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I am interested in singularity theory by topology. I want to understand following results.

f is a smooth map of a closed surface M which has only fold points and cusps as its singularities. Suppose that a closed curve c in M intersects a singular set S(f) transversely at a finite number of points.

Then the number of intersection points is odd if and only if c is orientation reversing:

i.e., if and only if {w_1(M), [c]}= 1, where w_1(M) in H^1(M; Z_2) is the first Stiefel-Whitney class of M, [c] in H_1(M;Z_2) is the Z_2-homology class represented by c, and {,} is the Kronecker product. H^1 is first cohomology and H_1 is first homology and Z_2 is order 2 cyclic group.

Above statement is Thom's result which states that the Poincare dual to the Z_2-homology class represented by S(f) coincides with w_1(M).

Question

How Thom's result is used for above statement? I want to know in detail. However, I do know little characteristic classes.

Thank you for your considerations.

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1 Answer 1

up vote 3 down vote accepted

Let me attempt to answer your question as I understand it.

Let $x\in H^1(M)$ be the Poincaré dual of $[c]\in H_1(M)$ (all (co)homology groups are with $\mathbb{Z}_2$ coefficients). The result of Thom you state is that $w_1(M)\in H^1(M)$ is the Poincaré dual of $[S(f)]\in H_1(M)$.

It is well known that the cup product in cohomology coincides under Poincaré duality with the intersection of transverse representatives of homology classes. It follows that the parity of the number of intersection points of $S(f)$ with $c$ equals the Kronecker product $\langle w_1(M)\cup x,[M]\rangle\in\mathbb{Z}_2$.

Now use the identity $\langle \alpha\cup \beta,\gamma \rangle = \langle \alpha, \beta \cap \gamma\rangle$ relating cup, cap and Kronecker products (see any good Algebraic Topology book, eg Switzer) to conclude that $$\langle w_1(M)\cup x,[M]\rangle = \langle w_1(M),x\cap [M]\rangle = \langle w_1(M),c\rangle$$ as required.

Added in response to OP's comment: One can think of the first Stiefel-Whitney class $$w_1(M)\in H^1(M;\mathbb{Z}_2)\cong\mathrm{Hom}(H_1(M),\mathbb{Z}_2)\cong\mathrm{Hom}(\pi_1(M),\mathbb{Z}_2)$$ as the homomorphism which assigns the value $1$ (resp. $0$) to a loop in $M$ if it is orientation reversing (resp. orientation preserving). I don't know a written reference for this off the top of my head, but I'm sure there are many.

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Very thanks! I have realized much. In addition, How we understand for orientation reversing? Do we use the nature of the characteristic class? –  muta yasushi Jul 22 '11 at 10:33
    
Nice answer!Thanks. –  muta yasushi Jul 22 '11 at 11:42
    
The thing with the loop is simply that you realize $w_1$ as a "principal $\mathbb{Z}_2$-bundle with connection over $M$". The homomorphism is its holonomy, which is trivial if and only if the bundle is trivial. –  Konrad Waldorf Jul 22 '11 at 15:29
    
Thanks!I will study the characteristic class in view point of the connection. Above bundle is trivial if and only if the loop is orientation no-reversing? –  muta yasushi Jul 23 '11 at 10:07

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