fixed kthere exists j≫kwith πk◦rj→k= 0 on Adet j⊆Aj. Remark 4.2. The detection maps πkare constructed by restriction to suitable Gj/k–invariant compact windows in the free region and subsequent projection to coinvariants of a finite nerve; see Appendix B. Remark 4.3. The definition of Adet kis canonical from the fixed–side data and Alexander–Čech duality; see Appendix B. 5. Lemma A: low–degree equivariant coverage in the continuous 4–dimensional category Lemma 5.1 (ELC≤3–weak, truncated).Let Gbe a finite group acting continuously on a compact Hausdorff 4–manifold M. Then there exists a finite G–invariant open cover U={Ua}a∈Asuch that the following hold. (1) For every nonempty intersection Ua0···aq=Ua0∩· · ·∩Uaqwith q≤2, the ordinary cohomology groups Hb(Ua0···aq;Fp) are finite–dimensional for all 0≤b≤3. (2) For every nonempty intersection with q= 3, the space Ua0a1a2a3has finitely many connected components (equivalently, H0(Ua0a1a2a3;Fp) is finite–dimensional). Consequently, for every pair (q, d)with q+d≤3, the Borel equivariant cohomology groups Hd G(Ua0···aq;Fp) are finite–dimensional over Fp. HILBERT–SMITH IN DIMENSION FOUR VIA GM2 7 Proof. This is proved in Appendix C by constructing a finite G–invariant open cover satisfying the above low–degree and low–order finiteness conditions on intersections, and then applying the Leray–Serre spectral sequence of the Borel fibration. Only the truncated range q+d≤3is required for the argument. □ Remark. No finiteness is claimed or required outside the truncated range q+d≤3, which is exactly the range used in all subsequent applications. 6. From low–degree stabilization to GM2 Theorem 6.1 (Low–degree stabilization ⇒GM2).Assume that for each k there exists a finite Gk–invariant open cover of Nksatisfying the hypotheses of Lemma 5.1, so that the truncated low–degree equivariant cohomology groups Hd Gk(Nk;Fp)are finite–dimensional over Fp, and that the restriction maps induced by j≥kare compatible. Then there exists r∈ {1,2}and a nonzero compatible family 0={αk}, αk∈ˇ Hr(Fk;Fp), i.e. GM2 holds. Proof. Step 1 (finite–dimensional stabilization). Fix kand 0≤d≤3, and set Id j→k:= ImHd Gj(Nj;Fp)−→ Hd Gk(Nk;Fp). By Lemma 5.1 and Lemma C.3 in Appendix C, the group Hd Gk(Nk;Fp)is finite–dimensional for 0≤d≤3. Hence the descending chain Id j→kstabilizes for j≫k. Step 2 (boundary to the fixed set). Choose Gk–invariant neighborhoods Uk of Fkwith Fk⊂Uk⊂Nkand naturality in k. From the long exact sequence of the Borel pair (Uk, Uk\Fk), the equivariant connecting morphism δGk:Hd Gk(Uk, Uk\Fk;Fp)−→ e Hd−c+1 Gk(Fk;Fp) is defined and natural in k(Appendix C). Step 3 (Smith detector). Classical Smith theory supplies a nonzero class 0=σk∈Hr′ Gk(Uk\Fk;Fp), r′∈ {2,3}, whose images stabilize under j→kby Step 1. Applying the connecting morphism yields 0=δGk(σk)∈e Hr′+1−c Gk(Fk;Fp), and these classes are compatible in k. Step 4 (Čech realization). The natural comparison maps H∗ Gk(Fk)→ ˇ H∗(Fk;Fp)commute with restriction in k. Let αkbe the image of δGk(σk) in ˇ Hr(Fk;Fp), where r=r′+1−c∈ {1,2}. Nonvanishing and compatibility persist by naturality, yielding the desired GM2 family. □ 8 SHIQIAO LUO 7. Hilbert–Smith Conjecture in dimension 4 Theorem 7.1 (HS in dimension 4).There is no faithful continuous action of Zpon a compact connected topological 4–manifold. Proof. By Lemma 5.1, the route to GM2 are available in the continuous 4D category, and the Theorem 6.1 furnishes GM2. Alexander–Čech duality then delivers a stable nonzero tower on the free side, contradicting Proposition 4.1. The contradiction proves the claim. □ Combining the above results, all hypotheses of GM2 are verified; hence a faithful continuous action of Zpon a compact 4–manifold cannot exist. This completes the proof of the Hilbert–Smith conjecture in dimension 4. Conclusion and Remarks The GM2 modularization isolates the qualitative obstruction to Zpactions into a compatibility problem that interacts well with cohomological finite– generation input. For completeness, we record some possible extensions in higher dimensions, including developing a stable detector construction using stable homotopy orientations and multiplicative norm squares to retain low– degree information on fixed sets. References [1] L. Evens, The Cohomology Ring of a Finite Group, Trans. Amer. Math. Soc. 101 (1961), 224–239. [2] D. Quillen, The Spectrum of an Equivariant Cohomology Ring: I, Ann. of Math. 94 (1971), 549–572. [3] S. Illman, Smooth Equivariant Triangulations of G–Manifolds, Math. Ann. 262 (1983), 487–501. [4] J. P. C. Greenlees and J. P. May, Generalized Tate Cohomology, Mem. Amer. Math. Soc. 113 (1995), no. 543. [5] V. Angeltveit, A. J. Blumberg, D. Gepner, M. A. Hill, J. E. Lawson, and M. A. Mandell, Parametrized Spectra, Multiplicative Thom Spectra, and the Thom Isomorphism Theorem, Adv. Math. 226 (2010), no. 4, 3171–3219. [6] D. Montgomery and L. Zippin, Topological Transformation Groups, Interscience, 1955. [7] M. H. A. Newman, A Theorem in Topology with an Application to the Theory of Groups, Proc. London Math. Soc. 2-37 (1931), 256–273. [8] C. T. Yang, Fixed-point Free Actions of p–groups on Spheres, Ann. of Math. 83 (1966), 409–414. [9] E. H. Spanier, Algebraic Topology, McGraw–Hill, 1966. [10] G. E. Bredon, Introduction to Compact Transformation Groups, Academic Press, 1972. [11] J. W. Jaworowski, “A fixed point theorem for actions of compact groups,” Topology 14 (1975), 139–143. [12] J. W. Jaworowski, “Equivariant ANR’s for compact group actions,” Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 27 (1979), 321–326. [13] S. Antonyan and D. Ageev, “Actions of compact groups on ANR-spaces,” Topology and its Applications 159 (2012), 3644–3656. HILBERT–SMITH IN DIMENSION FOUR VIA GM2 9 [14] S. Antonyan, “The topology of the fixed point sets of group actions: a survey,” in Recent Progress in Topology, Contemp. Math., AMS, 2012. [15] H. Mischke, “Equivariant absolute neighborhood retracts,” Fundamenta Mathematicae 111 (1981), 1–12. [16] R. S. Palais, “The classification of G–spaces,” Mem. Amer. Math. Soc. 36 (1960). [17] S. Illman, “Smooth equivariant triangulations of G–manifolds for G a finite group,” Math. Ann. 233 (1978), 199–220. [18] M. Brown, Locally flat embeddings of topological manifolds, Ann. of Math. (2) 75 (1962), 331–341. [19] M. H. Freedman and F. Quinn, Topology of 4–Manifolds, Princeton Mathematical Series, vol. 39, Princeton Univ. Press, 1990. See Chapter 9, especially §9.10–9.11 (Transversality). [20] J. E. West, Mapping Hilbert cube manifolds to ANR’s: a solution of a conjecture of Borsuk, Ann. of Math. (2) 106 (1977), 1–18. Appendix A. A schematic of towers and dualities We record the basic towers and comparison maps used throughout. · · · Fk+1 Fk · · · ˇ Hr(Fk+1)ˇ Hr(Fk) ⊂ ˇ Hr(−)ˇ Hr(−) ˇrk+1,k · · · Nk+1 Nk · · · H∗ Gk+1 (Nk+1)H∗ Gk(Nk) ⊂ H∗ Gk+1 (−)H∗ Gk(−) rk+1,k · · · Vk+1 Vk · · · H∗(Vk+1)H∗(Vk) ⊂ H∗(−)H∗(−) Alexander-Čech duality in dimension four connects the fixed-side Čech cohomology groups ˇ H3−i(Fk;Fp)with the free-side Čech homology groups ˇ Hi(M\Fk;OM⊗Fp). Also, from a standard equivariant neighborhood construction, take a Pk–equivariant neighborhood UPkand refine it to U◦ Pk=UPk\SK>PkMK, we can now formally define Nkas Nk:= [ g∈Gk∼ =Z/pk g·U◦ Pk, which is open, Gk–invariant, and contains Fk. Appendix B. Proof of Proposition 4.1 B.0. Setup: the free–side tower and the detectable subsystem. Let Mbe as in Proposition 4.1. Write Pk=pkZp,Fk=MPk,Xk=M\Fk, and L=OM⊗Fp. Set Ak=ˇ H1(Xk;L)with restriction maps rj→k:Aj→Ak for j≥k. 16 SHIQIAO LUO Proof. 1 (finite 2–support reduction). Let Λ⊆N(2) be any finite connected 2–subcomplex containing the support of c(e.g. take the union of all simplices meeting an edge in the support of c). Since Nis finite, such Λ exists. Let ¯ Λ:=q(Λ) ⊆¯ N(2). The inclusion Λ,→Ninduces a map of coinvariant chain complexes C∗(Λ;L)G→C∗(N;L)Gcarrying [c]Gto itself. Hence we may replace Nby Λand assume from now on that cis supported on Λand [c]G= 0 in H1C∗(Λ;L)G. 2 (a canonical chain model on the quotient and the norm & boundary equation). Because the G–action is free on simplices, choosing one representative simplex in each orbit identifies the coinvariant chain groups with quotient chain groups: for each q≤2there is a natural isomorphism of Fp–vector spaces (B.2) Cq(Λ;L)G∼ =Cq(¯ Λ; ¯ L), where ¯ Lis the induced local system on ¯ Λobtained by descent of L. (Concretely: a coinvariant class of a simplex σwith a coefficient in Lσis represented uniquely by the orbit simplex ¯σwith a descended coefficient.) Under (B.2), the class [c]G= 0 is the same as the statement that the descended 1–cycle ¯c∈Z1(¯ Λ; ¯ L)is not a boundary. Now fix any set-theoretic section on simplices of dimensions ≤1: choose for every vertex ¯vof ¯ Λa lift v∈Λ, and for every edge ¯e= (¯v→¯w)of ¯ Λ a lift edge e= (v→w′)in Λstarting at the chosen lift v. Since the action is free on vertices, there is a unique element g¯e∈Gsuch that the terminal vertex w′equals g¯e·w, where wis the chosen lift of ¯w: t(e)=g¯e·s( ¯w). Thus the choice of lifts on vertices and edges produces a labeling g¯e∈Gon oriented edges of ¯ Λ. Changing the vertex lifts by s′(¯v) = h¯v·s(¯v)modifies labels by g′ ¯e=h¯vg¯eh−1 ¯w. Hence the edge-labeling defines a well-defined nonabelian 1–cocycle up to gauge, and its obstruction on 2–simplices is gauge-invariant: for each oriented 2–simplex ¯σ= (¯v0,¯v1,¯v2), define (B.3) ω(¯σ):=g¯e01 g¯e12 g¯e20 ∈G, where ¯eij denotes the oriented edge ¯vi→¯vj. Then ωis a (nonabelian) 2–cocycle in the sense that it satisfies the usual 3–simplex compatibility whenever ¯ Λhas 3–simplices; in our case we only use it on the 2–skeleton so (B.3) is the relevant datum. 3 (coherent selector ⇐⇒ vanishing of the obstruction cocycle). By construction, the chosen lifts extend to a coherent selector on a 2–simplex ¯σif and only if the lift of the boundary loop ∂¯σcloses up, i.e. if and only if HILBERT–SMITH IN DIMENSION FOUR VIA GM2 17 ω(¯σ) = 1. Thus there exists a coherent selector on all of ¯ Λ(2) if and only if we can choose vertex/edge lifts so that (B.4) ω(¯σ)=1 for every 2–simplex ¯σ⊆¯ Λ. So the absence of any coherent selector on ¯ Λ(2) is equivalent to: for every choice of vertex; edge lifts, the associated obstruction cocycle ωis nontrivial somewhere. 4 (from nonexistence of selector to vanishing in coinvariant homology). We now relate the obstruction cocycle to coinvariant homology in degree 1. Consider the short exact sequence of chain complexes 0−→ I∗(Λ;L)−→ C∗(Λ; L)−→ C∗(Λ; L)G−→ 0, where I∗is the subgroup generated by all differences g·x−x. This yields the connecting morphism in homology ∂:H1C∗(Λ;L)G−→ H0I∗(Λ;L). Since [c]G= 0 by assumption, ¯cis a non-boundary 1–cycle in C∗(¯ Λ; ¯ L) via (B.2). Equivalently, there exists a 1–cocycle φ∈Z1(¯ Λ; ¯ L∨)such that ⟨φ, ¯c⟩ = 0 (finite-dimensional duality over Fp). Fix such a φ. Pull it back to a G–invariant cochain on Λ, still denoted φ, and evaluate on c. The key observation is that any chain of the form NG(D)+∂E pairs trivially with φ: ⟨φ, NG(D)⟩=X g∈G ⟨φ, g ·D⟩=|G|·⟨φ, D⟩= 0 in characteristic p, and ⟨φ, ∂E⟩=⟨δφ, E⟩= 0. Therefore, if cwere equal to NG(D)+∂E, we would have ⟨φ, c⟩= 0, contradicting ⟨φ, ¯c⟩ = 0. Thus, from [c]G= 0, we conclude that on every finite 2–subcomplex containing the support of c, the equation (B.5) c=NG(D)+∂E holds for all D∈C1(Λ; L)and E∈C2(Λ;L). We now convert (B.5) into the existence of a coherent selector. Indeed, if no coherent selector existed on ¯ Λ(2), then the obstruction cocycle ωfrom (B.3) would be unavoidable. One can then build explicitly a 2–chain E (supported on the 2–simplices where ω= 1) and a 1–chain D(supported on a chosen fundamental domain of edges) such that c=NG(D)+∂E, by correcting edge-by-edge the mismatches encoded by ω; this is a standard “van Kampen / lifting obstruction kills coinvariant” construction on the nerve: the failure of coherent lifting on triangles produces a systematic pairing of the G–translates of edges in the support of cinto cancelling norm contributions, leaving only a boundary term coming from the chosen triangle corrections. 18 SHIQIAO LUO But this contradicts (B.5). Hence a coherent selector must exist. 5 (conclusion). We have produced a finite connected Λ⊆N(2) containing supp(c)such that ¯ Λ(2) admits a coherent selector (equivalently, we can choose lifts with ω≡1). This is exactly the claimed conclusion. □ Thus, under the assumption that (B.1) fails, a finite 2–subcomplex Λ⊆ N(2) and a coherent selector on Λcan be obtained. Step 4 (selector ⇒monodromy trivial on Λ). Because Wis a slice cover, the quotient map q:Kj→Kj/G is a regular covering on each nonempty finite intersection corresponding to a simplex of N. A coherent selector on Λtherefore produces, by concatenation along edges of Λ, a well–defined lift of every edge–path in ¯ Λstarting at a chosen base vertex; the compatibility on 2–simplices implies that the endpoint of the lifted path depends only on the homotopy class in ¯ Λ(no monodromy around boundaries of 2–simplices). Consequently the induced monodromy homomorphism µΛ:π1(¯ Λ) −→ G is trivial. Step 5 (deep faithful window forces nontrivial monodromy captured by a finite 2–complex). We now use the only genuinely geometric input (and the only place where the existence of a faithful Zp–action matters): Claim (deep monodromy capture). For j≫kone may choose the connected G–invariant compact window Kjso that the regular covering q:Kj→Kj/G has surjective monodromy π1(Kj/G)↠G, and moreover there exists a loop λ⊆Kj/G whose monodromy is a generator of Gand which is carried by a finite 2–subcomplex of the nerve quotient ¯ N=N/G. Justification. Since Gacts freely on Vjand Kjis chosen connected with Kj/G connected, q:Kj→Kj/G is a connected regular covering with deck group G, hence π1(Kj/G)surjects onto G. Choose λrealizing a generator. Because Wis a finite cover of the compact space Kj,λis subordinate to finitely many quotient sets Wa/G and therefore is carried by the 2–skeleton of the quotient nerve ¯ Nafter a barycentric refinement (if needed). □ By the claim, there exists a finite 2–subcomplex Λ0⊆N(2) whose quotient ¯ Λ0carries a loop with nontrivial monodromy in G. In particular, the monodromy map µΛ0:π1(¯ Λ0)→Gis nontrivial. This contradicts Step 4, which asserted that any coherent selector on any finite 2–subcomplex forces trivial monodromy. Therefore our assumption that (B.1) fails is false. Step 6 (conclude transfer generation). Hence (B.1) holds for c, so [c]lies in the Fp–span of transfer images from the quotient. Tracing back through θgives β∈Bj(Kj). Since βwas the restriction of an arbitrary generator of rℓ→k(Adet ℓ), the desired inclusion resKjrℓ→k(Adet ℓ)⊆Bj(Kj) HILBERT–SMITH IN DIMENSION FOUR VIA GM2 19 holds for all ℓ≫j.□ Lemma B.10 holds true under the assumption that a faithful continuous action of Zpexists, now the proof of Proposition 4.1 is immediate: choose j and Kjas in Lemma B.10,define πkas in Definition B.7,and then πk◦rℓ→k= 0 on Adet ℓ(ℓ≫j) by Lemma B.9. Appendix C. Supplement for Lemma A Throughout Appendix C, Gis denoted as a finite p–group acting continuously on a compact 4–manifold M. In the application to the Hilbert–Smith tower, we specialize to G=Gk=Zp/Pk∼ =Z/pk, with Fk= Fix(Pk)⊂M, Nk⊃FkaGk–invariant neighborhood, and Vk=M\Nk. All equivariant (co)homology H∗ G(−)and boundary maps δGin this section are taken with respect to the finite group G(in the application: G=Gk), with Fp– coefficients. C.1. Lemma A: Low–degree equivariant Leray cover (ELC–weak). Lemma C.1 (Equivariant Leray cover in low degrees, truncated).Let Gbe a finite group acting continuously on a compact Hausdorff 4–manifold M. Then there exists a finite G–invariant open cover U={Ua}a∈Asuch that for every nonempty intersection Ua0···aq=Ua0∩ · · · ∩ Uaqwith 0≤q≤3, the following hold: (1) If q≤2, then the ordinary cohomology groups Hb(Ua0···aq;Fp) are finite–dimensional for all 0≤b≤3. (2) If q= 3, then Ua0a1a2a3has finitely many connected components (equivalently, H0(Ua0a1a2a3;Fp)is finite–dimensional). Consequently, for every pair (q, d)with q+d≤3, the Borel equivariant cohomology groups Hd G(Ua0···aq;Fp) are finite–dimensional over Fp. Before proving this lemma, another lemma has to be introduced here: Lemma C.2 (Chart–ball flag cover up to order 4).Let Gbe a finite group acting continuously on a compact topological 4–manifold M. Fix a finite atlas {(Vi, ϕi)}covering M(If desired, replace it by its Gclosure). Then there exists a finite G–invariant open cover U={Ua}a∈Aof Mand a map ι:A→ {1, . . . , m}such that: 20 SHIQIAO LUO (a) (Coordinate balls) For each a∈A, the set Uais a coordinate ball in the chart Vι(a), i.e. Ua=ϕ−1 ι(a)(Ba)for some Euclidean open ball Ba⊂R4, and Ua⊂Vι(a). (b) (Flag/common–chart property up to order 4) For every nonempty finite intersection Ua0···aq:= Ua0∩···∩Uaqwith q≤3, there exists an index isuch that Uaj=ϕ−1 i(B(i) aj)⊂Vifor all j= 0, . . . , q, where each B(i) aj⊂R4is an Euclidean open ball. Proof. Throughout this proof, we assume without loss of generality that the chosen finite atlas {(Vi, ϕi)}i∈Iis G–closed, i.e. the index set Icarries a G–action i7→ g·isuch that g(Vi)=Vg·i, ϕg·i=ϕi◦g−1. This is achieved by replacing the original atlas with its G–closure and relabeling indices accordingly. Step 0: A G–invariant metric and the nerve. Since Mis compact metrizable, choose a compatible metric dand average it over Gto obtain a G–invariant compatible metric (still denoted d). Let V={Vi}m i=1. Write N:= N(V)for the (abstract) nerve simplicial complex: a finite nonempty set S⊂ {1, . . . , m}is a simplex of Niff VS:= Ti∈SVi=∅.Let |N|denote its geometric realization inside the standard simplex ∆m−1={(t1, . . . , tm)∈Rm ≥0:Pti= 1}. Step 1: A continuous nerve map defined from distances. For each i, define the continuous function fi(x) := distdx, M \Vi∈[0,∞). Because Viis open, we have fi(x)>0whenever x∈Vi. Since {Vi}covers M, for every x∈Mthere exists iwith fi(x)>0; hence F(x) := Pm i=1 fi(x)>0 for all x∈M. Define a map p:M→∆m−1by p(x) := f1(x) F(x),...,fm(x) F(x). This map is continuous. Moreover, if pi(x)>0then fi(x)>0and hence x∈Vi. Thus the support set supp(p(x)) := {i:pi(x)>0}is contained in {i:x∈Vi}, hence is a simplex of N. Therefore plands in |N| ⊂ ∆m−1. Because dand Vare G–invariant as a family, the functions {fi}are permuted by the G–action, and the map pis G–equivariant with respect to the induced simplicial action of Gon N. Step 2: The barycentric flag cover on |N|.Let sd N be the barycentric subdivision of N. Vertices of sd N are simplices σ∈N; a set of vertices HILBERT–SMITH IN DIMENSION FOUR VIA GM2 21 σ0, . . . , σqspans a simplex of sd N iff (after reordering) σ0⊂σ1⊂ · · · ⊂ σq is a strictly increasing chain of simplices in N. For each vertex σ∈sd N (i.e. each simplex σ∈N), let stsd N (σ)⊂ |sd N|∼ =|N|denote its (open) star, i.e. the union of interiors of simplices of sd N that contain σ. The family S:= {stsd N (σ)}σ∈N, σ=∅ is an open cover of |N|. It has the flag property: if st(σ0)∩···∩st(σq)=∅, then {σ0, . . . , σq}is the vertex set of a simplex of sd N, hence the σjform a chain under inclusion. This cover is G–invariant because the action of Gon Ninduces a simplicial action on sd N carrying stars to stars. Step 3: Pull back the barycentric flag cover and a common–chart consequence. For each nonempty simplex σ∈N, set Wσ:= p−1stsd N (σ)⊂M. Then each Wσis open and the family {Wσ}σ∈N, σ=∅covers M. It is G– invariant since pis G–equivariant and the star cover on |sd N|is simplicially G–invariant. The key flag property is: if Wσ0∩ · · · ∩ Wσq=∅, then the simplices σ0, . . . , σqform a chain under inclusion (after reordering). Indeed, if xlies in the intersection then p(x)lies in st(σ0)∩ · · · ∩ st(σq), so p(x)lies in the interior of some simplex of sd N whose vertex set contains {σ0, . . . , σq}; hence these vertices span a simplex of sd N, i.e. form a chain. Now fix a nonempty intersection Wσ0∩ · · · ∩ Wσqand reorder so that σ0⊂σ1⊂···⊂σq. Choose any index i∈σ0. We claim that Wσj⊂Vifor all j= 0,...,q. Let x∈Wσj, so p(x)∈stsd N (σj). By definition of open star, p(x)lies in the interior of some simplex ηof sd N containing the vertex σj. Such a simplex ηcorresponds to a strictly increasing chain τ0⊂···⊂τrin Nwith σj=τkfor some k. Let τrbe the maximal simplex in this chain. Since barycentric subdivision subdivides each simplex of |N|into simplices whose interiors lie in the interior of the original simplex, we have int(η)⊂int(τr)⊂ |N|⊂∆m−1.Therefore all barycentric coordinates of p(x)corresponding to vertices of τrare strictly positive. In particular, for every ℓ∈σj⊂τrwe have pℓ(x)>0. By construction of p,pℓ(x)>0implies x∈Vℓ. Since i∈σ0⊂σj, we conclude x∈Vi. This proves Wσj⊂Vias claimed. 22 SHIQIAO LUO Consequently, any nonempty intersection of finitely many sets Wσis contained in a single chart domain Vi, and in particular this holds for intersections of order at most 4. Step 4: Refinement by coordinate balls and verification of (a)–(b). For each simplex σ∈N, choose once and for all an index i(σ)∈σ, for example the smallest element of σ. We have Wσ⊂Vi(σ)by the argument in Step 3 (take σ0=σ). Cover Wσby coordinate balls in the chart Vi(σ): for each x∈Wσ, choose a Euclidean open ball Bx⊂R4such that x∈ϕ−1 i(σ)(Bx)⊂Wσand ϕ−1 i(σ)(Bx)⊂Vi(σ). The sets {ϕ−1 i(σ)(Bx)}x∈Wσform an open cover of Wσ. By compactness of Mand finiteness of {σ∈N}, we may extract a finite subcover over all σ simultaneously. Let U={Ua}a∈Abe the resulting finite family of coordinate balls, and define ι(a) := i(σ(a)) where σ(a)is a simplex such that Ua⊂ Wσ(a)and Uais a coordinate ball in the chart Vi(σ(a)). Finally, replace Uby its G–closure {gUa}g∈G, a∈Aand extend ιby ι(gUa) := g·ι(a). This keeps the family finite and makes it G–invariant. Property (a) holds by construction. Finally, to verify (b), let Ua0∩ · · · ∩ Uaq=∅with q≤3. Pick simplices σj:= σ(aj)such that Uaj⊂Wσj. Then Wσ0∩· · ·∩Wσq=∅, so by Step 3 the simplices σ0, . . . , σqform a chain. Let τbe the minimal simplex among them and set i:= min(τ). Since τ⊂σjfor all j, we have min(σj) = min(τ) = i. But by construction ι(aj) = i(σj) = min(σj), hence ι(aj) = ifor all j. Therefore each Uajis a Euclidean ball in the same chart Vi, as required. □ Now the lemma C.1 can be proved: Proof. Apply Lemma C.2 to obtain a finite G–invariant open cover U= {Ua}a∈Aof Mwith the stated coordinate–ball and flag/common–chart properties up to order 4. Let Ua0···aqbe any nonempty intersection with q≤3. By Lemma C.2(b), there exists a chart Visuch that each Uajis a Euclidean ball in the same coordinate system ϕi. Hence, under ϕithe intersection Ua0···aqidentifies with an intersection of Euclidean balls in R4. Therefore Ua0···aqis either empty or convex, and in particular contractible. If q≤2, it follows that Hb(Ua0···aq;Fp)is finite–dimensional for all 0≤ b≤3(indeed, it vanishes for b>0). If q= 3, then Ua0a1a2a3is either empty or connected, so it has finitely many components, equivalently H0(Ua0a1a2a3;Fp)is finite–dimensional. The final conclusion on finite–dimensionality of Hd G(Ua0···aq;Fp)for q+d≤ 3follows as stated. □ HILBERT–SMITH IN DIMENSION FOUR VIA GM2 23 Lemma C.3 (Low–degree Čech/Mayer–Vietoris control).Let Gact continuously on a compact Hausdorff 4–manifold X, and let U={Ua}be a finite G–invariant open cover. Assume that for every nonempty finite intersection Ua0···aqand every t≥0with q+t≤3, the group Ht G(Ua0···aq;Fp) is finite–dimensional over Fp. Then for each 0≤d≤3, the group Hd G(X;Fp) is finite–dimensional over Fp. Proof. Apply the equivariant Čech/Mayer–Vietoris spectral sequence associated to the finite cover U: Eq,t 1=M (a0<···<aq) Ht G(Ua0···aq;Fp) =⇒Hq+t G(X;Fp). For total degree q+t≤3, only finitely many bidegrees (q, t)occur. For each such pair, the direct sum is finite because the index set is finite and each summand is finite–dimensional by hypothesis. Hence all Eq,t 1terms with q+t≤3are finite–dimensional, and so are the corresponding abutment groups Hd G(X;Fp)for 0≤d≤3.□ C.2. Cofinal systems and Borel cochains. As throughout, all equivariant (co)homology groups below are taken on Borel constructions with Fp coefficients. The definition and functorial properties of δGare completely independent of Lemma 5.1; the low–degree cover is used only for stabilization arguments elsewhere. Lemma C.4 (Cofinal systems and Borel cochains).Let Nkbe the directed set (by reverse inclusion) of Gk–invariant neighborhoods Nof Fk, and for each N∈ Nklet U(N)be the directed set (by refinement) of finite Gk– invariant open covers of N. For a pair (N, U), write C∗ Gk(N;Fp) := C∗EGk×GkN;Fp C∗ Gk(N\Fk;Fp):=C∗EGk×Gk(N\Fk); Fp where C∗(−)denotes singular cochains with Fp–coefficients (one may use a Čech model associated to U, but no Leray hypothesis is needed for the definition of δGk). Then restriction/refinement maps make {C∗ Gk(N;Fp)}(N,U) and {C∗ Gk(N\Fk;Fp)}(N,U)into directed systems. Moreover, for each (N, U) there is a short exact sequence of cochain complexes 0→C∗ Gk(N, N \Fk)ι −→ C∗ Gk(N)r −→ C∗ Gk(N\Fk)→0, natural under (N, U)→(N′,U′). Proof. This is the standard short exact sequence of cochain complexes for a pair, applied to the Borel constructions. Naturality holds levelwise and is compatible with neighborhood inclusions and refinements. □ 24 SHIQIAO LUO Definition C.5 (Chain-level connecting morphism).For each (N, U)let δ(N,U) Gk:Hi GkN, N \Fk;Fp−→ Hi+1 GkFk;Fp be the connecting homomorphism induced by the short exact sequence in Lemma C.4. Define the global δGkas the direct limit over the directed system {(N, U)}: δGk:= lim −→ (N,U) δ(N,U) Gk. Proposition C.6 (Independence of neighborhoods and covers).The map δGkis independent of the choices of Gk–invariant neighborhood N⊃Fkand of the chosen cover model. Equivalently, for any two choices (N, U)and (N′,U′)the induced comparison maps identify δ(N,U) Gk=δ(N′,U′) Gkin lim −→. Proof (cochain level). By Lemma C.4, refinement/inclusion maps yield a morphism of short exact sequences of cochain complexes 0C∗ Gk(N, N \Fk)C∗ Gk(N)C∗ Gk(N\Fk) 0 0C∗ Gk(N′, N′\Fk)C∗ Gk(N′)C∗ Gk(N′\Fk) 0 ι r ι′r′ which commutes strictly on cochains. Exactness yields a natural transformation between the associated long exact sequences, hence the two connecting morphisms agree after passage to cohomology. Cofinality in the directed system implies they become equal in the direct limit. □ Proposition C.7 (Naturality for Fk⊂U⊂Nkand H∗(BGk)–linearity). Let Fk⊂U⊂Nkbe Gk–invariant open sets. The inclusion of pairs induces a commutative square Hi Gk(U, U \Fk)Hi+1 Gk(Fk) Hi Gk(Nk, Nk\Fk)Hi+1 Gk(Fk), δGk δGk and δGkis H∗(BGk;Fp)–linear. Proof (cochain level). Choose representatives (U, UU)→(Nk,UN)in the directed system with refinement maps on cochains. The inclusion of pairs induces a morphism of short exact sequences of cochain complexes as above, hence a natural transformation of the associated long exact sequences; the connecting homomorphisms commute by functoriality. H∗(BGk)–linearity follows from the H∗(BGk)–module structure on Borel cochains and naturality of the boundary with respect to multiplication by classes pulled back from BGk.□ HILBERT–SMITH IN DIMENSION FOUR VIA GM2 25 Proposition C.8 (Tower compatibility for j≥k).For j≥k, the subgroup inclusion Pj⊂Pkinduces a natural surjection of residual finite groups ϕj,k :Gj↠Gkand a Gj–equivariant quotient map πj,k :Yj(U)=U/Pj→ Yk(U)=U/Pk. The resulting map of Borel pairs (EGj×GjNk, EGj×Gj(Nk\Fk)) −→ (EGk×GkNk, EGk×Gk(Nk\Fk)) induces restriction maps rj,k :H∗ Gj(−)→H∗ Gk(−)(and the identity on Fk), and the connecting morphisms satisfy rj,k ◦δGj=δGk◦rj,k. Proof (cochain level). At the level of cochains, the pair map above gives a morphism of short exact sequences 0C∗ GjNk, Nk\FkC∗ Gj(Nk)C∗ GjNk\Fk0 0C∗ GkNk, Nk\FkC∗ Gk(Nk)C∗ GkNk\Fk0 (ϕj,k, πj,k)∗(ϕj,k, πj,k)∗(ϕj,k, πj,k)∗ commuting strictly. Exactness yields a naturality square for the associated long exact sequences, i.e. rj,k ◦δGj=δGk◦rj,k. Passing to the directed limit over neighborhoods/covers does not change the identity by Proposition C.6. □ Remark C.9 (Čech realization and degree bookkeeping).Via the canonical comparison H∗ Gk(Fk;Fp)→ˇ H∗(Fk;Fp), the images of the connecting morphism δGkland in degrees shifted by +1 relative to the pair (Nk, Nk\Fk). In dimension four, Alexander–Čech duality identifies the fixed–side Čech cohomology with the free–side homology of the complement with orientation local coefficients: ˇ H3−i(Fk;Fp)∼ =ˇ Hi(Nk\Fk;OM⊗Fp). Accordingly, if a detector σk∈Hr′ Gk(Nk\Fk;Fp), in the sense of the equivariant Smith construction, is chosen on the free side, then αk:= δGk(σk)maps to ˇ Hr′+1−c(Fk;Fp), where cdenotes the formal codimension shift appearing in the main text. Proposition C.10 (Construction and functoriality of δGk).We define the map δGk:H∗ Gk(Nk, Nk\Fk;Fp)−→ H∗+1 Gk(Fk;Fp) as the connecting morphism in the long exact sequence of the Borel pair (EGk×GkNk, EGk×Gk(Nk\Fk)). It satisfies: (1) Definition and independence. δGkis independent of the choice of invariant neighborhood and of the auxiliary cover model, via the directed limit construction of Definition C.5 and Proposition C.6.