The cohomology ring of the orientable Seifert manifolds, II

Bryden, John; Zvengrowski, Peter

doi:10.1016/s0166-8641(02)00061-5

Cited by 14 publications

(17 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…It is instructive to compare this resolution with the case |G| = ∞, treated in [6,7,9], for which C j = 0, j ≥ 4. Here the finiteness of G is reflected by the new class σ 4 0 ∈ C 4 whose boundary generates Ker(d 3 ), which is no longer {0}.…”

Section: Seifert Structurementioning

confidence: 99%

Remarks on the cohomology of finite fundamental groups of 3–manifolds

Tomoda¹,

Zvengrowski²

2008

Geometry and Topology Monographs

View full text Add to dashboard Cite

Remarks on the cohomology of finite fundamental groups of 3-manifolds SATOSHI TOMODA PETER ZVENGROWSKI Computations based on explicit 4-periodic resolutions are given for the cohomology of the finite groups G known to act freely on S 3 , as well as the cohomology rings of the associated 3-manifolds (spherical space forms) M = S 3 /G. Chain approximations to the diagonal are constructed, and explicit contracting homotopies also constructed for the cases G is a generalized quaternion group, the binary tetrahedral group, or the binary octahedral group. Some applications are briefly discussed.Satoshi Tomoda and Peter Zvengrowski particular, construction of a chain approximation to the diagonal (which we simply call a "diagonal") suffices to determine the ring structure with arbitrary coefficients.Most Seifert manifolds have infinite fundamental group: any Seifert manifold with orbit surface not S 2 or RP 2 , or having at least four singular fibres, will have G infinite. Nevertheless, the relatively small class of Seifert manifolds having finite fundamental group is extremely important, indeed all known 3-manifolds with finite fundamental group are Seifert, and pending recent work of Perelman [36], Kleiner-Lott [29], Morgan-Tian [33] and Cao-Zhu [10], it seems very likely there are no others. These Seifert manifolds all arise from free orthogonal actions of G on S 3 , and the resulting manifolds M = S 3 /G, known as spherical space forms, have been of great interest to differential geometers since the nineteenth century; see Clifford [12], Killing [27], Klein [28] and the book of Wolf [46]. In this paper we attempt, in a certain sense, to complete the aforementioned programme of Zieschang and his colleagues to the orientable Seifert manifolds with finite fundamental group, ie to the spherical space forms. (The nonorientable case has little interest here, since a theorem of D B A Epstein [15] asserts that Z 2 is the only finite group that can be the fundamental group of a nonorientable 3-manifold.)It is important to note that, in contrast to the case where G is infinite, M is no longer aspherical. Thus, H * (M) and H * (G) are no longer isomorphic; indeed by a classical theorem (see ), H * (G) is now 4-periodic. The collection of all finite groups acting freely and orthogonally on S 3 is clearly listed by Milnor [32], based on earlier work of Hopf [26] and Seifert-Threlfall [39]. Ideally, for each such group, one would like to have a 4-periodic resolution C together with a contracting homotopy s and a diagonal ∆. For example, for the cyclic group C n , this is done (here C is 2-periodic) in [11].In Section 2, we give some preliminaries about the groups involved and about the cohomology of groups, also setting up necessary definitions and notation. The generalized quaternion groups Q 4n are considered in Section 3. In this case, a 4periodic resolution was given in [11], together with the somewhat cryptic statement "the verification that the homology groups are trivial involves some computations which will be omitted." ...

show abstract

Section: Seifert Structurementioning

confidence: 99%

Remarks on the cohomology of finite fundamental groups of 3–manifolds

Tomoda¹,

Zvengrowski²

2008

Geometry and Topology Monographs

View full text Add to dashboard Cite

show abstract

“…In the next section we will show, by naive calculations, that the rational cohomology ring gives more severe restrictions. The cohomology rings with Z p r coefficients of Seifert fibered spaces have been calculated [1,4]. The techniques used in these calculations are quite intricate (involving a diagonal approximation of an equivariant chain complex of the universal cover).…”

Section: Basic Invariants Of Homology Cobordismmentioning

confidence: 99%

Homology cobordism and Seifert fibered $3$-manifolds

Cochran

Tanner

2014

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

It is known that every closed oriented 3-manifold is homology cobordant to a hyperbolic 3-manifold. By contrast we show that many homology cobordism classes contain no Seifert fibered 3-manifold. This is accomplished by determining the isomorphism type of the rational cohomology ring of all Seifert fibered 3-manifolds with no 2-torsion in their first homology. Then we exhibit families of examples of 3-manifolds (obtained by surgery on links), with fixed linking form and cohomology ring, that are not homology cobordant to any Seifert fibered space, These are shown to represent distinct homology cobordism classes using higher Massey products and Milnor's µinvariants for links. 2; while • if β 1 (M ) is even, the rational cohomology ring of M is isomorphic to that of # β 1 (M ) (S 1 ×S 2 ).Corollary 1.2. Under the hypotheses of Theorem 1.1, if β 1 (M ) is odd and at least 3 then for any non-zero α ∈ H 1 (M ; Q) there exists β ∈ H 1 (M ; Q) such that α ∪ β = 0. If β 1 (M ) is even then, for any α, β ∈ H 1 (M ; Q), α ∪ β = 0.

show abstract

“…This was done, for example, for the orientable Seifert manifolds with infinite fundamental group in work of Bryden, Hayat-Legrand, Zieschang, and Zwengrowski (cf. [3,4]), and a little later for Seifert manifolds with infinite fundamental group in the work of Tomoda and Zvengrowski [10]. In the latter case the manifolds are not K(π, 1) spaces, nevertheless the group cohomology (which is 4-periodic) is still closely related to the manifold cohomology.…”

Section: Introductionmentioning

confidence: 99%