In[24], the universal complex of singular fibers was introduced as a refinement of Vassiliev’s universal complex of multi-singularities (see Vassiliev[26], Kazaryan[9]
or Ohmoto [18]), and it was shown that its cohomology classes give invariants of cobordisms of singular maps in the sense of Rim´anyi and Sz˝ucs[20]. In this section, we study the universal complex (with integer coefficients) of singular fibers corresponding to chiral singular fibers and give an interpretation of our main theorem in terms of the theory of universal complex of singular fibers.
We can define the universal complex of chiral singular fibers for proper C1 stable maps of oriented 5–manifolds into4–manifolds by exactly the same procedure as in [24]as follows.
For with 34, letC be the freeZ–module generated by theC0 equivalence classes modulo regular fibers of chiral singular fibers of codimension . Note that rankC3 D3 and rankC4 D14 according to Proposition 6.1. Since there exist no chiral singular fibers of codimension ¤3;4, we put CD0 for¤3;4. Note that forD4, we take the C0 equivalence classes modulo regular fibers of chiral singular
fibers with positive signs as generators, and we consider those with negative signs to be 1 times the corresponding class with positive sign.
The coboundary homomorphismı3W C3!C4 is defined by ı3.G/D X
.F/D4
ŒGWFF
for every generator G of C3, whereŒGWF2Z is the incidence coefficient which can be defined by exactly the same method as forŒIII8WF (seeSection 6). Note that all the other coboundary homomorphisms ı, ¤3, are necessarily trivial.
We call the resulting cochain complex .C; ı/ the universal complex of chiral singular fibersfor proper C1 stable maps of oriented 5–manifolds into 4–manifolds.
Note that its unique cohomology group that makes sense is its third cohomology group, and is nothing but the kernel of the coboundary homomorphismı3.
Then we get the following.
Proposition 8.1 The 3–dimensional cohomology group of the universal complex of chiral singular fibers for properC1 stable maps of oriented 5–manifolds into4– manifolds is an infinite cyclic group generated by theC0 equivalence class modulo regular fibers of III8 type fibers.
Proof Recall that we have exactly three C0 equivalence classes modulo regular fibers of chiral singular fibers of codimension 3, namely, III5, III7 and III8, byProposition 6.1. ByLemma 6.3, the incidence coefficients involvingIII8 are all zero and hence III8 is a cocycle. On the other hand, for the other two equivalence classes of chiral singular fibers, we have, for example,
ŒIII5WIV11¤0; ŒIII5WIV10D0; ŒIII7WIV11D0; ŒIII7WIV10¤0:
Therefore, a linear combination of III5, III7 and III8 is a cocycle if and only if the coefficients of III5 and III7 both vanish. Therefore, the kernel of the coboundary homomorphism ı3 is infinite cyclic and is generated by III8. This completes the proof.
According toProposition 8.1, we can interpret our main theorem (Theorem 5.5) as follows. The3–dimensional cohomology class represented by the cocycle III8 of the universal complex of chiral singular fibers for proper C1 stable maps of oriented 5–manifolds into 4–manifolds gives a complete invariant of the oriented cobordism
class of the source4–manifold. In particular, forN DR3, it gives a complete invariant of the oriented bordism class of a C1 stable map of a closed oriented 4–manifold into R3.
For related discussions, see[24].
We also see that the fiber which satisfies the property as in Theorem 5.5 should necessarily be the fiber of type III8. This explains the reason why the III8 type fiber appeared in the modulo two Euler characteristic formula in[24](seeCorollary 5.6of the present paper).
Remark 8.2 If we can realize the proof of our main theorem as mentioned inRemark 7.6for arbitrary stable maps of closed oriented null-cobordant 4–manifolds into3– manifolds, then that would imply thatIII8 is a cocycle of the universal complex (see [24, Section 12.2]). That is, it might be possible to prove that III8 is a cocycle without even classifying the singular fibers.
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Faculty of Mathematics, Kyushu University Hakozaki, Fukuoka 812-8581, Japan
Department of Mathematics, Hokkaido University Sapporo 060-0810, Japan
saeki@math.kyushu-u.ac.jp, taku_chan@math.sci.hokudai.ac.jp http://www.math.kyushu-u.ac.jp/~saeki/
Proposed: Yasha Eliashberg Received: 8 October 2004
Seconded: Shigeyuki Morita, Rob Kirby Accepted: 12 January 2006