Stable classification of manifolds

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1 Introduction

Two closed smooth manifolds $M$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> {{Stub}} == Introduction == <wikitex>; Two closed smooth manifolds $M$ and $N$ of dimension n$ are called '''stably diffeomorphic''' if there is an integer $r$ such that $M \sharp_r S^n \times S^n$ is diffeomorphic to $N \sharp_r S^n \times S^n$. By $\sharp_r S^n \times S^n$ we mean the connected sum with $r$ copies of $S^n \times S^n$. Note that since $S^n \times S^n$ has an orientation reversing diffeomorphism the connect sum with it is well defined (see [[Parametric connected sum]]). We present a method which reduces the stable classification to a bordism problem. </wikitex> == The normal k-type == <wikitex>; Consider the [[B-Bordism#B-structures and bordisms|stable normal Gauss map]] $\nu: M \to BO$. We consider an $r$-factorization $$M \stackrel{\nu_r}{\to}B_r \stackrel{p_r}{\to}BO $$ of $\nu$ \cite{Spanier1981}, page 440. This means that $p_r: B_r \to BO$ is a fibration, where $B_r$ is a $CW$-complex, the map $p_r$ is $r$-connected, i.e. the homotopy groups of the fibre vanish in degree $\ge r$, and the map $\bar \nu_r$ is an $r$-equivalence. {{beginthm|Definition}} The fibre homotopy type of the fibration $p_{k+1}: B_{k+1} \to BO$ is an invariant of the map $\nu$ and is called the '''normal $k$-type''' of $M$ denoted $p^k(M):B^{k}(M)\to BO$ \cite{Kreck1999}. {{endthm}} For example the normal <script type="math/tex">M$ and $N$$N$ of dimension $2n$$2n$ are called stably diffeomorphic if there is an integer $r$$r$ such that $M \sharp_r S^n \times S^n$$M \sharp_r S^n \times S^n$ is diffeomorphic to $N \sharp_r S^n \times S^n$$N \sharp_r S^n \times S^n$. By $\sharp_r S^n \times S^n$$\sharp_r S^n \times S^n$ we mean the connected sum with $r$$r$ copies of $S^n \times S^n$$S^n \times S^n$. Note that since $S^n \times S^n$$S^n \times S^n$ 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 $\nu: M \to BO$$\nu: M \to BO$. We consider an $r$$r$-factorization

$\displaystyle M \stackrel{\nu_r}{\to}B_r \stackrel{p_r}{\to}BO$ $$\displaystyle M \stackrel{\nu_r}{\to}B_r \stackrel{p_r}{\to}BO$$

of $\nu$$\nu$ [Spanier1981], page 440. This means that $p_r: B_r \to BO$$p_r: B_r \to BO$ is a fibration, where $B_r$$B_r$ is a $CW$$CW$-complex, the map $p_r$$p_r$ is $r$$r$-connected, i.e. the homotopy groups of the fibre vanish in degree $\ge r$$\ge r$, and the map $\bar \nu_r$$\bar \nu_r$ is an $r$$r$-equivalence.

Definition 2.1. The fibre homotopy type of the fibration $p_{k+1}: B_{k+1} \to BO$$p_{k+1}: B_{k+1} \to BO$ is an invariant of the map $\nu$$\nu$ and is called the normal $k$$k$-type of $M$$M$ denoted $p^k(M):B^{k}(M)\to BO$$p^k(M):B^{k}(M)\to BO$ [Kreck1999].

For example the normal $0$$0$-type of an oriented manifold is the universal covering

$\displaystyle BSO \to BO,$ $$\displaystyle BSO \to BO,$$

whereas the normal $0$$0$-type of a non-orientable manifold is the identity map

$\displaystyle BO \to BO.$ $$\displaystyle BO \to BO.$$

The normal $1$$1$-type of a simply connected manifold $M$$M$ is the fibration

$\displaystyle BSpin \to BSO$ $$\displaystyle BSpin \to BSO$$

,if $M$$M$ admits a $Spin$$Spin$-structure (if and only if the Stiefel-Whintey class $w_2(M)$$w_2(M)$ vanishes) and the fibration

$\displaystyle BSO \to BO,$ $$\displaystyle BSO \to BO,$$

if $M$$M$ does not admit a $Spin$$Spin$ structure. More generally, the normal $1$$1$-type of a Spin-manifold $M$$M$ is

$\displaystyle p: K(\pi_1(M),1) \times BSpin \to BO,$ $$\displaystyle p: K(\pi_1(M),1) \times BSpin \to BO,$$

where $p$$p$ is the composition of the projection to $BSpin$$BSpin$ and the projection $BSpin \to BO$$BSpin \to BO$. The normal $1$$1$-type of a manifold $M$$M$ such that the universal covering $\tilde M$$\tilde M$ does not admit a $Spin$$Spin$-structure, is

$\displaystyle p: K(\pi_1(M),1) \times BS= \to BO,$ $$\displaystyle p: K(\pi_1(M),1) \times BS= \to BO,$$

where $p$$p$ is the composition of the projection to $BSO$$BSO$ and the projection $BSO \to BO$$BSO \to BO$. The case where $\tilde M$$\tilde M$ admits a $Spin$$Spin$-structure but $M$$M$ doesn't, is treated in Stable classification of 4-manifolds.

If $B^k \to BO$$B^k \to BO$

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

$-type of an oriented manifold is the universal covering $$BSO \to BO,$$ whereas the normal $M$ and $N$$N$ of dimension $2n$$2n$ are called stably diffeomorphic if there is an integer $r$$r$ such that $M \sharp_r S^n \times S^n$$M \sharp_r S^n \times S^n$ is diffeomorphic to $N \sharp_r S^n \times S^n$$N \sharp_r S^n \times S^n$. By $\sharp_r S^n \times S^n$$\sharp_r S^n \times S^n$ we mean the connected sum with $r$$r$ copies of $S^n \times S^n$$S^n \times S^n$. Note that since $S^n \times S^n$$S^n \times S^n$ 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 $\nu: M \to BO$$\nu: M \to BO$. We consider an $r$$r$-factorization

$\displaystyle M \stackrel{\nu_r}{\to}B_r \stackrel{p_r}{\to}BO$ $$\displaystyle M \stackrel{\nu_r}{\to}B_r \stackrel{p_r}{\to}BO$$

of $\nu$$\nu$ [Spanier1981], page 440. This means that $p_r: B_r \to BO$$p_r: B_r \to BO$ is a fibration, where $B_r$$B_r$ is a $CW$$CW$-complex, the map $p_r$$p_r$ is $r$$r$-connected, i.e. the homotopy groups of the fibre vanish in degree $\ge r$$\ge r$, and the map $\bar \nu_r$$\bar \nu_r$ is an $r$$r$-equivalence.

Definition 2.1. The fibre homotopy type of the fibration $p_{k+1}: B_{k+1} \to BO$$p_{k+1}: B_{k+1} \to BO$ is an invariant of the map $\nu$$\nu$ and is called the normal $k$$k$-type of $M$$M$ denoted $p^k(M):B^{k}(M)\to BO$$p^k(M):B^{k}(M)\to BO$ [Kreck1999].

For example the normal $0$$0$-type of an oriented manifold is the universal covering

$\displaystyle BSO \to BO,$ $$\displaystyle BSO \to BO,$$

whereas the normal $0$$0$-type of a non-orientable manifold is the identity map

$\displaystyle BO \to BO.$ $$\displaystyle BO \to BO.$$

The normal $1$$1$-type of a simply connected manifold $M$$M$ is the fibration

$\displaystyle BSpin \to BSO$ $$\displaystyle BSpin \to BSO$$

,if $M$$M$ admits a $Spin$$Spin$-structure (if and only if the Stiefel-Whintey class $w_2(M)$$w_2(M)$ vanishes) and the fibration

$\displaystyle BSO \to BO,$ $$\displaystyle BSO \to BO,$$

if $M$$M$ does not admit a $Spin$$Spin$ structure. More generally, the normal $1$$1$-type of a Spin-manifold $M$$M$ is

$\displaystyle p: K(\pi_1(M),1) \times BSpin \to BO,$ $$\displaystyle p: K(\pi_1(M),1) \times BSpin \to BO,$$

where $p$$p$ is the composition of the projection to $BSpin$$BSpin$ and the projection $BSpin \to BO$$BSpin \to BO$. The normal $1$$1$-type of a manifold $M$$M$ such that the universal covering $\tilde M$$\tilde M$ does not admit a $Spin$$Spin$-structure, is

$\displaystyle p: K(\pi_1(M),1) \times BS= \to BO,$ $$\displaystyle p: K(\pi_1(M),1) \times BS= \to BO,$$

where $p$$p$ is the composition of the projection to $BSO$$BSO$ and the projection $BSO \to BO$$BSO \to BO$. The case where $\tilde M$$\tilde M$ admits a $Spin$$Spin$-structure but $M$$M$ doesn't, is treated in Stable classification of 4-manifolds.

If $B^k \to BO$$B^k \to BO$

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

$-type of a non-orientable manifold is the identity map $$BO \to BO.$$ The normal M and $N$$N$ of dimension $2n$$2n$ are called stably diffeomorphic if there is an integer $r$$r$ such that $M \sharp_r S^n \times S^n$$M \sharp_r S^n \times S^n$ is diffeomorphic to $N \sharp_r S^n \times S^n$$N \sharp_r S^n \times S^n$. By $\sharp_r S^n \times S^n$$\sharp_r S^n \times S^n$ we mean the connected sum with $r$$r$ copies of $S^n \times S^n$$S^n \times S^n$. Note that since $S^n \times S^n$$S^n \times S^n$ 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 $\nu: M \to BO$$\nu: M \to BO$. We consider an $r$$r$-factorization

$\displaystyle M \stackrel{\nu_r}{\to}B_r \stackrel{p_r}{\to}BO$ $$\displaystyle M \stackrel{\nu_r}{\to}B_r \stackrel{p_r}{\to}BO$$

of $\nu$$\nu$ [Spanier1981], page 440. This means that $p_r: B_r \to BO$$p_r: B_r \to BO$ is a fibration, where $B_r$$B_r$ is a $CW$$CW$-complex, the map $p_r$$p_r$ is $r$$r$-connected, i.e. the homotopy groups of the fibre vanish in degree $\ge r$$\ge r$, and the map $\bar \nu_r$$\bar \nu_r$ is an $r$$r$-equivalence.

Definition 2.1. The fibre homotopy type of the fibration $p_{k+1}: B_{k+1} \to BO$$p_{k+1}: B_{k+1} \to BO$ is an invariant of the map $\nu$$\nu$ and is called the normal $k$$k$-type of $M$$M$ denoted $p^k(M):B^{k}(M)\to BO$$p^k(M):B^{k}(M)\to BO$ [Kreck1999].

For example the normal $0$$0$-type of an oriented manifold is the universal covering

$\displaystyle BSO \to BO,$ $$\displaystyle BSO \to BO,$$

whereas the normal $0$$0$-type of a non-orientable manifold is the identity map

$\displaystyle BO \to BO.$ $$\displaystyle BO \to BO.$$

The normal $1$$1$-type of a simply connected manifold $M$$M$ is the fibration

$\displaystyle BSpin \to BSO$ $$\displaystyle BSpin \to BSO$$

,if $M$$M$ admits a $Spin$$Spin$-structure (if and only if the Stiefel-Whintey class $w_2(M)$$w_2(M)$ vanishes) and the fibration

$\displaystyle BSO \to BO,$ $$\displaystyle BSO \to BO,$$

if $M$$M$ does not admit a $Spin$$Spin$ structure. More generally, the normal $1$$1$-type of a Spin-manifold $M$$M$ is

$\displaystyle p: K(\pi_1(M),1) \times BSpin \to BO,$ $$\displaystyle p: K(\pi_1(M),1) \times BSpin \to BO,$$

where $p$$p$ is the composition of the projection to $BSpin$$BSpin$ and the projection $BSpin \to BO$$BSpin \to BO$. The normal $1$$1$-type of a manifold $M$$M$ such that the universal covering $\tilde M$$\tilde M$ does not admit a $Spin$$Spin$-structure, is

$\displaystyle p: K(\pi_1(M),1) \times BS= \to BO,$ $$\displaystyle p: K(\pi_1(M),1) \times BS= \to BO,$$

where $p$$p$ is the composition of the projection to $BSO$$BSO$ and the projection $BSO \to BO$$BSO \to BO$. The case where $\tilde M$$\tilde M$ admits a $Spin$$Spin$-structure but $M$$M$ doesn't, is treated in Stable classification of 4-manifolds.

If $B^k \to BO$$B^k \to BO$

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

$-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 $N$$N$ of dimension $2n$$2n$ are called stably diffeomorphic if there is an integer $r$$r$ such that $M \sharp_r S^n \times S^n$$M \sharp_r S^n \times S^n$ is diffeomorphic to $N \sharp_r S^n \times S^n$$N \sharp_r S^n \times S^n$. By $\sharp_r S^n \times S^n$$\sharp_r S^n \times S^n$ we mean the connected sum with $r$$r$ copies of $S^n \times S^n$$S^n \times S^n$. Note that since $S^n \times S^n$$S^n \times S^n$ 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 $\nu: M \to BO$$\nu: M \to BO$. We consider an $r$$r$-factorization

$\displaystyle M \stackrel{\nu_r}{\to}B_r \stackrel{p_r}{\to}BO$ $$\displaystyle M \stackrel{\nu_r}{\to}B_r \stackrel{p_r}{\to}BO$$

of $\nu$$\nu$ [Spanier1981], page 440. This means that $p_r: B_r \to BO$$p_r: B_r \to BO$ is a fibration, where $B_r$$B_r$ is a $CW$$CW$-complex, the map $p_r$$p_r$ is $r$$r$-connected, i.e. the homotopy groups of the fibre vanish in degree $\ge r$$\ge r$, and the map $\bar \nu_r$$\bar \nu_r$ is an $r$$r$-equivalence.

Definition 2.1. The fibre homotopy type of the fibration $p_{k+1}: B_{k+1} \to BO$$p_{k+1}: B_{k+1} \to BO$ is an invariant of the map $\nu$$\nu$ and is called the normal $k$$k$-type of $M$$M$ denoted $p^k(M):B^{k}(M)\to BO$$p^k(M):B^{k}(M)\to BO$ [Kreck1999].

For example the normal $0$$0$-type of an oriented manifold is the universal covering

$\displaystyle BSO \to BO,$ $$\displaystyle BSO \to BO,$$

whereas the normal $0$$0$-type of a non-orientable manifold is the identity map

$\displaystyle BO \to BO.$ $$\displaystyle BO \to BO.$$

The normal $1$$1$-type of a simply connected manifold $M$$M$ is the fibration

$\displaystyle BSpin \to BSO$ $$\displaystyle BSpin \to BSO$$

,if $M$$M$ admits a $Spin$$Spin$-structure (if and only if the Stiefel-Whintey class $w_2(M)$$w_2(M)$ vanishes) and the fibration

$\displaystyle BSO \to BO,$ $$\displaystyle BSO \to BO,$$

if $M$$M$ does not admit a $Spin$$Spin$ structure. More generally, the normal $1$$1$-type of a Spin-manifold $M$$M$ is

$\displaystyle p: K(\pi_1(M),1) \times BSpin \to BO,$ $$\displaystyle p: K(\pi_1(M),1) \times BSpin \to BO,$$

where $p$$p$ is the composition of the projection to $BSpin$$BSpin$ and the projection $BSpin \to BO$$BSpin \to BO$. The normal $1$$1$-type of a manifold $M$$M$ such that the universal covering $\tilde M$$\tilde M$ does not admit a $Spin$$Spin$-structure, is

$\displaystyle p: K(\pi_1(M),1) \times BS= \to BO,$ $$\displaystyle p: K(\pi_1(M),1) \times BS= \to BO,$$

where $p$$p$ is the composition of the projection to $BSO$$BSO$ and the projection $BSO \to BO$$BSO \to BO$. The case where $\tilde M$$\tilde M$ admits a $Spin$$Spin$-structure but $M$$M$ doesn't, is treated in Stable classification of 4-manifolds.

If $B^k \to BO$$B^k \to BO$

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

$-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 $N$$N$ of dimension $2n$$2n$ are called stably diffeomorphic if there is an integer $r$$r$ such that $M \sharp_r S^n \times S^n$$M \sharp_r S^n \times S^n$ is diffeomorphic to $N \sharp_r S^n \times S^n$$N \sharp_r S^n \times S^n$. By $\sharp_r S^n \times S^n$$\sharp_r S^n \times S^n$ we mean the connected sum with $r$$r$ copies of $S^n \times S^n$$S^n \times S^n$. Note that since $S^n \times S^n$$S^n \times S^n$ 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 $\nu: M \to BO$$\nu: M \to BO$. We consider an $r$$r$-factorization

$\displaystyle M \stackrel{\nu_r}{\to}B_r \stackrel{p_r}{\to}BO$ $$\displaystyle M \stackrel{\nu_r}{\to}B_r \stackrel{p_r}{\to}BO$$

of $\nu$$\nu$ [Spanier1981], page 440. This means that $p_r: B_r \to BO$$p_r: B_r \to BO$ is a fibration, where $B_r$$B_r$ is a $CW$$CW$-complex, the map $p_r$$p_r$ is $r$$r$-connected, i.e. the homotopy groups of the fibre vanish in degree $\ge r$$\ge r$, and the map $\bar \nu_r$$\bar \nu_r$ is an $r$$r$-equivalence.

Definition 2.1. The fibre homotopy type of the fibration $p_{k+1}: B_{k+1} \to BO$$p_{k+1}: B_{k+1} \to BO$ is an invariant of the map $\nu$$\nu$ and is called the normal $k$$k$-type of $M$$M$ denoted $p^k(M):B^{k}(M)\to BO$$p^k(M):B^{k}(M)\to BO$ [Kreck1999].

For example the normal $0$$0$-type of an oriented manifold is the universal covering

$\displaystyle BSO \to BO,$ $$\displaystyle BSO \to BO,$$

whereas the normal $0$$0$-type of a non-orientable manifold is the identity map

$\displaystyle BO \to BO.$ $$\displaystyle BO \to BO.$$

The normal $1$$1$-type of a simply connected manifold $M$$M$ is the fibration

$\displaystyle BSpin \to BSO$ $$\displaystyle BSpin \to BSO$$

,if $M$$M$ admits a $Spin$$Spin$-structure (if and only if the Stiefel-Whintey class $w_2(M)$$w_2(M)$ vanishes) and the fibration

$\displaystyle BSO \to BO,$ $$\displaystyle BSO \to BO,$$

if $M$$M$ does not admit a $Spin$$Spin$ structure. More generally, the normal $1$$1$-type of a Spin-manifold $M$$M$ is

$\displaystyle p: K(\pi_1(M),1) \times BSpin \to BO,$ $$\displaystyle p: K(\pi_1(M),1) \times BSpin \to BO,$$

where $p$$p$ is the composition of the projection to $BSpin$$BSpin$ and the projection $BSpin \to BO$$BSpin \to BO$. The normal $1$$1$-type of a manifold $M$$M$ such that the universal covering $\tilde M$$\tilde M$ does not admit a $Spin$$Spin$-structure, is

$\displaystyle p: K(\pi_1(M),1) \times BS= \to BO,$ $$\displaystyle p: K(\pi_1(M),1) \times BS= \to BO,$$

where $p$$p$ is the composition of the projection to $BSO$$BSO$ and the projection $BSO \to BO$$BSO \to BO$. The case where $\tilde M$$\tilde M$ admits a $Spin$$Spin$-structure but $M$$M$ doesn't, is treated in Stable classification of 4-manifolds.

If $B^k \to BO$$B^k \to BO$

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

$-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 $N$$N$ of dimension $2n$$2n$ are called stably diffeomorphic if there is an integer $r$$r$ such that $M \sharp_r S^n \times S^n$$M \sharp_r S^n \times S^n$ is diffeomorphic to $N \sharp_r S^n \times S^n$$N \sharp_r S^n \times S^n$. By $\sharp_r S^n \times S^n$$\sharp_r S^n \times S^n$ we mean the connected sum with $r$$r$ copies of $S^n \times S^n$$S^n \times S^n$. Note that since $S^n \times S^n$$S^n \times S^n$ 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 $\nu: M \to BO$$\nu: M \to BO$. We consider an $r$$r$-factorization

$\displaystyle M \stackrel{\nu_r}{\to}B_r \stackrel{p_r}{\to}BO$ $$\displaystyle M \stackrel{\nu_r}{\to}B_r \stackrel{p_r}{\to}BO$$

of $\nu$$\nu$ [Spanier1981], page 440. This means that $p_r: B_r \to BO$$p_r: B_r \to BO$ is a fibration, where $B_r$$B_r$ is a $CW$$CW$-complex, the map $p_r$$p_r$ is $r$$r$-connected, i.e. the homotopy groups of the fibre vanish in degree $\ge r$$\ge r$, and the map $\bar \nu_r$$\bar \nu_r$ is an $r$$r$-equivalence.

Definition 2.1. The fibre homotopy type of the fibration $p_{k+1}: B_{k+1} \to BO$$p_{k+1}: B_{k+1} \to BO$ is an invariant of the map $\nu$$\nu$ and is called the normal $k$$k$-type of $M$$M$ denoted $p^k(M):B^{k}(M)\to BO$$p^k(M):B^{k}(M)\to BO$ [Kreck1999].

For example the normal $0$$0$-type of an oriented manifold is the universal covering

$\displaystyle BSO \to BO,$ $$\displaystyle BSO \to BO,$$

whereas the normal $0$$0$-type of a non-orientable manifold is the identity map

$\displaystyle BO \to BO.$ $$\displaystyle BO \to BO.$$

The normal $1$$1$-type of a simply connected manifold $M$$M$ is the fibration

$\displaystyle BSpin \to BSO$ $$\displaystyle BSpin \to BSO$$

,if $M$$M$ admits a $Spin$$Spin$-structure (if and only if the Stiefel-Whintey class $w_2(M)$$w_2(M)$ vanishes) and the fibration

$\displaystyle BSO \to BO,$ $$\displaystyle BSO \to BO,$$

if $M$$M$ does not admit a $Spin$$Spin$ structure. More generally, the normal $1$$1$-type of a Spin-manifold $M$$M$ is

$\displaystyle p: K(\pi_1(M),1) \times BSpin \to BO,$ $$\displaystyle p: K(\pi_1(M),1) \times BSpin \to BO,$$

where $p$$p$ is the composition of the projection to $BSpin$$BSpin$ and the projection $BSpin \to BO$$BSpin \to BO$. The normal $1$$1$-type of a manifold $M$$M$ such that the universal covering $\tilde M$$\tilde M$ does not admit a $Spin$$Spin$-structure, is

$\displaystyle p: K(\pi_1(M),1) \times BS= \to BO,$ $$\displaystyle p: K(\pi_1(M),1) \times BS= \to BO,$$

where $p$$p$ is the composition of the projection to $BSO$$BSO$ and the projection $BSO \to BO$$BSO \to BO$. The case where $\tilde M$$\tilde M$ admits a $Spin$$Spin$-structure but $M$$M$ doesn't, is treated in Stable classification of 4-manifolds.

If $B^k \to BO$$B^k \to BO$

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