ATMW Algebraic Structures on Manifolds (2016)

(I) Aim of the workshop and syllabus
The ATM workshop aims to understand certain algebraic constructions on free loop space of manifolds. These constructions have algebraic counterpart in Hochschild and cyclic homology of associative algebras. The syllabus of the workshop is broadly divided into four sections:

(A) Preliminaries: We will discuss manifolds, Poincaré duality, cap product. We will also review basic homotopy theory, including fibrations and simplicial objects. We will review Milnor’s construction of classifying spaces and then discuss bar and cobar constructions of algebras and coalgebras respectively.
References: Algebraic Topology by Hatcher, Construction of universal bundles I, II by Milnor,
The geometric realization of a semi-simplicial complex by Milnor, Algebraic Operads by Loday,
Valette.

(B) Hochschild homology: We will discuss Hochschild (co)homology of associative algebras. We will study deformations of algebras leading to Gerstenhaber algebras. This further leads to Batalin-Vilkovisky algebra. The algebra of smooth differential forms on a Poisson manifold provides an example of Gerstenhaber algebra.
References: On the cohomology groups of an associative algebra by Hochschild, On the deformation of rings and algebras I, II by Gerstenhaber, From Poisson algebras to Gerstenhaber algebras by Kosmann-Schwarzbach.

(C) Cyclic homology: We will discuss cyclic sets and circle action leading to cyclic homology of algebras as well as S¹ -equivariant homology of spaces on which the circle acts. We will discuss the connection of cyclic homology of cochains on a manifold with the S¹ -equivariant homology of the free loop space of the manifolds.
References: Cyclic homology by Loday, Free loop space and homology by Loday, Cyclic homology and equivariant homology by Jones, Cyclic homology, derivations and the free loop space by Goodwillie.

(D) Loop spaces: We will describe loop product on the homology of free loop space of a manifold. We will discuss how this homology becomes a BV algebra. The case of surfaces and Goldman bracket will be discussed. We will finish by describing the connection of these with topological conformal field theory.
References: String Topology by Chas, Sullivan, A homotopy theoretic realization of string topology by Cohen, Jones, String topology and cyclic homology by Cohen, Hess, Voronov, Open and closed string field theory interpreted in classical algebraic topology by Sullivan.
 

(II) Time-Table

Topic A (Basic Material)
A1) Simplicial objects and classifying spaces [Goutam Mukherjee]
A2) Bar and cobar constructions [Samik Basu]
A3) Manifolds and Poincaré duality [Amiya Mukherjee]
A4) Fibrations and homotopy theory [Amiya Mukherjee]

Topic B (Hochschild Homology)
B1) Hochschild (co)homology of algebras [Ashis Mandal]
B2, B3) Deformation of algebras (Gerstenhaber and BV algebras) [Anita Naolekar]
B4) Examples of the above (forms on Poisson manifolds etc.) [Mahuya Datta]

Topic C (Cyclic Homology)
C1) Cyclic homology of algebras [Ashis Mandal]
C2, C3) S¹ -equivariant homology, cyclic sets and S ¹ -actions [Debashis Sen]
C4) Relation between cyclic and S¹ -equivariant homology [Debasis Sen]
C5) Relation between Hochschild homology and H _∗ ( LM) [Debasis Sen]

Topic D (Loop Spaces and Field Theory)
D1) Loop product and string bracket [Samik Basu]
D2) Analogous structures on Hochschild homology [Samik Basu]
D3) Example of surfaces [Somnath Basu]
D4, D5) Connections with homological conformal field theory [Somnath Basu]

(III) Report of the talks by individual lecturers

Speaker: Prof. Amiya Mukherjee
I. Manifolds and Poincare Duality Theorem
Homology characterization of orientation, Exact sequence. Cup Product and Cap product. Direct limit group. Poincare duality theorem and its proof for non-compact oriented manifolds.
II. Homotopy Theory and Fibrations.
H-group and H-cogroup. Exact sequences of sets of homotopy classes of maps. Fibrations. Homotopy exact sequence of a Serre fibration.

Speaker: Dr. Anita Naolekar
In these lectures we introduce deformation theory of associative algebras over a field of char zero, with base a commutative unital ring with a fixed augmentation. We show that deformations of an associative algebra with base a local commutative unital ring, is controlled by a differential graded Lie algebra (dgla), via its Maurer-Cartan elements. This dgla arises from the Hochschild cochain complex of the algebra with coefficients in itself.
We also talk about Kontevich’s result on deformation quantization of Poisson manifolds. Later, in the context of deformation theory, we introduce the definitions of Gerstenhaber algebras and BV algebras, and give some examples.

Speaker: Dr. Ashis Mandal
I. On Hochchild (co)homology of algebras:
First we recall some basics on associative algebras, its modules and bimodules. Then we define Hochschild chain complex and deduce the homology for an associative algebra with coefficients in a bimodule. Few basic properties for the homology are discussed with an emphasis on low dimensional homologies. Next we introduce the Hochschild cochain complex and define the Hochschild cohomology for an associative algebra with coefficient in a bimodule. Then we present the derived functor description of Hochschild homology and cohomology by introducing the Hochschild standrad resolution and Bar complex for an associative algebra.
II. Cyclic homology of algebras:
We define cyclic bicomplex and cyclic homology groups of an associative algebra. Then we present the Connes complex for associative algebras and its homology. We derive the b-B bicomplex by recalling the Connes operator B and comparison of homologies in these bicomplexes. Next we discuss about mixed complex, its associated Connes’s bicomplex and Connes’s long exact sequence. For commutative algebras we show the relation between cyclic homology and differential forms. In the sequel we find that the Connes’s operator B is the noncommutative analogue of the the de Rham differential and the cyclic homology is the noncommutative version of de Rham cohomology.

Speaker: Dr. Debasis Sen
We introduced Cyclic Category, Cyclic Sets, and gave some examples. Then we proved that geometric realization of cyclic sets is naturally equipped with a circle action. We showed that Cyclic homology of cyclic set is isomorphic to S¹-equivariant Borel homology of the geometric realization. Next we proved that geometric realization of Cyclic bar construction of a group is homotopy equivalent to the free loop space on the classifying space of the group. In the end we gave a short survey comparing homotopy theory of cyclic sets and spaces with circle action.

Speaker: Prof. Goutam Mukherjee
Algebraic topology may be described as the study of those functors from the category of topological spaces to that of groups which are invariants of homotopy type. From this point of view, the category of topological spaces which are of the homotopy type of a CW complex is equivalent to the category of the category of simplicial sets satisfying Kan extension condition. This simplicialization is an efficient way to construct manifolds combinatorially which permits us to study them up to homotopy. Moreover, this simplicial technique is also useful to construct classifying space of a given a topological group G, which classifies principal G-bundles up to homotopy. This is a particular case of a more general construction known as bar construction. We introduced simplicial objects and described briefly construction of classifying space of a given group.

Speaker: Prof. Mahuya Dutta
Gerstenhaber algebra and BV Algebra structure on Multivector fields. Correspondence between Lie algebroids Gerstenhaber algebras. Poisson manifolds. Correspondence between Poisson bivector field on a manifold and Poisson bracket on the space smooth functions on it. Koszul bracket on the space of 1-forms on a Poisson manifold. Lie algebroid structure on the cotangent bundle of a Poisson manifold. BV algebra structure on graded algebra of differential forms on Poisson manifolds. Correspondence between Lie Algebroids with flat connections and BV Algebras.

Speaker: Dr. Samik Basu
I. Bar and Cobar constructions
We discussed bar constructions for differential graded algebras and cobar constructions for differential graded coalgebras. These were shown to be adjoint using the space of twisting morphisms. A number of examples from topology were discussed during the lecture.
II. Loop product and String Bracket
We constructed the Chas-Sullivan loop product on the homology of the free loop space and made some computations for S¹ and S³. We discussed how this gives rise to a BV algebra structure on the homology of the free loop space. For the S¹ equivariant homology this gives rise to a graded Lie bracket, called the string bracket.
III. Algebraic structures and string topology
We saw how the homology of the free loop space was the Hochschild cohomology of the cochain algebra for a manifold M. It was also indicated how in the case of a manifold the Gerstenhaber algebra structure on the Hochschild cohomology of the cochain algebra refined to a BV algebra structure. In these cases the isomorphism between the homology of the free loop space and the Hochschild cohomology of the cochain algebra becomes an isomorphism of Gerstenhaber algebras over any field. For fields of characteristic 0, this becomes an isomorphism of BV algebras. Among other implications, this means that the string topology operations are homotopy invariant. We further discussed how to view all these isomorphisms from the perspective of stable homotopy theory.

Speaker: Dr. Somnath Basu
I. String topology for surfaces
We discussed the computations associated with the free loop space of surfaces. The free loop space was shown to decompose essentially into a union of covering spaces, indexed by conjugacy classes of the fundamental group. Using this, the generators for the equivariant homology were computed and visualized. We saw that there is also a structure of Lie coalgebra which together turns the string topology (for surfaces) into a Lie bialgebra, due to work of Goldman and Turaev.
II. Topological Quantum Field Theory
After setting up the definitions of the category nCob, monoidal categories and functors between them, we defined what a Topological Quantum Field Theory (TQFT) was. We argues our way to classifying all 2 dimensional TQFT’s by realizing that such a TQFT is given by a commutative Frobenius algebra. We also discussed several examples as well equivalent definitions of a Frobenius algebra.
III. Conformal Field Theory and free loop spaces
We defined what a Conformal Field Theory (CFT) is, starting with a discussion of conformal structures and moduli space of such structures on Riemann surfaces. When we restrict to the genus 0 part of a CFT, we get relations with the operad of framed ltitle disks and little disks. These are known, due to Getzler and Cohen respectively, to govern Gerstenhaber an BV algebras. Thus, the genus 0 part of a CFT will be a BV algebra. However, the homology of the free loop space, although being a BV algebra, is far from being a CFT and relevant recent results towards this were discussed.

Local participants
Debabrata Adak (Jadavpur University), Soham Chanda (CMI), Shuvojit Mandal (Jadavpur University), Aritra Bhowmick (ISI Kolkata), Suvrajit Bhattacharjee (ISI Kolkata)
 

How to reach

SI Kolkata Campus

The Kolkata campus of the Indian Statistical Institute is located in a sprawling 30-acre estate on the Barrackpore Trunk Road (B.T. Road) in the Baranagore suburb of Greater Kolkata. It consists of two approximately equal parts - the office complex and the residential complex, - separated by a public road. This road (Girish Chand Ghosh Street) connects B.T. Road with Gopal Lal Tagore Road, a road which runs along the western boundary of the main campus. The office complex bears door numbers 203, 204, and the residential complex, 205. There is a connection between the two parts of the campus through a subway; to move between the residential and office complexes.
 
How to Reach ISI

The ISI campus lies five kilometers North of the Shyambazar five-point crossing. If you are coming from the airport, drive towards Dunlop Bridge (via Nager Bazar and Chiria More). If you are coming from the downtown area or one of the railway stations, you will have to drive past Shyambazar and Chiria More. As you approach Dunlop Bridge, the boundary wall of the ISI campus will be visible on your left. The residential campus is at the Bus Stop called Bon Hooghly and the office campus is at the Bus Stop called ISI or Statistical. The Bus Stop further to the Statistical Bus Stop is Dunlop Bridge, which is an important landmark. Dunlop Bridge is also a useful way of describing the general destination when you are asking for directions.

Taxi

Kolkata has a huge fleet of Black and Yellow taxis, most of which are Ambassadors. They run on a meter (Electronic and Flag meter). The Electronic meter starts at Rs. 10/- and the Flag meter at Rs. 5/-. The taxi driver is supposed to carry a conversion chart with the validation date clearly marked on it. The best thing to do is ask for a tariff conversion card and pay according to that.

Taxi Fare Conversion Rate:

Electronic Meter: 1.8 times actual meter reading + 2

Flag meter : 3.6 times actual meter reading + 2

Please ensure that the meter is started at the beginning of the journey.

If you are coming from the airport or one of the railway stations, it is better to hire a taxi from the prepaid taxi booth. Mention the destination as Dunlop Bridge at the pre-paid Taxi booth counter. In this case you have to pay the total fare at the taxi booth. The prepaid taxi fare is marginally higher than normal, but it is a surer and safer mode of transport.

Buses

There are a large number of buses and minibuses that pass by the Institute campus. Almost any bus that goes to Dunlop Bridge passes by the ISI gates. The buses that pass by the ISI gates are 3, 32A, 34B/1, 78, 78/1, 78A, 78B, 78C, 201, 214, 230, 234, DN9, DN9/1, 9A, L20, S11, S32 and GL32. The minibuses plying from Howrah to Nimta/Belgharia/Rathtala, from BBD Bag to DunlopBridge, from Babughat to Sodepur/New Barrackpore, and Garia to Belur Math (Via Ultadanga, E M Bypass) go past the ISI gates before reaching Dunlop Bridge. While riding the bus from the city, the major bus stops on your way (from South to North) would be Shyambazar, Chiria More, Sinthi More and Tobin Road. The Bon Hooghly Bus Stop is just after the Ananya Cinema stoppage. This is where you have to get down in order to enter the residential campus at 205 B T Road. The next Bus Stop is ISI / Statistical, which is located just outside the entrance of the main office campus at 203 B.T. Road.