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A Symplectic Instanton Homology via Traceless Character Varieties.

Record Type:	Language materials, manuscript : Monograph/item
Title/Author:	A Symplectic Instanton Homology via Traceless Character Varieties./
Author:	Horton, Henry T.
Description:	1 online resource (216 pages)
Notes:	Source: Dissertation Abstracts International, Volume: 79-01(E), Section: B.
Contained By:	Dissertation Abstracts International79-01B(E).
Subject:	Mathematics. -
Online resource:	click for full text (PQDT)
ISBN:	9780355146325

A Symplectic Instanton Homology via Traceless Character Varieties.
Horton, Henry T.

A Symplectic Instanton Homology via Traceless Character Varieties. - 1 online resource (216 pages)

Source: Dissertation Abstracts International, Volume: 79-01(E), Section: B.

Thesis (Ph.D.)

Includes bibliographical references

Since its inception, Floer homology has been an important tool in low-dimensional topology. Floer theoretic invariants of 3-manifolds tend to be either gauge theoretic or symplecto-geometric in nature, and there is a general philosophy that each gauge theoretic Floer homology should have a corresponding symplectic Floer homology and vice-versa. In this thesis, we construct a Lagrangian Floer invariant for any closed, oriented 3-manifold Y (called the symplectic instanton homology of Y and denoted SI(Y)) which is conjecturally equivalent to a Floer homology defined using a certain variant of Yang-Mills gauge theory. The crucial ingredient for defining SI( Y) is the use of traceless character varieties in the symplectic setting, which allow us to avoid the debilitating technical hurdles present when one attempts to define a symplectic version of instanton Floer homologies.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2018

Mode of access: World Wide Web

ISBN: 9780355146325Subjects--Topical Terms:

527692
Mathematics.
Index Terms--Genre/Form:

554714
Electronic books.

A Symplectic Instanton Homology via Traceless Character Varieties.
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245 1 2 $a A Symplectic Instanton Homology via Traceless Character Varieties.
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300 $a 1 online resource (216 pages)
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500 $a Source: Dissertation Abstracts International, Volume: 79-01(E), Section: B.
500 $a Advisers: Paul A. Kirk; Dylan P. Thurston.
502 $a Thesis (Ph.D.) $c Indiana University $d 2017.
504 $a Includes bibliographical references
520 $a Since its inception, Floer homology has been an important tool in low-dimensional topology. Floer theoretic invariants of 3-manifolds tend to be either gauge theoretic or symplecto-geometric in nature, and there is a general philosophy that each gauge theoretic Floer homology should have a corresponding symplectic Floer homology and vice-versa. In this thesis, we construct a Lagrangian Floer invariant for any closed, oriented 3-manifold Y (called the symplectic instanton homology of Y and denoted SI(Y)) which is conjecturally equivalent to a Floer homology defined using a certain variant of Yang-Mills gauge theory. The crucial ingredient for defining SI( Y) is the use of traceless character varieties in the symplectic setting, which allow us to avoid the debilitating technical hurdles present when one attempts to define a symplectic version of instanton Floer homologies.
520 $a Floer theories are also expected to roughly satisfy the axioms of a topological quantum field theory (TQFT), and furthermore Dehn surgeries on knots should induce exact triangles of Floer homologies. Following a strategy used by Ozsvath and Szabo in the context of Heegaard Floer homology, we prove that our theory is functorial with respect to connected 4-dimensional cobordisms, so that cobordisms induce homomorphisms between symplectic instanton homologies. By studying the effect of Dehn surgeries on traceless character varieties, we establish a surgery exact triangle using work of Seidel that relates the geometry of Lefschetz fibrations with exact triangles in Lagrangian Floer theory. We further prove that Dehn surgeries on a link L in a 3-manifold Y induce a spectral sequence of symplectic instanton homologies -- the E2-page is isomorphic to a direct sum of symplectic instanton homologies of all possible combinations of 0- and 1-surgeries on the components of L, and the spectral sequence converges to SI(Y). For the branched double cover Sigma(L) of a link L in S3, we show there is a link surgery spectral sequence whose E 2-page is isomorphic to the reduced Khovanov homology of L and which converges to the symplectic instanton homology of Sigma( L).
533 $a Electronic reproduction. $b Ann Arbor, Mich. : $c ProQuest, $d 2018
538 $a Mode of access: World Wide Web
650 4 $a Mathematics. $3 527692
655 7 $a Electronic books. $2 local $3 554714
690 $a 0405
710 2 $a ProQuest Information and Learning Co. $3 1178819
710 2 $a Indiana University. $b Mathematics. $3 1180679
773 0 $t Dissertation Abstracts International $g 79-01B(E).
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=10606862 $z click for full text (PQDT)