Hi, I am trying to show that for M a 4-manifold,(adsbygoogle = window.adsbygoogle || []).push({});

and [a]_2 a class in H_2(M,Z) , there is always

a surface that represents [a]_2 , i.e., there

exists a surface S , and an embedding i of S into

M , with [ioS]_2 =[a]_2.

(Equiv.: there exists S, and an embedding i of S of M , so that a triangulation of S

induces the class [a]_2)

** What I have **

If M is simply-connected, so that Pi_1(M)=0

(Notation: Pi_1:=Fund. Grp.)

Then, by the Hurewicz Theorem (Hip, Hip Hurewicz!)

Pi_2(M) is actually Isomorphic to H_2(M;Z) , so that

every class in H_2(M;Z) can be represented as an

embedded sphere S^2 (possibly with self-intersections,

which can be smoothed away ).

**BUT** I can't think of what can be done if

M is not simply-connected.

Any Ideas.?

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# For M a 4-mfld., every class in H_2(M;Z) can be represented by a Surface.

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