Atlas: Non-stability of smooth 6-manifolds with $\pi_1=\mathbf{Z}^3$ by Alexei V. Zhubr

Atlas home || Conferences | Abstracts | about Atlas

Topology and Dynamics: Rokhlin Memorial
August 19-25, 1999
Steklov Institute of Mathematics at St. Petersburg
St. Petersburg, Russia

Organizers
N. Netsvetaev, A. Vershik, O. Viro

View Abstracts
Conference Homepage

Non-stability of smooth 6-manifolds with p₁=Z³
by
Alexei V. Zhubr
Syktyvkar State University

<>We say that smooth 2n -manifolds M and M₁ are stably diffeomorphic, if their connected sums with some number of copies of Sⁿ×Sⁿ (the same for both manifolds) are diffeomorphic:

M # rSⁿ×Sⁿ ≈ M₁ # rSⁿ×Sⁿ ;

we denote this by M\mathrel s
( ≈ )

M₁. We say that a manifold M is stable, if from M\mathrel s
( ≈ )

M₁ it follows M ≈ M₁. The stability property for closed simply connected 6 -manifolds had been first proved in []; later this theorem was extended (using a different approach) to all (not necessarily closed) simply connected (4k+2) -manifolds [] (in dimensions 4k there is an evident algebraic obstruction to stability - namely, the non -stability of quadratic forms). Here we describe an example showing non -stability of closed 6 -manifolds in the non -simply connected case. Actually, this example is well known in another context - it is the Casson -Siebenmann example (described in []) of a manifold Q which is homotopy equivalent, however not diffeomorphic, to S³×T³; we just show that Q\mathrel s
( ≈ )

S³×T³. This is an immediate consequence of theorems 1 and 2 below.

Let X be a finite connected CW -space. By (X, spin, 6) -manifold we understand a closed 6 -dimensional smooth spin manifold M and a map f:M→ X which is a 3 -equivalence (that is, f induces isomorphisms p_i(M)→p_i(X) for i=1, 2, and epimorphism for i=3). There is an evident definition of diffeomorphism and oriented diffeomorphism of (X, spin, 6) -manifolds. Moreover, if p₃(X)=0, then we can define also the stable diffeomorphism of (X, spin, 6) -manifolds. Denote (for any X with p₃(X)=0) by \fam\eufmfam M^s(X) the set of oriented stable diffeomorphism classes of (X, spin, 6) -manifolds. We have an evident map

J:\fam\eufmfam M^s(X) → Ø^spin₆(X) .

The following theorem can be proved by easy surgery in dimensions ≤ 2

(a much more general statement is proved in []).

Theorem 1 The map J is a bijection.

The next theorem is an easy exercise on Atiyah -Hirzebruch spectral sequence. Let x₁, x₂ and x₃ be the standard generators of H¹(T³).

Theorem 2 The map Ø^spin₆(T³)→Z³, defined by the formula

(M, f) → 1
48
æ
è \ < p₁, f^*(x₁x₂) , \ < p₁, f^*(x₂x₃) , \ < p₁, f^*(x₁x₃) ö
ø ,

where p₁=p₁(M) is the Pontrjagin class, is an isomorphism.

Now, applying [] (Theorem 5.2), we get the following

Corollary 1 A bordism class (M, f) ∈ Ø^spin₆(T³) does not depend on spin structure on M, and is homotopy invariant.

Combining this with theorem 1, we finally obtain

Corollary 2 The manifolds Q and S³×T³ are stably diffeomorphic.

Remarks

1. Instead of using the homotopy equivalence of Q and S³×T³ and referring to [], we could use the fact that these manifolds are actually homeomorphic (see []), and refer to the topological invariance of (rational) Pontrjagin classes.

2. Using the technique described in [] (Essay III, Appendix C), it is easy to transfer the results of both [] and [] cited above to the TOP category. In this connection it is natural to ask whether this TOP stability also fails for non -simply connected case (the counterexample above vanishes in TOP).

References

1 A. V. Zhubr Theorem on decomposition for simply connected six -dimensional manifolds LOMI seminar notes 36 1973 40-49 English transl. in J. Sov. Math 8 1977 554-561

2 N. Yu. Netsvetaev Diffeomorphism and stable diffeomorphism of simply connected manifolds Algebra i Analiz 2 1990 112-120 English transl. in Leningr. Math. J. 2 1991 313-320

3 R. C. Kirby, L. C. Siebenmann Foundational essays on topological manifolds, smoothings and triangulations Princeton University, Princeton, New Jersey 1977

4 M. Kreck Surgery and duality Annals of mathematics

5 V. A. Rokhlin The Pontrjagin -Hirzebruch class of codimension 2 Izv. AN SSSR (ser.mat.) 30 1966 705-718

Date received: May 21, 1999