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normal bundle

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id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

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Contents

Definition

Definition

For i:XYi : X \hookrightarrow Y an immersion (notably an embedding) of smooth manifolds, the normal bundle of XX in YY relative to ii is the the vector bundle

N iXX N_i X \to X

defined as the fiberwise quotient bundle

N iX=i *TYTX N_i X =\frac{i^* T Y}{T X}

If YY is equipped with Riemannian metric structure then this is equivalently the space of tangent vectors to YY which a normal vectors to the tangent vectors to XX, whence the name.

The pullback i *TYi^* T Y can be of course interpreted as the restriction TY| XT Y|_X. The normal bundle is fiberwise the quotient of the fiber of the tangent bundle of YY by the fiber of the tangent bundle of XX: for xXx \in X

N i(X) x=T i(x)Y/T x(X). N_i (X)_x = T_{i(x)}Y/T_x(X) \,.

The dual notion is that of conormal bundle. The notion also makes sense for some other contexts, e.g. for smooth algebraic varieties.

Remark

There is always an isomorphism

TXN iXTY| X, T X \oplus N_i X \simeq T Y|_X \,,

but it is not canonically given, hence in particular cannot it general be chosen naturally. Except if certain additional structure is given:

If YY is equipped with a Riemannian metric gg, then we may identify the normal bundle with the bundle of vectors that are orthogonal to (“normal to”) the vectors in TXT X:

N i(X)(TX) :={(x,v)TY| X|wT xX:g(v,w)=0}. N_i(X) \simeq (T X)^\perp := \{ (x,v) \in T Y|_{X} | \forall w \in T_x X : g(v,w) = 0 \} \,.

Let M nM^n be a smooth compact nn-dimensional manifold without boundary, then the question of triviality of the normal bundle for an embedding M nR n+rM^n\hookrightarrow \mathbf{R}^{n+r} for sufficiently large rr does not depend on the embedding. For this one uses the fact that any two such embeddings are regularly homotopic (this means the existence of a smooth homotopy H(x,t)H(x,t) which is immersion for every t[0,1]t \in [0,1] and which induces on the level of differentials a homotopy for the tangent bundles) and that any two regular homotopies are themselves homotopic through regular homotopies leaving end points fixed. Then one just uses the homotopy invariance of vector bundles. Thus, if M nM^n admits an embedding into R n+r\mathbf{R}^{n+r} with a trivial normal bundle then one says that M nM^n has a stably trivial normal bundle. In that case, if M + nM^n_+ is the union of MM with a disjoint base point, then there is a homeomorphism T(M n×R r)Σ rM + nT (M^n\times \mathbf{R}^{r})\cong \Sigma^r M^n_+ where Σ r\Sigma^r denotes the rr-fold (reduced) suspension of based spaces (S r×M + n)/(S rM +)(S^r\times M^n_+)/(S^r\wedge M_+).

Literature

  • Victor Snaith, Stable homotopy around the Arf-Kervaire invariant, Birkhauser 2009

Revised on May 26, 2017 05:51:04 by Urs Schreiber (92.218.150.85)