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1 Introduction
-
Two closed smooth manifolds and of dimension are called stably diffeomorphic if there is an integer such that is diffeomorphic to . By we mean the connected sum with copies 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
-
Consider the
stable normal Gauss map . We consider an
-factorization
of
[
Spanier1981], page 440. This means that
is a fibration, where
is a
-complex, the map
is
-connected, i.e. the homotopy groups of the fibre vanish in degree
, and the map
is an
-equivalence.
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
is determined by its second Stiefel-Whitney class
which in turn determines whether it is spinable or not: if
admits a
-structure its normal
-type is the fibration
,
if does not admit a -structure then it's normal -type is the fibration
More generally, the normal
-type of a Spin-manifold
is
where
is the composition of the projection to
and the projection
. The normal
-type of a manifold
such 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
doesn't, is treated in
Stable classification of 4-manifolds.
If
References
$-type of an oriented manifold is the universal covering
$$BSO \to BO,$$ whereas the normal and
of dimension
are called
stably diffeomorphic if there is an integer
such that
is diffeomorphic to
. By
we mean the connected sum with
copies 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
-
Consider the
stable normal Gauss map . We consider an
-factorization
of
[
Spanier1981], page 440. This means that
is a fibration, where
is a
-complex, the map
is
-connected, i.e. the homotopy groups of the fibre vanish in degree
, and the map
is an
-equivalence.
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
is determined by its second Stiefel-Whitney class
which in turn determines whether it is spinable or not: if
admits a
-structure its normal
-type is the fibration
,
if does not admit a -structure then it's normal -type is the fibration
More generally, the normal
-type of a Spin-manifold
is
where
is the composition of the projection to
and the projection
. The normal
-type of a manifold
such 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
doesn't, is treated in
Stable classification of 4-manifolds.
If
References
$-type of a non-orientable manifold is the identity map
$$BO \to BO.$$ The normal M and
of dimension
are called
stably diffeomorphic if there is an integer
such that
is diffeomorphic to
. By
we mean the connected sum with
copies 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
-
Consider the
stable normal Gauss map . We consider an
-factorization
of
[
Spanier1981], page 440. This means that
is a fibration, where
is a
-complex, the map
is
-connected, i.e. the homotopy groups of the fibre vanish in degree
, and the map
is an
-equivalence.
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
is determined by its second Stiefel-Whitney class
which in turn determines whether it is spinable or not: if
admits a
-structure its normal
-type is the fibration
,
if does not admit a -structure then it's normal -type is the fibration
More generally, the normal
-type of a Spin-manifold
is
where
is the composition of the projection to
and the projection
. The normal
-type of a manifold
such 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
doesn't, is treated in
Stable classification of 4-manifolds.
If
References
$-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 M and
of dimension
are called
stably diffeomorphic if there is an integer
such that
is diffeomorphic to
. By
we mean the connected sum with
copies 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
-
Consider the
stable normal Gauss map . We consider an
-factorization
of
[
Spanier1981], page 440. This means that
is a fibration, where
is a
-complex, the map
is
-connected, i.e. the homotopy groups of the fibre vanish in degree
, and the map
is an
-equivalence.
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
is determined by its second Stiefel-Whitney class
which in turn determines whether it is spinable or not: if
admits a
-structure its normal
-type is the fibration
,
if does not admit a -structure then it's normal -type is the fibration
More generally, the normal
-type of a Spin-manifold
is
where
is the composition of the projection to
and the projection
. The normal
-type of a manifold
such 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
doesn't, is treated in
Stable classification of 4-manifolds.
If
References
$-type is the fibration
$$BSpin \to BSO$$,
if $M$ does not admit a $Spin$-structure then it's normal M and
of dimension
are called
stably diffeomorphic if there is an integer
such that
is diffeomorphic to
. By
we mean the connected sum with
copies 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
-
Consider the
stable normal Gauss map . We consider an
-factorization
of
[
Spanier1981], page 440. This means that
is a fibration, where
is a
-complex, the map
is
-connected, i.e. the homotopy groups of the fibre vanish in degree
, and the map
is an
-equivalence.
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
is determined by its second Stiefel-Whitney class
which in turn determines whether it is spinable or not: if
admits a
-structure its normal
-type is the fibration
,
if does not admit a -structure then it's normal -type is the fibration
More generally, the normal
-type of a Spin-manifold
is
where
is the composition of the projection to
and the projection
. The normal
-type of a manifold
such 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
doesn't, is treated in
Stable classification of 4-manifolds.
If
References
$-type is the fibration
$$BSO \to BO.$$ More generally, the normal M and
of dimension
are called
stably diffeomorphic if there is an integer
such that
is diffeomorphic to
. By
we mean the connected sum with
copies 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
-
Consider the
stable normal Gauss map . We consider an
-factorization
of
[
Spanier1981], page 440. This means that
is a fibration, where
is a
-complex, the map
is
-connected, i.e. the homotopy groups of the fibre vanish in degree
, and the map
is an
-equivalence.
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
is determined by its second Stiefel-Whitney class
which in turn determines whether it is spinable or not: if
admits a
-structure its normal
-type is the fibration
,
if does not admit a -structure then it's normal -type is the fibration
More generally, the normal
-type of a Spin-manifold
is
where
is the composition of the projection to
and the projection
. The normal
-type of a manifold
such 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
doesn't, is treated in
Stable classification of 4-manifolds.
If
References
$-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 M and
of dimension
are called
stably diffeomorphic if there is an integer
such that
is diffeomorphic to
. By
we mean the connected sum with
copies 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
-
Consider the
stable normal Gauss map . We consider an
-factorization
of
[
Spanier1981], page 440. This means that
is a fibration, where
is a
-complex, the map
is
-connected, i.e. the homotopy groups of the fibre vanish in degree
, and the map
is an
-equivalence.
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
is determined by its second Stiefel-Whitney class
which in turn determines whether it is spinable or not: if
admits a
-structure its normal
-type is the fibration
,
if does not admit a -structure then it's normal -type is the fibration
More generally, the normal
-type of a Spin-manifold
is
where
is the composition of the projection to
and the projection
. The normal
-type of a manifold
such 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
doesn't, is treated in
Stable classification of 4-manifolds.
If
References
$-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 $B^k \to BO$
== References ==
{{#RefList:}}
[[Category:Theory]]M and
of dimension
are called
stably diffeomorphic if there is an integer
such that
is diffeomorphic to
. By
we mean the connected sum with
copies 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
-
Consider the
stable normal Gauss map . We consider an
-factorization
of
[
Spanier1981], page 440. This means that
is a fibration, where
is a
-complex, the map
is
-connected, i.e. the homotopy groups of the fibre vanish in degree
, and the map
is an
-equivalence.
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
is determined by its second Stiefel-Whitney class
which in turn determines whether it is spinable or not: if
admits a
-structure its normal
-type is the fibration
,
if does not admit a -structure then it's normal -type is the fibration
More generally, the normal
-type of a Spin-manifold
is
where
is the composition of the projection to
and the projection
. The normal
-type of a manifold
such 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
doesn't, is treated in
Stable classification of 4-manifolds.
If
References