Stable classification of manifolds

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1 Introduction

Two closed smooth manifolds $M$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}M$ and $N$$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 Lemma 2.1 of 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: M \to BO$$\nu_M: M \to BO$. We consider an $r$$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_M$$\nu_M$ [Spanier1981], page 440. This means that $p_r: B_r \to BO$$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 1.1. The fibre homotopy type of the fibration $p_{k+1}: B_{k+1} \to BO$$p_{k+1}: B_{k+1} \to BO$ is an invariant of $M$$M$ 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].

In particular, the normal $k$$k$-type does not depend on the choice of an embedding.

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

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

There are only two possibilities for the normal $1$$1$-type of a simply connected manifold $M$$M$ depending on whether the second Stiefel-Whitney class vanishes or not: If $w_2(M) =0$$w_2(M) =0$ (M admits a $Spin$$Spin$-structure) its normal $1$$1$-type is the fibration

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

if $w_2(M)\neq 0$$w_2(M)\neq 0$ ($M$$M$ does not admit a $Spin$$Spin$-structure) then it's normal $1$$1$-type is the fibration

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

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 simply connected 4-manifolds.

Definition 1.2. If $p:B \to BO$$p:B \to BO$ is the normal $k$$k$-type of a manifold $M$$M$, the choice of a homotopy class of lifts $\bar \nu_M$$\bar \nu_M$, which is a $k$$k$-equivalence is called a normal $k$$k$-smoothing. We say that two normal $k$$k$-smoothings $(M, \bar \nu_M)$$(M, \bar \nu_M)$ and $(N, \bar \nu_N)$$(N, \bar \nu_N)$ with same normal $k$$k$-type are diffeomorphic, if there is a diffeomorphism $f: M \to N$$f: M \to N$ compatible with the normal $k$$k$-smoothings. The latter means that if we have embedded $N$$N$ into $\mathbb R^{n+N}$$\mathbb R^{n+N}$, we embed $M$$M$ via the composition with $f$$f$, so that $\nu _M = \nu _Nf$$\nu _M = \nu _Nf$. Then we require that $\bar \nu _N f$$\bar \nu _N f$ id fibre homotopic to $\bar \nu _M)$$\bar \nu _M)$.

The group of homotopy classes of fibre homotopy self equivalences $Aut(B))$$Aut(B))$ acts by composition on the normal $k$$k$-smoothings and this action is free and transitive [Kreck1999]. Thus if one fixes a normal $k$$k$-smoothing the composition with elements of $Aut(B)$$Aut(B)$ is a bijection from $Aut(B)$$Aut(B)$ to the different normal $k$$k$-smoothings.

The stable classification of normal (k-1)-smoothings on 2k-dimensional manifolds

Above we defined stable diffeomorphisms. We can define stable diffeomorphisms of normal $(k-1)$$(k-1)$-smoothings of $2k$$2k$-dimensional manifolds $(M,\bar \nu)$$(M,\bar \nu)$ as follows. Suppose that we have two $2k$$2k$-dimensional Manifolds $M$$M$ and $N$$N$ with same normal ${k-1}$${k-1}$-type $p:B \to BO$$p:B \to BO$. We consider $S^k$$S^k$ as the boundary of $D^{k+1}$$D^{k+1}$ and equip it with the restriction of the the unique (up to homotopy) lift of the stable normal bundle to $B$$B$. Using this we obtain a lift of the stable normal bundle of $S^k \times D^{k+1}$$S^k \times D^{k+1}$ to $B$$B$ and consider its restriction to $S^k \times S^k$$S^k \times S^k$. If $\bar \nu_M : M \to B$$\bar \nu_M : M \to B$ is a normal $(k-1)$$(k-1)$-smoothing of $M$$M$ this together with the constructed lift of the stable normal bundle of $S^k \times S^k$$S^k \times S^k$ to $B$$B$ induces a well defined normal $(k-1)$$(k-1)$-smoothing of $M \sharp_r S^k \times S^k$$M \sharp_r S^k \times S^k$ which we call $\bar \nu _{M \sharp_r S^k \times S^k}$$\bar \nu _{M \sharp_r S^k \times S^k}$. We say that two normal $(k-1)$$(k-1)$-smoothings in $B$$B$ are stably diffeomorphic if the stabilized normal $(k-1)$$(k-1)$-smoothings are diffeomorphic.

Theorem 3.1. [Kreck1999] Let $M$$M$ and $N$$N$ be $2k$$2k$-dimensional closed smooth manifolds with same normal $(k-1)$$(k-1)$-type $B$$B$. Then two normal $(k-1)$$(k-1)$smoothings $(M, \bar \nu _M)$$(M, \bar \nu _M)$ and $(N, \bar \nu _N)$$(N, \bar \nu _N)$ are stably diffeomorphic if and only if the bordism classes of $(M, \bar \nu_M)$$(M, \bar \nu_M)$ and $(N,\bar \nu_N)$$(N,\bar \nu_N)$ agree in the $B$$B$-bordism group $\Omega_{2k} (B)$$\Omega_{2k} (B)$ and the Euler characteristics agree: $e(M) = e(N)$$e(M) = e(N)$.

If $M$$M$ and $N$$N$ are compact manifolds with boundary and $f : \partial M \to \partial N$$f : \partial M \to \partial N$ is a diffeomorphism compatible wirth the restriction of the normal $(k-1)$$(k-1)$-smoothings to the boundaries, then $f$$f$ extends to a stable diffeomorphism of the normal $(k-1)$$(k-1)$-smoothings if and only if $e(M) = e(N)$$e(M) = e(N)$ and the closed manifold obtained by gluing $M$$M$ to $-N$$-N$ via $f$$f$ together with the normal structure given by $\bar \nu_M$$\bar \nu_M$ and $\bar \nu_N$$\bar \nu_N$ is zero bordant in $\Omega _{2k}(B)$$\Omega _{2k}(B)$. Here by $-N$$-N$ we mean $N$$N$ equipped with the normal $(k-1)$$(k-1)$-smoothing which is given by the restriction to $N \times 1$$N \times 1$ of the obvious normal structure on $N \times [0,1]$$N \times [0,1]$ extending the given structure on $N \times 0$$N \times 0$.

Thus two $2k$$2k$-dimensional closed manifolds $M$$M$ and $N$$N$ are stably diffeomorphic if and only if $e(M) = e(N)$$e(M) = e(N)$, they have the same normal $(k-1)$$(k-1)$-type $(B)$$(B)$ and admit bordant normal $(k-1)$$(k-1)$-smoothings in $B$$B$.

For simply connected closed smooth $4$$4$-manifolds Wall [Wall1964] showed that they are stably diffeomorphic if and only both admit a $Spin$$Spin$-structure or both don't admit a $Spin$$Spin$-structure, the signatures and the Euler characteristics agree (he uses a different formulation in terms of the intersection forms but by the stable classification of unimodular quadratic forms this is equivalent to our conditions). This is a consequence of the Theorem 3.1. Namely under Wall's condition the normal $1$$1$-types agree and the $B$$B$-bordisms groups correspond to $Spin$$Spin$ bordism resp. oriented bordism groups which are detected by the signature. Thus we obtain

Corollary 3.2. [Wall1964] Let $M$$M$ and $N$$N$ be simply connected closed smooth $4$$4$-manifolds, then $M$$M$ and $N$$N$ are stably diffeomorphic if and only if they are both spinnable or both not spinnable and the signature and Euler characteristics agree.

Wall proves his theorem in two steps, the first is an (unstable) classification of simply connected $4$$4$-manifolds up top $h$$h$-cobordism. The second is the proof of a stable $h$$h$-cobordism theorem in dimension $4$$4$. The first part is a very special application of a big theory: Surgery. The proof of Theorem 3.1 is a special case of a modified surgery theory [Kreck1999].

There are versions of the concepts and results of this page for $PL$$PL$-manifolds or topological manifolds. Everything is the same if one replaces the normal bundles by the $PL$$PL$ or topological normal bundles and $BO$$BO$ by the corresponding classifying spaces $BPL$$BPL$ an $BTOP$$BTOP$. Then one replaces the smooth $B$$B$-bordism groups by the corresponding $PL$$PL$ or topological $B$$B$-bordism groups B-Bordism. Then Theorem 3.1. holds for $PL$$PL$ or topological manifolds where we classify up to stable $PL$$PL$-homeomorphisms or stable homeomorphisms.

An unstable classification of some normal (k-1)-smoothings on 2k-dimensional manifolds

If $k =1$$k =1$ the normal $0$$0$-type is the same if and only if the surfaces are both orientable or both non-orientable and one has a better Theorem, namely an unstable classification by the Euler characteristic (in this case the bordism class is determined by the Euler characteristic mod $2$$2$). It is an interesting question, under which conditions one gets an unstable classification in higher dimensions. For manifolds with finite fundamental group one has the following result for $k >2$$k >2$:

Theorem 6.1. [Kreck1999] Let $M$$M$ and $N$$N$ be $2k$$2k$-dimensional compact smooth manifolds with same normal $(k-1)$$(k-1)$-type $B$$B$ and $k >2$$k >2$. Suppose that if the fundamental group is trivial and, if $k$$k$ is even, $M$$M$ is of the form $M' \sharp S^k \times S^k$$M' \sharp S^k \times S^k$, or if the fundamental group is finite and $M$$M$ is of the form $M' \sharp _2(S^k \times S^k)$$M' \sharp _2(S^k \times S^k)$.

Then two normal $(k-1)$$(k-1)$smoothings $(M, \bar \nu _M)$$(M, \bar \nu _M)$ and $(N, \bar \nu _N)$$(N, \bar \nu _N)$ are diffeomorphic extending a diffeomorphism of the boundaries compatible with the normal $(k-1)$$(k-1)$-smoothings, if and only if $e(M) =$$e(M) =$e(N) and the bordism classes of $(M, \bar \nu_M)$$(M, \bar \nu_M)$ and $(N,\bar \nu_N)$$(N,\bar \nu_N)$the closed manifold obtained by gluing $M$$M$ to $-N$$-N$ via $f$$f$ together with the normal structure given by $\bar \nu_M$$\bar \nu_M$ and $\bar \nu_N$$\bar \nu_N$ is zero bordant in $\Omega _{2k}(B)$$\Omega _{2k}(B)$.

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
• [Wall1964] C. T. C. Wall, Diffeomorphisms of $4$$4$-manifolds, J. London Math. Soc. 39 (1964), 131–140. MR0163323 (29 #626) Zbl 0121.18101