Integer graded instanton homology groups for homology three spheres
Integer graded instanton homology groups for homology three spheres
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同调三球的整数分级瞬子同调群
DOI：
10.1016/0040-9383(92)90053-k
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发表时间：
1992
期刊：
影响因子：
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通讯作者：
R. Stern
中科院分区：
文献类型：
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作者：
R. Fintushel;R. Stern
设Z是一个有向的积分同调3球。在本文中，我们将一个离散集Ax C R与C联系起来，并对每个p E [wx= Iw\Ax]与一个自然Z分级的阿贝尔群S ',)(C)联系起来。此外，fx (Y $ ' ' (Z))= A (C)，其中n (E)为Casson不变量（见[1]）。特别地，4x (9$ " (Z)) exp (Z)(mod 2)，其中p (Z)是Z的Kervaire-Milnor-Rochlin不变量。这些群只在p所在的Iwr区间内依赖于n；即如果区间[pO， pl] c lWr，则为分级组Y？”(C) = Y“美元”(C)。此外，集合AZ在平移r~ r+ 1和9$ ' + " (Z)= 9$ ' \,(Z)下是不变的。群9 t′(C)可以看作是在14-J中引入的花同源群9、（YE）、+ E&的整数提升。在以下意义上。设W (C)表示a: nl (C)+ SU(2)的共轭类空间，设8表示平凡表示的类。为简单起见，假设所有非平凡的表示都是规则的，即H ' (Z; ad (a))= 0对于每个表示a~ 9？*(Z)= gP (Z)\{0}。通过仔细分析c上平凡束或规范等价类空间的无限循环覆盖&z上的chen - simons不变量c: 5 r-+ r的行为，我们在$2中分别将ES (c)和FE (I)关联为一个定义良好的整数P ' (a)，并定义9$ ' (E)= Z {aE44 (; r.)\{0} 11 ' ' (a)= ' I}。设R，(X)，+ E Z8表示Floer链基C41，则，er% ' ~~~ ~(Z) Z R,(E)，对于无Za。在定义边界运算符8 ":& ' $ ' (Z)+ W！$,(Z)与Fleer的[4]相似，并且在$2中显示d ' ' ' d ' ' = 0，我们得到了9$ ' ' (E)的同源群。一般来说，对于* E he， Z,, zS $ ' ！8 (Z) #我(Z)。然而，在$5中，我们构造了一个以It ‘ (C)作为其E ’项并收敛于I，(C)的谱序列。并非所有的UEW (E)都是正则的，因此在第3节中，我们将展示如何通过扰动chen - simons函数来一般定义这些瞬子同调群。然后在第4节中，我们证明了这些群是与扰动无关的，因此是拓扑不变量。
LET Z be an oriented integral homology 3-sphere. In this paper we associate to C a discrete set Ax c R and for each p E [wx= Iw\Ax an abelian group S’,)(C) with a natural Z grading. Furthermore, fx (Y $“(Z))= A (C), where n (E) is Casson’s invariant (see [1)). In particular, 4x (9$“(Z)) E p (Z)(mod 2). where p (Z) is the Kervaire-Milnor-Rochlin invariant of Z. These groups will depend on n only through the interval in Iwr in which p lies; ie if the interval [pO, pl] c lWr, then as graded groups Y?‘(C)= Y’$‘(C). Furthermore, the set AZ is invariant under the translation r~ r+ 1 and 9$‘+“(Z)= 9$‘\,(Z). The groups 9 t’(C) can be viewed as integer lifts of the Floer homology groups 9,(YE),+ E&, introduced in 14-J. in the following sense. Let W (C) denote the space of conjugacy classes of representations a: nl (C)+ SU (2). and let 8 denote the class of the trivial representation. For simplicity assume that all nontrivial representations are regular, ie that H’(Z; ad (a))= 0 for every representation a~ 9?*(Z)= gP (Z)\{O}. By carefully analyzing the behavior of the Chern-Simons invariant c: 5 r-+ R on the infinite cyclic cover &z of the space of gauge equivalence classes of connections Or in the trivial bundle over C, we associate in $2 to each a ES (C) and FE 9I’ea well-defined integer P’(a) and define 9$‘(E)= Z {aE44 (; r.)\{0} 11”‘(a)=‘I}. If we let R,(X),+ E Z8 denote Floer’s chain groups C41, then, er%‘~~~(Z) z R,(E), for no Za. After defining a boundary operator 8”: & ‘$‘(Z)+ W! $,(Z) similar to that of Fleer’s [4] and showing in $2 that d’“‘d”’= 0, we have the resulting homology groups 9$“(E). In general, for* E he, Z,, zS $‘! 8,(Z)# I,(Z). However, in $5 we construct a spectral sequence with It’(C) as its E’term and converging to I,(C).It is not always the case that all UEW (E) are regular, so in $3 we show how to define these instanton homology groups in general by perturbing the Chern-Simons function. Then in $4 we show that these groups are independent of perturbation and are thus topological invariants.