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The Novikov Covering
Kenneth Blakey
Brown University
June 3, 2021
Prepared While Participating at DIMACS REU 2021
Kenneth Blakey (Brown University) The Novikov Covering June 3, 2021 1 / 17
Introduction
This is a brief overview of the material needed to define the Novikovcovering of the space of paths between Lagrangians. The following can befound in [1], [2] and is followed closely. The author makes no claims oforiginality.
Kenneth Blakey (Brown University) The Novikov Covering June 3, 2021 2 / 17
Outline
1 The Space of Paths Between Lagrangians
2 The Universal Cover of ⌦
3 The �-Equivalence
4 The Novikov Covering of ⌦
Kenneth Blakey (Brown University) The Novikov Covering June 3, 2021 3 / 17
The Space of Paths
Let (L0, L1) be a pair of compact Lagrangian subamanifolds of (M,!).Consider
⌦(L0, L1) = {[0, 1] `�! M : `(0) 2 L0, `(1) 2 L1}. (1)
By specifying a base path `0 2 ⌦(L0, L1) we get the connected component
⌦(L0, L1; `0). (2)
Hence we may assume (L0, L1) are connected.
Kenneth Blakey (Brown University) The Novikov Covering June 3, 2021 4 / 17
Why can we assume connected?
Kenneth Blakey (Brown University) The Novikov Covering June 3, 2021 5 / 17
Choosing an Io actually chooses connected components
of Lo L since
Io G C Lo
I il C L
So D Lo L Io is the space of paths between these
connected components
Action 1-form
We have the action 1-form given by
↵`(Y ) =
Z 1
0!( ˙̀(t),Y (t))dt (3)
for Y 2 T`⌦(L0, L1; `0).
Kenneth Blakey (Brown University) The Novikov Covering June 3, 2021 6 / 17
By viewings tangent vectorsin IL Lo h lo
as equivalence classesof curves we may view
YE Toh Cho L Io as a rector Greed Yfy along d
Universal Cover
Consider set of pairs (`,w) such that ` 2 ⌦(L0, L1; `0) and
[0, 1]2 w�! M (4)
subject to
If we consider w as a map s 7! w(s, ·) then the fiber at ` of the universalcover ⌦(L0, L1; `0) can be represented by the set of path homotopy classesof w relative to its ends s = 0, 1.
Kenneth Blakey (Brown University) The Novikov Covering June 3, 2021 7 / 17
GqWco I do well a I
w Cs o c Lo w s t EL HseG if
The Map w
Let (`,w), (`,w 0) be two such pairs. Then the concatenation
[0, 1]2 w#w 0����! M (5)
induces maps:
Kenneth Blakey (Brown University) The Novikov Covering June 3, 2021 8 / 17
si s n a o.fi do
sina.tn
Some Homomorphisms
Since the symplectic area
I!(c) =
Z
C! (6)
is independent of the homotopy of C we have a homomorphism
Kenneth Blakey (Brown University) The Novikov Covering June 3, 2021 9 / 17
it GA Lo Li Io s R
Some Homomorphisms Cont.
Also we have that C associates a symplectic bundle pair
VC = C ⇤TM,�C =a
c⇤i TLi (7)
where S1 ci�! Li is
Since (VC ,�C ) are independent of the homotopy of C this induces ahomomorphism
Kenneth Blakey (Brown University) The Novikov Covering June 3, 2021 10 / 17
Cics s.ie ci On
it A Lali Io sz
Why are I!, Iµ independent of homotopy?
Kenneth Blakey (Brown University) The Novikov Covering June 3, 2021 11 / 17
T
General fact that if A B Kmanifolds w A B
homotopic smooth mapsde 2 37 closed
fa f's Jagsuse Stoke's thin4 w
the homotopyt
General fact that pullbackof fibrebundles as invariant
under homotopy
use theuniversal property
The �-Equivalence
We have that w#w 0 induces maps c ,C as before.
DefinitionTwo pairs (`,w), (`,w 0) are said to be �-Equivalent if
I!(w#w 0) = 0 = Iµ(w#w 0) (8)
Kenneth Blakey (Brown University) The Novikov Covering June 3, 2021 12 / 17
The Novikov Covering of ⌦
DefinitionThe Novikov covering ⌦̃(L0, L1; `0) is the set of �-Equivalent classes[`,w ].
We note `0 has a natural lift
and so ⌦̃(L0, L1; `0) has a natural base point.
Kenneth Blakey (Brown University) The Novikov Covering June 3, 2021 13 / 17
do do w UT set Io Ct
Deck Transformations of ⌦̃
Let ⇧(L0, L1; `0) denote the deck transformation group of ⌦̃(L0, L1; `0).Then I!, Iµ push down to homomorphisms
Kenneth Blakey (Brown University) The Novikov Covering June 3, 2021 14 / 17
E
TI Lo 4 Io R
TI lo L Io Is a
E g Iw Ec ve g Ive c
It canbeshown 1T Lo L lo is abelian since themapExve
TT Lo L lo RX TI
is an injective gp morphism
Deck Transformations of ⌦̃ Cont.
Kenneth Blakey (Brown University) The Novikov Covering June 3, 2021 15 / 17
what is C Wellit facto t lol
TI ChoiLi Io EP CtGIGO 4 lol
where pa is projection of Novikov covering So
g is some loop in A ti lo up to composition
w hoops satisfying Iw Ive o Then
CI is this class of gas identification
References
[1] D. McDuff and D. Salamon. Introduction to Symplectic Topology.New York: Oxford University Press, 1995.
[2] K. Fukaya [et al.] Lagrangian Intersection Floer Theory. AMS/IPStudies in Advanced Mathematics, 2009.
Kenneth Blakey (Brown University) The Novikov Covering June 3, 2021 16 / 17
Acknowledgements
Work supported by the Rutgers Department of Mathematics, NSF grantDMS-1711070, and the DIMACS REU program.
Kenneth Blakey (Brown University) The Novikov Covering June 3, 2021 17 / 17
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