Degree-one normal map for the surgery program

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $X$ be a connected finite CW complex of dimension $n$ together with a vector bundle $\xi$ over $X$, and let $M$ be a closed oriented smooth $n$-manifold (possibly disconnected) with fundamental class $[M]$ (Fundamental class of a compact oriented manifold, Smooth manifolds and their smooth charts). A normal map $(f,b):M\to X$ with respect to $(X,\xi )$ consists of a continuous map $f:M\to X$ together with a stable isomorphism of stable vector bundles $b:{\nu }_M⟶f^{\ast }\xi ,$ in the sense of Stable normal bundle of a compact smooth manifold and Bundle maps, sections, subbundles, and isomorphisms: bundles are identified by adding trivial real summands, and $b$ is represented by an isomorphism $\nu \oplus {\epsilon }^a\cong f^{\ast }{\xi }^{\mathrm{\prime }}\oplus {\epsilon }^b$ between genuine representative bundles $\nu$ of the smooth stable normal bundle ${\nu }_M$ and ${\xi }^{\mathrm{\prime }}$ of $\xi$. The target datum $(X,\xi )$ is part of the normal structure, not an invariant of $f$.

The normal map is a degree-one normal map if the target datum also includes a class $[X]\in H_n(X;\mathbb{Z})$ with $H_n(X;\mathbb{Z})\cong \mathbb{Z}$ generated by $[X]$, and $f_{\ast }[M]=[X]\text{in }{H}_n(X;\mathbb{Z})\mathrm{.}$ For a disconnected source, $[M]$ is the sum of its component fundamental classes; degree one means this total class maps to $[X]$, without asserting that each component has degree one.

The principal case is a connected closed oriented smooth $n$-manifold $X$ with fundamental class $[X]$ and $\xi$ representing its stable normal bundle ${\nu }_X$. For connected $M$ the condition is the degree-one condition of Degree of a map between oriented closed manifolds, the degree is computed from fundamental classes and is homotopy invariant (Manifold degree is functorial and detected in top cohomology), and a continuous map $M\to X$ may be replaced by a homotopic smooth map without changing the degree (Every continuous map between smooth manifolds is homotopic to a smooth map). The generality of a finite CW complex $X$, with the class $[X]$ fixed as part of the target datum, is the setting in which the below-middle surgery step is formulated when no smooth structure on the target is assumed.

For the manifold target $\xi ={\nu }_X$ the bundle datum may equivalently be given as a stable isomorphism of the tangent bundles, $TM\oplus {\epsilon }^k\cong f^{\ast }TX\oplus {\epsilon }^k$ for some $k\ge 0$: for a supplied Euclidean embedding of $M$ one has the stable splitting $TM\oplus {\nu }_M\cong {\epsilon }^N$, and the same identity holds for $X$, so the two formulations are exchanged by adding ${\epsilon }^N$ to both sides. The stable normal class is intrinsic on a compact manifold and does not depend on the chosen embedding, by Stable normal bundle is independent of the embedding.

In the unoriented setting the orientations are dropped: degrees and fundamental classes are taken with ${\mathbb{F}}_2$ coefficients, using the canonical ${\mathbb{F}}_2$-orientation of every manifold (Every manifold is F2-orientable and orientability is componentwise), the class $[X]$ generates $H_n(X;{\mathbb{F}}_2)\cong {\mathbb{F}}_2$, and $b$ remains a stable isomorphism of real stable normal bundles. Only the homology coefficients and degree condition change to ${\mathbb{F}}_2$; the real bundle datum is not replaced by a vector bundle over ${\mathbb{F}}_2$.

A normal bordism between degree-one normal maps $(f_0,b_0):M_0\to X$ and $(f_1,b_1):M_1\to X$ with respect to the same target datum $(X,[X],\xi )$ is data consisting of an oriented bordism $(W;M_0,M_1)$ in the sense of Oriented smooth cobordism (or an unoriented bordism Unoriented smooth cobordism of closed manifolds in the mod-two setting), a continuous map $F:W\to X$ restricting to $f_0$ and $f_1$ under the collar identifications of the two faces, and a stable isomorphism $B:{\nu }_W\to F^{\ast }\xi$ whose restrictions to the faces are the data determining $b_0$ and $b_1$, under the standard identification of the stable normal bundle of a face with ${\nu }_W$ restricted along an inward normal field. The two normal maps are normally bordant when such data exist. This is Lück's bordism relation of degree-one normal maps, specialised to a common target bundle; Lück's definition allows the two maps to use bundles that agree with the bordism bundle only after adding trivial summands, and the transitivity of the relation is part of the normal bordism calculus rather than reproved here.

The definition records the input of the surgery programme. It neither constructs the space $G\mathrm{/}O$ of normal invariants nor the surgery obstruction group, which are separate constructions. Countable Choice is inherited from Stable normal bundle of a compact smooth manifold and is the only choice used.