Pardon implicit atlases and virtual fundamental class package

Let $Z$ be compact Hausdorff and $d\in \mathbb{Z}$. A Pardon implicit atlas of virtual dimension $d$ has an index set $\mathcal{A}$ and the following data. For each $\alpha \in \mathcal{A}$ choose a finite group ${\mathrm{\Gamma }}_{\alpha }$ and a finite-dimensional real ${\mathrm{\Gamma }}_{\alpha }$-representation $E_{\alpha }$. Put ${\mathrm{\Gamma }}_I={\prod }_{\alpha \in I}{\mathrm{\Gamma }}_{\alpha }$ and $E_I={⨁}_{\alpha \in I}{E}_{\alpha }$. For each finite $I\subset \mathcal{A}$ choose a Hausdorff ${\mathrm{\Gamma }}_I$-space $Z_I$, with $Z_{\mathrm{\varnothing }}=Z$, and compatible equivariant component maps $s_{\alpha }:Z_I\to E_{\alpha }$. Write $s_K=(s_{\alpha }{)}_{\alpha \in K}$ on any thickening containing $K$. For $I\subset J$ choose a ${\mathrm{\Gamma }}_I$-invariant open footprint $U_{IJ}\subset Z_I$ and a ${\mathrm{\Gamma }}_J$-equivariant map ${\psi }_{IJ}:s_{J\setminus I}^{-1}(0)\subset Z_J\to U_{IJ}$, where ${\mathrm{\Gamma }}_J$ acts on $U_{IJ}$ through its projection to ${\mathrm{\Gamma }}_I$. Finally choose ${\mathrm{\Gamma }}_I$-invariant regular loci $Z_I^{\mathrm{r}\mathrm{e}\mathrm{g}}\subset Z_I$.

The compatibility axioms are ${\psi }_{II}=\mathrm{i}\mathrm{d}$, ${\psi }_{IJ}{\psi }_{JK}={\psi }_{IK}$, $s_I{\psi }_{IJ}=s_I$, $U_{II}=Z_I$, $U_{I{J}_1}\cap U_{I{J}_2}=U_{I,J_1\cup J_2}$, and ${\psi }_{IJ}^{-1}(U_{IK})=U_{JK}\cap s_{J\setminus I}^{-1}(0)$, with all maps restricted to their defined domains and nested indices as appropriate. The footprint map induces a homeomorphism $s_{J\setminus I}^{-1}(0)\mathrm{/}{\mathrm{\Gamma }}_{J\setminus I}\cong U_{IJ}$.

The regularity axioms say that $Z_I^{\mathrm{r}\mathrm{e}\mathrm{g}}$ is open, ${\psi }_{IJ}^{-1}(Z_I^{\mathrm{r}\mathrm{e}\mathrm{g}})\subset Z_J^{\mathrm{r}\mathrm{e}\mathrm{g}}$, and the RELATIVE group ${\mathrm{\Gamma }}_{J\setminus I}$ acts freely on that preimage. Near every point of ${\psi }_{IJ}^{-1}(Z_I^{\mathrm{r}\mathrm{e}\mathrm{g}})$, the EXTRA obstruction map $s_{J\setminus I}:Z_J\to E_{J\setminus I}$ is topologically the coordinate projection ${\mathbb{R}}^{d+\dim E_I}\times {\mathbb{R}}^{\dim E_{J\setminus I}}\to {\mathbb{R}}^{\dim E_{J\setminus I}}$. Taking $I=J$ makes $Z_I^{\mathrm{r}\mathrm{e}\mathrm{g}}$ a topological manifold of dimension $d+\dim E_I$. Finally the regular zero loci cover the base: $Z={\bigcup }_I{\psi }_{\mathrm{\varnothing }I}(s_I^{-1}(0)\cap Z_I^{\mathrm{r}\mathrm{e}\mathrm{g}})$. An empty regular locus is allowed. A smooth implicit atlas additionally specifies compatible smooth structures, smooth group actions and section maps, and smooth versions of these relative submersion and zero-set local-diffeomorphism conditions.

In particular, the definition does NOT require $s_I:Z_I^{\mathrm{r}\mathrm{e}\mathrm{g}}\to E_I$ transverse to zero, nor the full ${\mathrm{\Gamma }}_I$ action free. In the stable-map application, Pardon's geometric regular locus instead requires surjectivity of the vertical linearization of the THICKENED Cauchy–Riemann equation in map AND obstruction directions, together with trivial automorphisms of the full auxiliary-marked thickened tuple. This is different from transversality of its obstruction-parameter recording map $s_I$ or trivial stabilizers for the entire covering-group action on isomorphism classes.

For an atlas with boundary specify invariant closed subsets $\mathrm{\partial }{Z}_I\subset Z_I$, with $\mathrm{\partial }{Z}_{\mathrm{\varnothing }}=\mathrm{\partial }Z$, compatible with the zero-set footprint maps. The local projection model may also be ${\mathbb{R}}_{\ge 0}\times {\mathbb{R}}^{d+\dim E_I-1}\times {\mathbb{R}}^{\dim E_{J\setminus I}}\to {\mathbb{R}}^{\dim E_{J\setminus I}}$, with $\mathrm{\partial }{Z}_J^{\mathrm{r}\mathrm{e}\mathrm{g}}$ its boundary. Restricting to $\mathrm{\partial }{Z}_I$ gives a dimension-$(d-1)$ boundary atlas.

Over $\mathbb{Q}$, let $o_E=H_{\dim E}(E,E\setminus 0;\mathbb{Q})$ be the orientation line of a real vector space, and let $o_{Z_I^{\mathrm{r}\mathrm{e}\mathrm{g}}}$ be the rank-one local orientation sheaf of its regular manifold (the stalk at an interior point $x$ is $H_{d+\dim E_I}(Z_I^{\mathrm{r}\mathrm{e}\mathrm{g}},Z_I^{\mathrm{r}\mathrm{e}\mathrm{g}}\setminus \{x\};\mathbb{Q})$; at boundary points use its extension from the interior). Local orientability requires the stabilizer $({\mathrm{\Gamma }}_I{)}_x$ to act trivially on $(o_{Z_I^{\mathrm{r}\mathrm{e}\mathrm{g}}}{)}_x\otimes o_{E_I}^{\vee }$ at every regular zero $x$. The compatible identifications under ${\psi }_{IJ}$ define the virtual orientation local system $o_Z$. An orientation is a specified trivialization $o_Z\cong {\mathbb{Q}}_Z$. For a boundary atlas put $o_{Z\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{\partial }}=j_{!}{j}^{\ast }{o}_Z$, where $j:Z\setminus \mathrm{\partial }Z\hookrightarrow Z$ and $j_{!}$ is extension by zero.

For the Čech cohomology and VFC clauses assume full AC: every set-indexed family of nonempty sets admits a choice function. This is used to well-order the open subsets and choose refinement functions in the adopted cohomology convention. The Čech cohomology convention is the refinement colimit of fixed-cover cohomology with its stated sheaf coefficients, as in the refinement-colimit Čech definition. Pardon's rational VFC package assigns a functional $[Z{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}\in {\mathrm{Hom}}_{\mathbb{Q}}\text{\hspace{0.17em}⁣}({\check{H}}^d(Z;o_Z),\mathbb{Q})$ for a boundaryless locally orientable atlas; for a boundary atlas replace $o_Z$ by $o_{Z\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{\partial }}$. Thus this is the dual of degree-$d$ Čech COHOMOLOGY, not the dual of Čech homology. A finite subatlas suffices; compactness and regular-zero coverage give such a finite subatlas, and the source package makes its functional independent of that choice. Rational coefficients invert every finite covering-group order. No reduction of these rational weights to characteristic two is part of this definition. For an oriented ordinary regular manifold the functional is $a\mapsto ⟨a,[Z]⟩$, the usual fundamental class through its pairing, rather than a change in cohomological variance. This item specifies the source package and its types; it does not reconstruct its virtual-cochain comparison, existence, or bordism proof.

For an oriented boundaryless atlas and a continuous map $f:Z\to Y$, its covariant pushforward is the functional $(f_{\ast }[Z{]}^{\mathrm{v}\mathrm{i}\mathrm{r}})(a)=[Z{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}(f^{\ast }a)$ on ${\check{H}}^d(Y;\mathbb{Q})$. In Pardon's stable-map application with $2g-2+n>0$, the source's canonically trivialized virtual orientation and implicit atlas, when supplied, define $[{\bar{\mathcal{M}}}_{g,n}^{\beta }(X,J){]}^{\mathrm{v}\mathrm{i}\mathrm{r}}$. The GW functional $G{W}_{g,n,\beta }^P(X,J)$ is its pushforward under stabilization and evaluation $\tau :{\bar{\mathcal{M}}}_{g,n}^{\beta }(X,J)⟶{\bar{\mathcal{M}}}_{g,n}\times X^n\mathrm{.}$ This is an analytic/topological cohomology-dual theory, not a Behrend–Fantechi Chow class. Its definition asserts neither construction of the needed geometric atlas nor the Kontsevich–Manin axioms.

Pardon Gromov topology on stable J-holomorphic map moduli

Refinement-colimit Čech cohomology

The Axiom of Choice