Stable classification of manifolds
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For example the normal $0$-type of an oriented manifold is the universal covering | For example the normal $0$-type of an oriented manifold is the universal covering | ||
$$BSO \to BO,$$ whereas the normal $0$-type of a non-orientable manifold is the identity map | $$BSO \to BO,$$ whereas the normal $0$-type of a non-orientable manifold is the identity map | ||
− | $$BO \to BO.$$ The normal $1$-type of a simply connected manifold $M$ is | + | $$BO \to BO.$$ The normal $1$-type of a simply connected manifold $M$ is determined by its second Stiefel-Whitney class $w_2(M)$ which in turn determines whether it is spinable or not: if $M$ admits a $Spin$-structure its normal $1$-type is the fibration |
− | $$ | + | $$BSpin \to BSO$$, |
− | $$ | + | if $M$ does not admit a $Spin$-structure then it's normal $1$-type is the fibration |
+ | $$BSO \to BO.$$ More generally, the normal $1$-type of a Spin-manifold $M$ is $$p: K(\pi_1(M),1) \times BSpin \to BO,$$ where $p$ is the composition of the projection to $BSpin$ and the projection $BSpin \to BO$. The normal $1$-type of a manifold $M$ such that the universal covering $\tilde M$ does not admit a $Spin$-structure, is $$p: K(\pi_1(M),1) \times BS= \to BO,$$ where $p$ is the composition of the projection to $BSO$ and the projection $BSO \to BO$. The case where $\tilde M $ admits a $Spin$-structure but $M$ doesn't, is treated in [[Stable classification of 4-manifolds]]. | ||
If $B^k \to BO$ | If $B^k \to BO$ | ||
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Revision as of 15:56, 26 November 2010
This page has not been refereed. The information given here might be incomplete or provisional. |
1 Introduction
Tex syntax errorand of dimension are called stably diffeomorphic if there is an integer
Tex syntax errorsuch that is diffeomorphic to . By we mean the connected sum with
Tex syntax errorcopies of . Note that since has an orientation reversing diffeomorphism the connect sum with it is well defined (see Parametric connected sum). We present a method which reduces the stable classification to a bordism problem.
2 The normal k-type
Tex syntax error-factorization
of [Spanier1981], page 440. This means that is a fibration, where is a -complex, the map is
Tex syntax error-connected, i.e. the homotopy groups of the fibre vanish in degree , and the map is an
Tex syntax error-equivalence.
Definition 1.1. The fibre homotopy type of the fibration is an invariant of the map and is called the normal -type of
Tex syntax errordenoted [Kreck1999].
For example the normal -type of an oriented manifold is the universal covering
whereas the normal -type of a non-orientable manifold is the identity map
The normal -type of a simply connected manifold
Tex syntax erroris determined by its second Stiefel-Whitney class which in turn determines whether it is spinable or not: if
Tex syntax erroradmits a -structure its normal -type is the fibration
,
if
Tex syntax errordoes not admit a -structure then it's normal -type is the fibration
More generally, the normal -type of a Spin-manifold
Tex syntax erroris
where is the composition of the projection to and the projection . The normal -type of a manifold
Tex syntax errorsuch that the universal covering does not admit a -structure, is
where is the composition of the projection to and the projection . The case where admits a -structure but
Tex syntax errordoesn't, is treated in Stable classification of 4-manifolds.
If
References
- [Kreck1999] M. Kreck, Surgery and duality, Ann. of Math. (2) 149 (1999), no.3, 707–754. MR1709301 (2001a:57051) Zbl 0935.57039
- [Spanier1981] E. H. Spanier, Algebraic topology, Springer-Verlag, 1981. MR666554 (83i:55001) Zbl 0810.55001
Tex syntax errorsuch that is diffeomorphic to . By we mean the connected sum with
Tex syntax errorcopies of . Note that since has an orientation reversing diffeomorphism the connect sum with it is well defined (see Parametric connected sum). We present a method which reduces the stable classification to a bordism problem.
2 The normal k-type
Tex syntax error-factorization
of [Spanier1981], page 440. This means that is a fibration, where is a -complex, the map is
Tex syntax error-connected, i.e. the homotopy groups of the fibre vanish in degree , and the map is an
Tex syntax error-equivalence.
Definition 1.1. The fibre homotopy type of the fibration is an invariant of the map and is called the normal -type of
Tex syntax errordenoted [Kreck1999].
For example the normal -type of an oriented manifold is the universal covering
whereas the normal -type of a non-orientable manifold is the identity map
The normal -type of a simply connected manifold
Tex syntax erroris determined by its second Stiefel-Whitney class which in turn determines whether it is spinable or not: if
Tex syntax erroradmits a -structure its normal -type is the fibration
,
if
Tex syntax errordoes not admit a -structure then it's normal -type is the fibration
More generally, the normal -type of a Spin-manifold
Tex syntax erroris
where is the composition of the projection to and the projection . The normal -type of a manifold
Tex syntax errorsuch that the universal covering does not admit a -structure, is
where is the composition of the projection to and the projection . The case where admits a -structure but
Tex syntax errordoesn't, is treated in Stable classification of 4-manifolds.
If
References
- [Kreck1999] M. Kreck, Surgery and duality, Ann. of Math. (2) 149 (1999), no.3, 707–754. MR1709301 (2001a:57051) Zbl 0935.57039
- [Spanier1981] E. H. Spanier, Algebraic topology, Springer-Verlag, 1981. MR666554 (83i:55001) Zbl 0810.55001
Tex syntax errorsuch that is diffeomorphic to . By we mean the connected sum with
Tex syntax errorcopies of . Note that since has an orientation reversing diffeomorphism the connect sum with it is well defined (see Parametric connected sum). We present a method which reduces the stable classification to a bordism problem.
2 The normal k-type
Tex syntax error-factorization
of [Spanier1981], page 440. This means that is a fibration, where is a -complex, the map is
Tex syntax error-connected, i.e. the homotopy groups of the fibre vanish in degree , and the map is an
Tex syntax error-equivalence.
Definition 1.1. The fibre homotopy type of the fibration is an invariant of the map and is called the normal -type of
Tex syntax errordenoted [Kreck1999].
For example the normal -type of an oriented manifold is the universal covering
whereas the normal -type of a non-orientable manifold is the identity map
The normal -type of a simply connected manifold
Tex syntax erroris determined by its second Stiefel-Whitney class which in turn determines whether it is spinable or not: if
Tex syntax erroradmits a -structure its normal -type is the fibration
,
if
Tex syntax errordoes not admit a -structure then it's normal -type is the fibration
More generally, the normal -type of a Spin-manifold
Tex syntax erroris
where is the composition of the projection to and the projection . The normal -type of a manifold
Tex syntax errorsuch that the universal covering does not admit a -structure, is
where is the composition of the projection to and the projection . The case where admits a -structure but
Tex syntax errordoesn't, is treated in Stable classification of 4-manifolds.
If
References
- [Kreck1999] M. Kreck, Surgery and duality, Ann. of Math. (2) 149 (1999), no.3, 707–754. MR1709301 (2001a:57051) Zbl 0935.57039
- [Spanier1981] E. H. Spanier, Algebraic topology, Springer-Verlag, 1981. MR666554 (83i:55001) Zbl 0810.55001
Tex syntax errorsuch that is diffeomorphic to . By we mean the connected sum with
Tex syntax errorcopies of . Note that since has an orientation reversing diffeomorphism the connect sum with it is well defined (see Parametric connected sum). We present a method which reduces the stable classification to a bordism problem.
2 The normal k-type
Tex syntax error-factorization
of [Spanier1981], page 440. This means that is a fibration, where is a -complex, the map is
Tex syntax error-connected, i.e. the homotopy groups of the fibre vanish in degree , and the map is an
Tex syntax error-equivalence.
Definition 1.1. The fibre homotopy type of the fibration is an invariant of the map and is called the normal -type of
Tex syntax errordenoted [Kreck1999].
For example the normal -type of an oriented manifold is the universal covering
whereas the normal -type of a non-orientable manifold is the identity map
The normal -type of a simply connected manifold
Tex syntax erroris determined by its second Stiefel-Whitney class which in turn determines whether it is spinable or not: if
Tex syntax erroradmits a -structure its normal -type is the fibration
,
if
Tex syntax errordoes not admit a -structure then it's normal -type is the fibration
More generally, the normal -type of a Spin-manifold
Tex syntax erroris
where is the composition of the projection to and the projection . The normal -type of a manifold
Tex syntax errorsuch that the universal covering does not admit a -structure, is
where is the composition of the projection to and the projection . The case where admits a -structure but
Tex syntax errordoesn't, is treated in Stable classification of 4-manifolds.
If
References
- [Kreck1999] M. Kreck, Surgery and duality, Ann. of Math. (2) 149 (1999), no.3, 707–754. MR1709301 (2001a:57051) Zbl 0935.57039
- [Spanier1981] E. H. Spanier, Algebraic topology, Springer-Verlag, 1981. MR666554 (83i:55001) Zbl 0810.55001
Tex syntax errorsuch that is diffeomorphic to . By we mean the connected sum with
Tex syntax errorcopies of . Note that since has an orientation reversing diffeomorphism the connect sum with it is well defined (see Parametric connected sum). We present a method which reduces the stable classification to a bordism problem.
2 The normal k-type
Tex syntax error-factorization
of [Spanier1981], page 440. This means that is a fibration, where is a -complex, the map is
Tex syntax error-connected, i.e. the homotopy groups of the fibre vanish in degree , and the map is an
Tex syntax error-equivalence.
Definition 1.1. The fibre homotopy type of the fibration is an invariant of the map and is called the normal -type of
Tex syntax errordenoted [Kreck1999].
For example the normal -type of an oriented manifold is the universal covering
whereas the normal -type of a non-orientable manifold is the identity map
The normal -type of a simply connected manifold
Tex syntax erroris determined by its second Stiefel-Whitney class which in turn determines whether it is spinable or not: if
Tex syntax erroradmits a -structure its normal -type is the fibration
,
if
Tex syntax errordoes not admit a -structure then it's normal -type is the fibration
More generally, the normal -type of a Spin-manifold
Tex syntax erroris
where is the composition of the projection to and the projection . The normal -type of a manifold
Tex syntax errorsuch that the universal covering does not admit a -structure, is
where is the composition of the projection to and the projection . The case where admits a -structure but
Tex syntax errordoesn't, is treated in Stable classification of 4-manifolds.
If
References
- [Kreck1999] M. Kreck, Surgery and duality, Ann. of Math. (2) 149 (1999), no.3, 707–754. MR1709301 (2001a:57051) Zbl 0935.57039
- [Spanier1981] E. H. Spanier, Algebraic topology, Springer-Verlag, 1981. MR666554 (83i:55001) Zbl 0810.55001
Tex syntax errorsuch that is diffeomorphic to . By we mean the connected sum with
Tex syntax errorcopies of . Note that since has an orientation reversing diffeomorphism the connect sum with it is well defined (see Parametric connected sum). We present a method which reduces the stable classification to a bordism problem.
2 The normal k-type
Tex syntax error-factorization
of [Spanier1981], page 440. This means that is a fibration, where is a -complex, the map is
Tex syntax error-connected, i.e. the homotopy groups of the fibre vanish in degree , and the map is an
Tex syntax error-equivalence.
Definition 1.1. The fibre homotopy type of the fibration is an invariant of the map and is called the normal -type of
Tex syntax errordenoted [Kreck1999].
For example the normal -type of an oriented manifold is the universal covering
whereas the normal -type of a non-orientable manifold is the identity map
The normal -type of a simply connected manifold
Tex syntax erroris determined by its second Stiefel-Whitney class which in turn determines whether it is spinable or not: if
Tex syntax erroradmits a -structure its normal -type is the fibration
,
if
Tex syntax errordoes not admit a -structure then it's normal -type is the fibration
More generally, the normal -type of a Spin-manifold
Tex syntax erroris
where is the composition of the projection to and the projection . The normal -type of a manifold
Tex syntax errorsuch that the universal covering does not admit a -structure, is
where is the composition of the projection to and the projection . The case where admits a -structure but
Tex syntax errordoesn't, is treated in Stable classification of 4-manifolds.
If
References
- [Kreck1999] M. Kreck, Surgery and duality, Ann. of Math. (2) 149 (1999), no.3, 707–754. MR1709301 (2001a:57051) Zbl 0935.57039
- [Spanier1981] E. H. Spanier, Algebraic topology, Springer-Verlag, 1981. MR666554 (83i:55001) Zbl 0810.55001