Outline for lectures on string topology for stacks
Overview: the first lecture will introduce the notion of geometric
and topological stacks and the construction of the bivariant theory
for topological stacks. The second lecture will deal with algebraic
structures on (co)homology of stacks focusing on string topology. In
particular we will develop the required notion of mapping stacks,
orientation for stacks and cover some examples (including some
specific to stacks such as inertia stacks).
Detailed plan of the lectures:
First lecture:
I) About Topological and Differentiable stacks:
A) Definitions and Examples: introduce the notion of differentiable
stacks (or geometric stacks) and give classical examples such as the
classifying spaces of groups viewed as a stack, orbifolds. Explain the
relationship with groupoids.
B) Vector bundles on stacks: explain how to generalize the notion of
vector bundles to stacks. The issue with tangent bundles shall be
mentioned.
II) Algebraic topology for (topological) stacks:
Introduce the notion of topological stacks focusing on why they are
a good notion for stacks with which to algebraic topology. In
particular, explain why they have a well-defined homotopy type,
singular (co)homology theories. Possibly, one shall recall the link
with equivariant (co)homology.
III) Bivariant Theory for topological stacks:
A) Explain what is a bivariant theory: the categorical setup and
axioms. Explain the slight generalization needed in the case of
stacks.
B) Gysin maps associated to bivariant theory. Describe how one can
construct Gysin maps in (co)homology and explain what kind of
functoriality they get.
C) The bivariant theory for stacks: Explain the definition,
construction and link with singular homology. The examples of Gysin
maps in equivariant homology shall be given if time permits.
References for part I are String Topology for Stacks, Parts 1,2,3 as
well as Behrang Noohi's Introduction to Topological Stacks or Behrend
and Xu's paper on differentiable stacks and Moerdijk and Mrunc's book
on Lie groupoids. Additional references for part II are Behrend's
paper on cohomology for stacks (ICTP lectures notes) and Noohi's paper
on homotopy type for paratopological stacks. Fulton and MacPherson's
book "Categorical framework for the study of singular spaces", memoir
of the AMS, is the important reference for bivariant theory.
Second lecture:
I) Orientation for stacks:
Introduce the notion of nns map, the link with orientation and give
the key example.
II) Mapping stacks:
Definition of the mapping stack as a stack over Topological spaces and
explain the condition under which it is a topological stack. Introduce
the notion of Hurewicz stack and explain why it solves the problem of
Map(-,Y) not commuting with pushouts. Give a groupoid presentation for
loop stacks.
III) Applications and Examples:
A) Explain the construction of the loop product first and maybe its
instance for ghost loops and or sphere product...
B) Explain PoincarĂ© duality and intersection pairing for orbifolds.
Introduce the inertia stacks as the ghost loop stack. Explain the
intersection pairing and gave its relationship with Chen-Ruan
cup-product.
C) Explain the others string topology operations for stacks.
D) Compute the example of the Frobenius structure for classifying
stack of compact Lie groups.