Preparing local tests
Assume a faithful jointly continuous \(\Zp\)-action on a connected \(n\)-manifold. This chapter finds a faithful orientation-preserving action inside one coordinate chart, builds a degree-one test from a compact sphere through the orbit space, and moves a positive-degree multiple of that test into any prescribed open region of the orbit space, using the odd-sphere homotopy calculation.
The source over \(X \setminus \{b\}\) of a restricted test is an oriented open subset of the manifold with a proper map: these are the manifold sources used for duality later, while the compact orbit space enters only through ordinary sheaf direct image. The positive degree from moving a test may depend on the region.
Chart and test source
There are an open subgroup \(G \cong \Zp\) and a nonempty connected \(G\)-invariant open subset \(O\) of a coordinate chart of \(M\) such that \(G\) acts faithfully and preserves orientation on \(O\). In particular \(O\) may be regarded as an actual open subset of \(\R^n\).
Proof sketch. Some point \(x\) has no open subgroup fixing a neighborhood pointwise: otherwise some interior fixed set \(F_j\) is nonempty, and it is closed because near a boundary point the action factors through a finite group to which Newman rigidity applies, contradicting faithfulness. Saturating a small ball at \(x\) gives a connected invariant \(O\); a nonzero restriction kernel would be an open subgroup fixing \(O\), impossible. Passing to squares preserves orientation (index two only for \(p = 2\)).
Dependencies
With \(W = O \times \R^{d-n} \subset \R^d\) carrying the product action (trivial in the added coordinates), the collapse map \(c : S^d \to W^+\) sending \(W\) identically and everything else to \(\infty\) is continuous. The action extends jointly continuously to \(W^+\) fixing \(\infty\). Both \(W^+\) and its orbit space by any closed subgroup of \(G\) are compact metrizable.
Proof sketch. Inverse images of basic neighborhoods are open on both sides. Saturation of compacts gives joint continuity at \(\infty\). Averaging a compatible metric over \(G\) and minimizing over orbits produces compatible (quotient) metrics.
Dependencies
After replacing \(G\) by an open subgroup identified with \(\Zp\), with \(q : W^+ \to Y = W^+/G\), there is a continuous map \(a : Y \to S^d\) with \(\deg(aqc) = 1\). Moreover \(aq\) is homotopic on \(W^+\) to a degree-one pinch supported in a round ball compactly contained in \(W\).
Proof sketch. A ball pinch \(P_{z,r}\) has degree one after \(c\). For a small enough open subgroup, the pinch is uniformly close to its Haar average \(A\); normalizing the straight-line homotopy gives an invariant sphere-valued map homotopic to the pinch, fixing \(\infty\) and constant outside a compact subset of \(W\). It descends to \(a\), and faithfulness plus orientation preservation persist.
Dependencies
Moving the test
Let \(E\) be compact metrizable with a continuous \(G\)-action and \(r : E \to E/G\) its orbit map. For any field \(k\), \(Rr_*\underline k_E \simeq r_*\underline k_E\) in the bounded-below derived category. For \(k = \Q\) the unit \(\underline\Q_{E/G} \to r_*\underline\Q_E\) has a natural sheaf retraction, so \(r^* : \check H^j(E/G;\Q) \to \check H^j(E;\Q)\) is injective for every \(j\).
Proof sketch. Each orbit is compact zero-dimensional, where global sections of vector-space sheaves are exact, so higher fiber cohomology vanishes and proper base change gives the first claim with no dimension bound. A locally constant rational function on a saturated preimage factors through a finite quotient, so Haar averaging is rational; a finite-cover argument shows the average is locally constant and descends, giving the sheaf retraction, and the Cech--sheaf comparison transfers split injectivity.
Dependencies
Let \(d \geq 3\) be odd. If \(K\) is compact metrizable and \(v : K \to S^d\) satisfies \(v^*\gamma_d = 0\), then there are a positive integer \(s\) and a degree-\(s\) map \(D_s : S^d \to S^d\) with \(D_s v\) nullhomotopic.
Proof sketch. Approximate \(v\) by a map factoring through a finite nerve, then refine so the finite-complex map is rationally trivial. The mapping telescope of degree \(2, 3, \ldots\) maps has type \(K(\Q, d)\): Serre finiteness plus the degree-map killing assertion kill all higher homotopy, and the degree-\(d\) direct limit is \(\Q\). Eilenberg--MacLane representability nullhomotopes the composite into the telescope; bounded height retracts to a finite stage, giving the nullhomotopy of \(D_s v\) for a product of the displayed degrees.
Dependencies
For every \(k \geq 1\), with \(m = p^k\) and \(\zeta = \exp(2\pi i/m)\), there is a nonempty open \(V \subset Y \setminus \{y_\infty\}\) over which the finite quotient \(W/(p^kG) \to W/G\) is a disjoint union of \(m\) homeomorphisms onto \(V\). There is a locally constant \(\beta : q^{-1}(V) \to \{1, \zeta, \ldots, \zeta^{m-1}\}\) with \(\beta(gw) = \zeta^{\bar g}\beta(w)\) for \(\bar g \in G/p^kG = \Z/m\).
Proof sketch. Some \(x \in W\) has stabilizer inside \(p^kG\), or \(p^{k-1}G\) would fix everything, contradicting faithfulness. On \(Z = W/p^kG\) the finite cyclic action is free at the image of \(x\); disjoint neighborhoods of its orbit intersected over translates give \(N\) with disjoint translates, and the open finite quotient map restricts to homeomorphisms on each. Constant values \(\zeta^j\) on the pulled-back sheets give \(\beta\).
Dependencies
Let \(Y_0\) be compact metrizable, \(F_0 \subset Y_0\) closed, and \(f : Y_0 \to S^d\) continuous. Every homotopy \(H : F_0 \times [0,1] \to S^d\) starting at \(f|_{F_0}\) extends to a homotopy on \(Y_0\) starting at \(f\).
Proof sketch. Paste \(H\) with \(f\) on the closed subset \((F_0 \times I) \cup (Y_0 \times \{0\})\), extend coordinates by Tietze, normalize where the extension is large, and damp the time parameter by a cutoff function equal to \(1\) on \(F_0\) and \(0\) outside the good neighborhood.
Dependencies
For every nonempty open \(V \subset Y \setminus \{y_\infty\}\), with \(F = Y \setminus V\), there are a positive integer \(s\), a degree-\(s\) map \(D_s : S^d \to S^d\), and a map \(h : Y \to S^d\) with \(h \simeq D_s a\) and \(h|_F \equiv b\) for some \(b \in S^d\). In particular \(h(y_\infty) = b\) and the nonconstant source lies over \(V\). The degree \(s\) may depend on \(V\) (and hence on \(k\) for sheet regions).
Proof sketch. The degree-one pinch can be transported along a path in \(W\) to a ball over \(V\), so \(aq\) is homotopic to a map constant on \(q^{-1}(F)\); the restricted orbit map meets the orbit pullback hypotheses, giving \((a|_F)^*\gamma_d = 0\). Odd-sphere nullhomotopy kills some \(D_s a|_F\), and homotopy extension promotes the nullhomotopy to \(Y\).