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| $$ of $\nu$ \cite{Spanier1981}, page 440. This means that $p_r: B_r \to BO$ is a fibration, where $B_r$ is a $CW$-complex, the map $p_r$ is $r$-connected, i.e. the homotopy groups of the fibre vanish in degree $\ge r$, and the map $\bar \nu_r$ is an $r$-equivalence. | | $$ of $\nu$ \cite{Spanier1981}, page 440. This means that $p_r: B_r \to BO$ is a fibration, where $B_r$ is a $CW$-complex, the map $p_r$ is $r$-connected, i.e. the homotopy groups of the fibre vanish in degree $\ge r$, and the map $\bar \nu_r$ is an $r$-equivalence. |
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− | {{beginthm|Definition}} The fibre homotopy type of the fibration $p_{k+1}: B_{k+1} \to BO$ is an invariant of the map $\nu$ and is called the '''normal $k$-type''' of $M$ \cite{Kreck1999} denoted $p^k(M):B^{k}(M)\to BO$. | + | {{beginthm|Definition}} The fibre homotopy type of the fibration $p_{k+1}: B_{k+1} \to BO$ is an invariant of the map $\nu$ and is called the '''normal $k$-type''' of $M$ denoted $p^k(M):B^{k}(M)\to BO$ \cite{Kreck1999}. |
| {{endthm}} | | {{endthm}} |
| | | |
Revision as of 14:14, 26 November 2010
This page has not been refereed. The information given here might be incomplete or provisional.
|
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 the fibration
,if
admits a
-structure (if and only if the Stiefel-Whintey class
vanishes) and the fibration
if
does not admit a
structure. 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 the fibration
,if
admits a
-structure (if and only if the Stiefel-Whintey class
vanishes) and the fibration
if
does not admit a
structure. 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 the fibration
,if
admits a
-structure (if and only if the Stiefel-Whintey class
vanishes) and the fibration
if
does not admit a
structure. 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 the fibration
$$BSpin \to BSO$$,if $M$ admits a $Spin$-structure (if and only if the Stiefel-Whintey class $w_2(M)$ vanishes) and the fibration
$$BSO \to BO,$$ if $M$ does not admit a $Spin$ structure. 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 the fibration
,if
admits a
-structure (if and only if the Stiefel-Whintey class
vanishes) and the fibration
if
does not admit a
structure. 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 the fibration
,if
admits a
-structure (if and only if the Stiefel-Whintey class
vanishes) and the fibration
if
does not admit a
structure. 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 the fibration
,if
admits a
-structure (if and only if the Stiefel-Whintey class
vanishes) and the fibration
if
does not admit a
structure. 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