Immersion
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Contents |
1 Definition
We work in a fixed category CAT of topological, piecewise linear, -differentiable or real analytic manifolds (second countable, Hausdorff, without boundary) and maps between them. denotes the open unit ball in .
Let be such a map between manifolds of the indicated dimensions .
Definition 1.1. is a local immersion at a point if there exist open neighbourhoods of and of in and , resp., such that and:
- there is a CAT--isomorphism (i.e. both and are CAT--maps) which maps onto ; and
- yields a CAT-isomorphism from onto .
We call f an immersion (and we write ) if is a local immersion at every point .
Thus an immersion looks locally like the inclusion of Euclidean spaces. It allows us to visualize a given manifold in a possibly more familar setting such as . E.g. the projective plane can be visualized in with the help of {\itshape Boy's surface}, the image of a --immersion (beautiful illustrations can be found in the internet). The following two questions play an important role.
- Existence: Given and , is there any immersion at all?
- Classification: How many `essentially different´ immersions exist?
2 The smooth case:CAT = DIFF
This section is about the category of smooth, i.e. , manifolds and maps. It follows from the inverse function theorem that a smooth map between smooth manifolds is a local immersion at precisely if the tangent map is injective. Thus is a smooth immersion if and only if it induces a vector bundle monomorphism . E.g. the figure cannot be the image of a smooth immersion, due to the two sharp corners which don't allow a welldefined tangent line. However there exists a smooth immersion with image the figure .
Theorem 1.2 (Smale--Hirsch 1959/Phillips 1967). If (i) , or if (ii) is open and , then the map
is a weak homotopy equivalence. Here the sets of all smooth immersions and of all vector bundle monomorphisms , resp., are endowed with the --topology and the compact--open topology, resp.
Remark 1.3. For a good exposition of Theorem 1.2 see [Adachi1993, pp.87 and 93].
Corollary 1.4. Under the assumptions of theorem 1.2 there exists an immersion if and only if there is a vector bundle monomorphism from the tangent bundle of to . E. g. if is parallelizable (i. e. ) then .
Theorem 1.5 [Whitney1944a] . If then there exists an immersion (E. g. any surface can be immersed into ).
Remark 1.6. See also e.g. [Adachi1993, p. 86ff].
Theorem 1.5 is best possible as long as we put no restrictions on .
Example 1.7. The real projective space cannot be immersed into if . This follows from an easy calculation using Stiefel--Whitney classes.
Definition 1.8. Two immersions are \textit{regularly homotopic} if there exists a smooth map which with satisfies the following:
- ;
- is a immersion for all .
Corollary 1.9. Assume . Two immersions are regularly homotopic if and only if their tangent maps are homotopic through vector bundle monomorphisms.
Example 1.10 , [Smale1959]. The regular homotopy classes of immersions , are in one--to--one correspondance with the elements of the homotopy group , where is the Stiefel manifold of --frames in . In particular, all immersions are regularly homotopic (since ). E. g. the standard inclusion is regularly homotopic to ; i. e. you can \textit{'turn the sphere inside out'} in , with possible selfintersections but without creating any crease. The Smale-Hirsch theorem makes existence and classification problems accessible to standard methods of algebraic topology such as classical obstruction theorem (cf. e.g. [Steenrod1951]), characteristic classes (cf. e.g. [Milnor&Stasheff1974]), Postnikov towers, the singularity method (cf. e.g. [Koschorke1981]) etc.
3 Selfintersections
It is a characteristic feature of immersions --- as compared to embeddings --- that --tuple selfintersections may occur for some , i. e. points in which are the image of at least distinct elements of M (e. g. the double point in the figure 8 immersion with image ). Generically the locus of r--tuple points of a smooth immersion is an immersed ()--dimensional manifold in . Its properties may yield a variety of interesting invariants which link immersions to other concepts of topology. E. g. let denote the number of ()--tuple points of a selftransverse immersion .
Theorem 2.1 [Eccles1981] . Given a natural number , there is an --dimensional closed smooth manifold and an immersion satisfying if and only if there exists a framed ()--dimensional manifold with Kervaire invariant .
According to [Hill&Hopkins&Ravenel2009] (and previous authors) this holds precisely when or possibly . If and the manifold in question cannot be orientable (cf. [Eccles1981]). Thus the figure 8 immersion plays a rather special role here.
4 References
- [Adachi1993] M. Adachi, Embeddings and immersions, Translated from the Japanese by Kiki Hudson. Translations of Mathematical Monographs, 124. Providence, RI: American Mathematical Society (AMS), 1993. MR1225100 (95a:57039) Zbl 0810.57001
- [Eccles1981] P. J. Eccles, Codimension one immersions and the Kervaire invariant one problem, Math. Proc. Cambridge Philos. Soc. 90 (1981), no.3, 483–493. MR628831 (83c:57015) Zbl 0479.57016
- [Hill&Hopkins&Ravenel2009] M. A. Hill, M. J. Hopkins and D. C. Ravenel, On the non-existence of elements of Kervaire invariant one, (2009). Available at the arXiv:0908.3724.
- [Koschorke1981] U. Koschorke, Vector fields and other vector bundle morphisms-a singularity approach, Springer, 1981. MR611333 (82i:57026) Zbl 0459.57016
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
- [Smale1959] S. Smale, Diffeomorphisms of the -sphere, Proc. Amer. Math. Soc. 10 (1959), 621–626. MR0112149 (22 #3004) Zbl 0118.39103
- [Steenrod1951] N. Steenrod, The topology of fibre bundles., (Princeton Mathematical Series No. 14.) Princeton: Princeton University Press. VIII, 224 p. , 1951. MR1688579 (2000a:55001) Zbl 0054.07103
- [Whitney1944a] H. Whitney, The singularities of a smooth -manifold in -space, Ann. of Math. (2) 45 (1944), 247–293. MR0010275 (5,274a) Zbl 0063.08238
Let be such a map between manifolds of the indicated dimensions .
Definition 1.1. is a local immersion at a point if there exist open neighbourhoods of and of in and , resp., such that and:
- there is a CAT--isomorphism (i.e. both and are CAT--maps) which maps onto ; and
- yields a CAT-isomorphism from onto .
We call f an immersion (and we write ) if is a local immersion at every point .
Thus an immersion looks locally like the inclusion of Euclidean spaces. It allows us to visualize a given manifold in a possibly more familar setting such as . E.g. the projective plane can be visualized in with the help of {\itshape Boy's surface}, the image of a --immersion (beautiful illustrations can be found in the internet). The following two questions play an important role.
- Existence: Given and , is there any immersion at all?
- Classification: How many `essentially different´ immersions exist?
2 The smooth case:CAT = DIFF
This section is about the category of smooth, i.e. , manifolds and maps. It follows from the inverse function theorem that a smooth map between smooth manifolds is a local immersion at precisely if the tangent map is injective. Thus is a smooth immersion if and only if it induces a vector bundle monomorphism . E.g. the figure cannot be the image of a smooth immersion, due to the two sharp corners which don't allow a welldefined tangent line. However there exists a smooth immersion with image the figure .
Theorem 1.2 (Smale--Hirsch 1959/Phillips 1967). If (i) , or if (ii) is open and , then the map
is a weak homotopy equivalence. Here the sets of all smooth immersions and of all vector bundle monomorphisms , resp., are endowed with the --topology and the compact--open topology, resp.
Remark 1.3. For a good exposition of Theorem 1.2 see [Adachi1993, pp.87 and 93].
Corollary 1.4. Under the assumptions of theorem 1.2 there exists an immersion if and only if there is a vector bundle monomorphism from the tangent bundle of to . E. g. if is parallelizable (i. e. ) then .
Theorem 1.5 [Whitney1944a] . If then there exists an immersion (E. g. any surface can be immersed into ).
Remark 1.6. See also e.g. [Adachi1993, p. 86ff].
Theorem 1.5 is best possible as long as we put no restrictions on .
Example 1.7. The real projective space cannot be immersed into if . This follows from an easy calculation using Stiefel--Whitney classes.
Definition 1.8. Two immersions are \textit{regularly homotopic} if there exists a smooth map which with satisfies the following:
- ;
- is a immersion for all .
Corollary 1.9. Assume . Two immersions are regularly homotopic if and only if their tangent maps are homotopic through vector bundle monomorphisms.
Example 1.10 , [Smale1959]. The regular homotopy classes of immersions , are in one--to--one correspondance with the elements of the homotopy group , where is the Stiefel manifold of --frames in . In particular, all immersions are regularly homotopic (since ). E. g. the standard inclusion is regularly homotopic to ; i. e. you can \textit{'turn the sphere inside out'} in , with possible selfintersections but without creating any crease. The Smale-Hirsch theorem makes existence and classification problems accessible to standard methods of algebraic topology such as classical obstruction theorem (cf. e.g. [Steenrod1951]), characteristic classes (cf. e.g. [Milnor&Stasheff1974]), Postnikov towers, the singularity method (cf. e.g. [Koschorke1981]) etc.
3 Selfintersections
It is a characteristic feature of immersions --- as compared to embeddings --- that --tuple selfintersections may occur for some , i. e. points in which are the image of at least distinct elements of M (e. g. the double point in the figure 8 immersion with image ). Generically the locus of r--tuple points of a smooth immersion is an immersed ()--dimensional manifold in . Its properties may yield a variety of interesting invariants which link immersions to other concepts of topology. E. g. let denote the number of ()--tuple points of a selftransverse immersion .
Theorem 2.1 [Eccles1981] . Given a natural number , there is an --dimensional closed smooth manifold and an immersion satisfying if and only if there exists a framed ()--dimensional manifold with Kervaire invariant .
According to [Hill&Hopkins&Ravenel2009] (and previous authors) this holds precisely when or possibly . If and the manifold in question cannot be orientable (cf. [Eccles1981]). Thus the figure 8 immersion plays a rather special role here.
4 References
- [Adachi1993] M. Adachi, Embeddings and immersions, Translated from the Japanese by Kiki Hudson. Translations of Mathematical Monographs, 124. Providence, RI: American Mathematical Society (AMS), 1993. MR1225100 (95a:57039) Zbl 0810.57001
- [Eccles1981] P. J. Eccles, Codimension one immersions and the Kervaire invariant one problem, Math. Proc. Cambridge Philos. Soc. 90 (1981), no.3, 483–493. MR628831 (83c:57015) Zbl 0479.57016
- [Hill&Hopkins&Ravenel2009] M. A. Hill, M. J. Hopkins and D. C. Ravenel, On the non-existence of elements of Kervaire invariant one, (2009). Available at the arXiv:0908.3724.
- [Koschorke1981] U. Koschorke, Vector fields and other vector bundle morphisms-a singularity approach, Springer, 1981. MR611333 (82i:57026) Zbl 0459.57016
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
- [Smale1959] S. Smale, Diffeomorphisms of the -sphere, Proc. Amer. Math. Soc. 10 (1959), 621–626. MR0112149 (22 #3004) Zbl 0118.39103
- [Steenrod1951] N. Steenrod, The topology of fibre bundles., (Princeton Mathematical Series No. 14.) Princeton: Princeton University Press. VIII, 224 p. , 1951. MR1688579 (2000a:55001) Zbl 0054.07103
- [Whitney1944a] H. Whitney, The singularities of a smooth -manifold in -space, Ann. of Math. (2) 45 (1944), 247–293. MR0010275 (5,274a) Zbl 0063.08238