In surgery theory, a branch of mathematics, the stable normal bundle of a differentiable manifold is an invariant which encodes the stable normal (dually, tangential) data. There are analogs for generalizations of manifold, notably PL-manifolds and topological manifolds. There is also an analogue in homotopy theory for Poincaré spaces, the Spivak spherical fibration, named after Michael Spivak.
Construction via embeddings Given an embedding of a manifold in Euclidean space (provided by the theorem of Hassler Whitney), it has a normal bundle. The embedding is not unique, but for high dimension of the Euclidean space it is unique up to isotopy, thus the (class of the) bundle is unique, and called the stable normal bundle. This construction works for any Poincaré space X: a finite CW-complex admits a stably unique (up to homotopy) embedding in Euclidean space, via general position, and this embedding yields a spherical fibration over X. For more restricted spaces (notably PL-manifolds and topological manifolds), one gets stronger data.
Details Two embeddings i , i ′ : X ↪ R m {\displaystyle i,i'\colon X\hookrightarrow \mathbb {R} ^{m}} are isotopic if they are homotopic through embeddings. Given a manifold or other suitable space X, with two embeddings into Euclidean space i : X ↪ R m , {\displaystyle i\colon X\hookrightarrow \mathbb {R} ^{m},} j : X ↪ R n , {\displaystyle j\colon X\hookrightarrow \mathbb {R} ^{n},} these will not in general be isotopic, or even maps into the same space ( m {\displaystyle m} need not equal n {\displaystyle n} ). However, one can embed these into a larger space R N {\displaystyle \mathbf {R} ^{N}} by letting the last N − m {\displaystyle N-m} coordinates be 0:
i : X ↪ R m ≅ R m × { ( 0 , … , 0 ) } ⊂ R m × R N − m ≅ R N {\displaystyle i\colon X\hookrightarrow \mathbb {R} ^{m}\cong \mathbb {R} ^{m}\times \left\{(0,\dots ,0)\right\}\subset \mathbb {R} ^{m}\times \mathbb {R} ^{N-m}\cong \mathbb {R} ^{N}} . This process of adjoining trivial copies of Euclidean space is called stabilization. One can thus arrange for any two embeddings into Euclidean space to map into the same Euclidean space (taking N = max ( m , n ) {\displaystyle N=\max(m,n)} ), and, further, if N {\displaystyle N} is sufficiently large, these embeddings are isotopic, which is a theorem. Thus there is a unique stable isotopy class of embedding: it is not a particular embedding (as there are many embeddings), nor an isotopy class (as the target space is not fixed: it is just "a sufficiently large Euclidean space"), but rather a stable isotopy class of maps. The normal bundle associated with this (stable class of) embeddings is then the stable normal bundle. One can replace this stable isotopy class with an actual isotopy class by fixing the target space, either by using Hilbert space as the target space, or (for a fixed dimension of manifold n {\displaystyle n} ) using a fixed N {\displaystyle N} sufficiently large, as N depends only on n, not the manifold in question. More abstractly, rather than stabilizing the embedding, one can take any embedding, and then take a vector bundle direct sum with a sufficient number of trivial line bundles; this corresponds exactly to the normal bundle of the stabilized embedding.
Construction via classifying spaces An n-manifold M has a tangent bundle, which has a classifying map (up to homotopy)
τ M : M → B O ( n ) . {\displaystyle \tau _{M}\colon M\to B{\textrm {O}}(n).}
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