Presentazione LaTeX - MatAppstaff.matapp.unimib.it/~ferrario/var/sem-2013-07-03.pdf · D.L....
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Presentazione LaTeX
D.L. Ferrario
2013-07-03U5-2109
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.. WYSIWYM1 Introduction
A necessary and sufficient condition will be given for a nontrivial homolo-gy class of a simply connected 4-manifold to be represented by a simple,topologically locally flat embedding of a compact Riemann surface.
2 Splittings of Hermitian Modules
We begin with an algebraic result.
Theorem 1. The following is a commutative diagram of pointed hermitian
modules.
(M,h, z)
π0
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α
∼=))S
S
S
S
S
S
S
S
S
S
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S
S
S
π1// (M1, h1, 0)
π1d
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α1
∼=))R
R
R
R
R
R
R
R
R
R
R
R
R
R
(M ′, h′, z′)⊕H(Λk)
π0
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π1// (M ′
1, h′
1, 0)⊕H(Λk
1)
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(M0, h0, z0)α0
∼=))S
S
S
S
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π0d// (Md, hd, 0)
αd
∼=))R
R
R
R
R
R
R
R
R
R
R
R
R
R
(M ′
0, h′
0, z′
0)⊕H(Λk
0)
β′
0⊕id∼=
��
π0d// (M ′
d, h′
d, 0)⊕H(Λkd)
β′
d⊕id∼=
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(M0, h0, z0)β0
∼=))S
S
S
S
S
S
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π0d// (Md, hd, 0)
βd
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R
R
R
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R
R
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R
R
R
R
R
R
(L, λ, x)⊕H(Λk0)
π0d// (Ld, λd, 0)⊕H(Λk
d)
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.. (WYSIWYM)\section{Introduction}
A necessary and sufficient condition will be given for a nontrivial homology
class of a simply connected \text{4-manifold} to be represented by a simple,
topologically locally flat embedding of a compact Riemann surface.
\section{Splittings of Hermitian Modules}
We begin with an algebraic result.
\begin{thm}
The following is a commutative diagram of pointed hermitian modules.\\
%\CompileMatrices
\[\xymatrix{
(M,h,z) \ar[dd]^{\pi_0} \ar[dr]^\alpha_\cong \ar[rr]^{\pi_1}
&& (M_1,h_1,0) \ar’[d]^-{\pi_{1d}}[dd] \ar[dr]^{\alpha_1}_\cong
\\
& (M’,h’,z’)\oplus H(\Lambda^k) \ar[dd]^<(.25){\pi_0} \ar[rr]^<(.25){\pi_1}
&& (M’_1,h’_1,0)\oplus H(\Lambda_1^k) \ar[dd]^{\pi_{1d}}
\\
(M_0,h_0,z_0) \ar@{=}[dd] \ar[dr]^{\alpha_0}_\cong \ar’[r]^<(.6){\pi_{0d}}[rr]
&& (M_d,h_d,0) \ar@{=}’[d][dd] \ar[dr]^{\alpha_d}_\cong
\\
& (M’_0,h’_0,z’_0)\oplus H(\Lambda_0^k) \ar[dd]^<(.25){\beta’_0\oplus\text{id}}_<
\ar[rr]^<(.25){\pi_{0d}}
&& (M’_d,h’_d,0)\oplus H(\Lambda_d^k) \ar[dd]^{\beta’_d\oplus\text{id}}_\cong
\\
(M_0,h_0,z_0) \ar[dr]^{\beta_0}_\cong \ar’[r]^<(.6){\pi_{0d}}[rr]
&& (M_d,h_d,0) \ar[dr]^{\beta_d}_\cong
\\
& (L,\lambda,x)\oplus H(\Lambda_0^k) \ar[rr]^{\pi_{0d}}
&& (L_d,\lambda_d,0)\oplus H(\Lambda_d^k)
}\]
\end{thm}
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b a c k n e x t h o m e
11/19/1863
Gettysburg Cemetery Dedication
Abraham Lincoln
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Fine
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