Geometry & Topology

Homology surgery and invariants of 3–manifolds

Stavros Garoufalidis and Jerome Levine

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We introduce a homology surgery problem in dimension 3 which has the property that the vanishing of its algebraic obstruction leads to a canonical class of π–algebraically-split links in 3–manifolds with fundamental group π. Using this class of links, we define a theory of finite type invariants of 3–manifolds in such a way that invariants of degree 0 are precisely those of conventional algebraic topology and surgery theory. When finite type invariants are reformulated in terms of clovers, we deduce upper bounds for the number of invariants in terms of π–decorated trivalent graphs. We also consider an associated notion of surgery equivalence of π–algebraically split links and prove a classification theorem using a generalization of Milnor’s μ̄–invariants to this class of links.

Article information

Geom. Topol., Volume 5, Number 2 (2001), 551-578.

Received: 31 May 2000
Revised: 2 May 2001
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx]
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

homology surgery finite type invariants 3–manifolds clovers


Garoufalidis, Stavros; Levine, Jerome. Homology surgery and invariants of 3–manifolds. Geom. Topol. 5 (2001), no. 2, 551--578. doi:10.2140/gt.2001.5.551.

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