In mathematics, a normal map is a concept in geometric topology due to William Browder which is of fundamental importance in surgery theory. Given a Poincaré complex X (more geometrically a Poincaré space), a normal map on X endows the space, roughly speaking, with some of the homotopy-theoretic global structure of a closed manifold. In particular, X has a good candidate for a stable normal bundle and a Thom collapse map, which is equivalent to there being a map from a manifold M to X matching the fundamental classes and preserving normal bundle information. If the dimension of X is ≥ {\displaystyle \geq } 5 there is then only the algebraic topology surgery obstruction due to C. T. C. Wall to X actually being homotopy equivalent to a closed manifold. Normal maps also apply to the study of the uniqueness of manifold structures within a homotopy type, which was pioneered by Sergei Novikov. The cobordism classes of normal maps on X are called normal invariants. Depending on the category of manifolds (differentiable, piecewise-linear, or topological), there are similarly defined, but inequivalent, concepts of normal maps and normal invariants. It is possible to perform surgery on normal maps, meaning surgery on the domain manifold, and preserving the map. Surgery on normal maps allows one to systematically kill elements in the relative homotopy groups by representing them as embeddings with trivial normal bundle.
Definition There are two equivalent definitions of normal maps, depending on whether one uses normal bundles or tangent bundles of manifolds. Hence it is possible to switch between the definitions which turns out to be quite convenient.
Given a Poincaré complex X (i.e. a CW-complex whose cellular chain complex satisfies Poincaré duality) of formal dimension n {\displaystyle n} , a normal map on X consists of a map f : M → X {\displaystyle f\colon M\to X} from some closed n-dimensional manifold M, a bundle ξ {\displaystyle \xi } over X, and a stable map from the stable normal bundle ν M {\displaystyle \nu _{M}} of M {\displaystyle M} to ξ {\displaystyle \xi } , and usually the normal map is supposed to be of degree one. That means that the fundamental class of M {\displaystyle M} should be mapped under f {\displaystyle f} to the fundamental class of X {\displaystyle X} : f ∗ ( [ M ] ) = [ X ] ∈ H n ( X ) {\displaystyle f_{*}([M])=[X]\in H_{n}(X)} . Given a Poincaré complex X {\displaystyle X} of formal dimension n {\displaystyle n} , a normal map on X {\displaystyle X} (with respect to the tangent bundle) consists of a map f : M → X {\displaystyle f\colon M\to X} from some closed n {\displaystyle n} -dimensional manifold M {\displaystyle M} , a bundle ξ {\displaystyle \xi } over X {\displaystyle X} , and a stable map from the stable tangent bundle τ M ⊕ ε k {\displaystyle \tau _{M}\oplus \varepsilon ^{k}} of M {\displaystyle M} to ξ {\displaystyle \xi } , and similarly as above it is required that the fundamental class of M {\displaystyle M} should be mapped under f {\displaystyle f} to the fundamental class of X {\displaystyle X} : f ∗ ( [ M ] ) = [ X ] ∈ H n ( X ) {\displaystyle f_{*}([M])=[X]\in H_{n}(X)} . Two normal maps are equivalent if there exists a normal bordism between them.
Role in surgery theory
Surgery on maps versus surgery on normal maps Consider the question:
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