- #1
Bacle
- 662
- 1
Hi, I am trying to show that for M a 4-manifold,
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.?
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.?