## MAT 566 DIFFERENTIAL TOPOLOGY

## SPRING 2013

**SYLLABUS.**

We are going to cover a subset
of the following topics. The emphasis will be on the ideas and concepts, not on the
proofs. However, some selected important proofs will be given.

**Differential forms:** Differential forms and
differential forms with compact supports.
Poincare' lemmas; degree of a proper map R^n-->;R^n.
Orientation, integration, Stokes' theorem. The Mayer-Vietoris principle
(case of finite good covers): de Rham cohomology is finite dimensional,
Poincare' duality, Kunneth formula,
Leray-Hirsch theorem. The Poincare' dual of a closed oriented
submanifold of an oriented manifold and its compact supports analogue.
Degree of
a proper map of oriented manifolds. The Thom isomorphism for the
compact vertical cohomology
of an orientable R-vector bundle via integration along the fibers; the
Thom class; Thom class and Poincare' duality (e.g. zero section of
oriented vector bundle). Global angular
form-Euler-Thom
class of an oriented rank 2 R-vector bundle. Aspects of the Hodge
theorem for: compact Riemannian,
compact Hermitean, and compact Kahler manifolds: Poincare' and
Kodaira-Serre duality,
Hodge (p,q)-decomposition.
**Vector fields.** The Lie bracket. The Lie
algebra of a Lie group. Flows. The Ehresmann fibration theorem
for proper submersions and applications to hypersurfaces of complex
projective space.
Frobenius theorem. Rudiments of Morse theory and the Lefschetz
hyperplane theorem. Covariant derivatives on the tangent bundle:
torsion, curvature, holonomy. Riemannian metrics: the Levi-Civita
connection, the Riemann
curvature tensor. Gauss-Bonnet for oriented Riemannian surfaces.
**Vector bundles.** Basic operations on vector
bundles. Chern classes of C-vector bundles via projectivization
and via induction.
The splitting principle. Properties. Pontryagin classes of R-vector
bundles.
Connections on vector bundles: curvature
and holonomy; zero curvature and the Frobenius theorem, flat
connections and the holonomy representation.
Chern-Weil: closed differential forms
of a connection, their relation
to the Chern and Pontryagin classes. Metric connections. The
generalized Gauss-Bonnet
formula for an oriented
even rank R-vector bundle with a metric connection. Classification of
vector bundles via
the infinite Grassmannians; universality of Chern classes.
Indentification
of Chern classes with Chern-Weil classes. ``Baby Trinity" on the
trivial complex line bundle
on a compact Riemann surface: C^* (resp. U(1)) representations of the
fundamental group =
flat (resp. unitary) connections = holomorphic line bundles with (resp.
without) a Higgs field.
**G-bundles.** Principal G-bundles, associated
G-bundles. ``Dependence" on G. Classification
for G discrete and relation with covering spaces. Connections,
curvature and holonomy on
principal G-bundles and relation with analogous notions on vector
bundles. Principal bundles as a menas to globalize Cartan's structure
equation.
Reduction of structure. Sphere bundles:
orientability, Euler class and existence of sections,
Euler # and local degrees; re-phrasing for vector bundles: Euler class
Poincare' dual to good zero section
(rank vector bundle=dimension base),
Hopf index theorem; Euler class and zero locus of a transversal section
(any rank).

**TEXTBOOK.**

I am going to
use various sources: [Mi]: Milnor's Morse theory, [Mi-St]:
Milnor-Stasheff's Characteristic classes,
[St]: Steenrod's The topology of fiber bundles, [Bo-Tu]: Bott-Tu's
Differential forms
in algebraic topology, [Wa]: Warner's Foundations of differentiable
manifolds
and Lie groups, [Hi]: Hirsch's
Differential topology, [Jo]: Joyce's Compact manifolds with special
holonomy,
[dC]: de Cataldo's The Hodge theory of projective manifolds, [De]:
Demailly's
Complex analytic and differential geometry, [Go-Xia]: W. Goldman and E.
Xia's
Rank one Higgs bundles and representations of fundamental groups of
Riemann surfaces, [Co]: Conlon's book Diffeential geometry, ...

** CALENDAR:**

First day of class TU JAN 29,
Last TH MAY 9. No classes MAR 18-24 (spring break);

Absences due to travel: No class on MAR 25-29 (KIAS workshop); No class
on APR 11 (SLC
colloquium).

** Actual schedule.**

In the first 10 lectures, Jan 29-Feb 28
we have covered the lecture notes Chapters 1-6 (which present the material in
an order different from the one of the syllabus and from the one of the tentative schedule): preliminaries, basics of smooth manifolds, vector fields, few facts about Lie groups, vector bundles, differential forms.
** Tentative schedule.** some of the lectures
below
have the density
of black holes. It is very likely that the schedule will be changed as
we move along.

Jan 29:
**Differential forms.** Differential forms and
differential forms
with
compact supports. Poincare' lemmas. [Bo-Tu].
Jan 31: **Differential forms.** Orientation,
integration, Stokes' theorem degree of
a proper map R^n-->R^n. [Bo-Tu].
Feb 5: **Differential forms.** The
Mayer-Vietoris
principle (case of finite good covers): de Rham cohomology is finite
dimensional, Poincare' duality, Kunneth formula, Leray-Hirsch theorem.
[Bo-Tu].
Feb 7: **Differential forms.** The Poincare'
dual of a closed oriented submanifold of an oriented
manifold and its compact supports analogue. Degree of a proper map of
oriented manifolds. The Thom isomorphism for the compact vertical
cohomology of an orientable R-vector bundle via integration along the
fibers; the Thom class; Thom class and Poincare' duality (e.g. zero
section of oriented vector bundle). Global angular form-Euler-Thom
class of an oriented rank 2 R-vector bundle. (?? pd of sub with c-supp
different??). [Bo-Tu].
Feb 12: above continued. [Bo-Tu].
Feb 14: **Differential forms.** Aspects of
the Hodge theorem for: compact Riemannian,
compact Hermitean, and compact Kahler manifolds: Poincare' and Kodaira
Serre duality, Hodge (p,q)-decomposition. [dC], [De].
Feb 19: **Vector fields.** Vector fields. The
Lie bracket. The Lie algebra of a Lie group.
Frobenius theorem. Flows. The Ehresmann fibration theorem for proper
submersions and
applications to hypersurfaces of complex projective space. [Wa], [De].
Feb 21:**Vector fields.** Rudiments of Morse
theory and the Lefschetz hyperplane theorem. [Mi[
Feb 26: **Vector fields.** Covariant
derivatives on the tangent bundle: torsion, curvature,
holonomy. Riemannian metrics: the Levi-Civita connection, the Riemann
curvature tensor. Gauss-Bonnet for oriented Riemannian surfaces. [Jo].
Feb 28: ** Vector bundles.** Basic operations
on vector bundles. Chern classes of
C-vector bundles via projectivization and via induction.
The splitting principle. Properties. Pontryagin classes of R-vector
bundles. [Bo-Tu], [Mi-St].
Mar 5: ** Vector bundles.** Classification of
vector bundles via the infinite Grassmannians; universality of Chern
classes. Identification of Chern classes with Chern-Weil classes.
[Bo-Tu], [Mi-St].
Mar 7:** Vector bundles.** ``Baby Trinity" on
the trivial complex line bundle on
a
compact Riemann surface: C^* (resp. U(1)) representations of the
fundamental group = flat (resp. unitary) connections = holomorphic
line bundles with (resp. without) a Higgs field. [Go-Xia] and maybe
some notes of mine.
Mar 12: ** G-bundles.** Principal G-bundles,
associated G-bundles. ``Dependence"
on G. Classification for G discrete and relation with covering spaces.
[St].
Mar 14: ** G-bundles.** Connections,
curvature and holonomy on principal G-bundles and
relation with analogous notions on vector bundles. Principal bundles as
a means to globalize Cartan's structure equation. Reduction of
structure. [Jo], [Co].
Apr 2: ** G-bundles.** Sphere bundles (need
SS?): orientability, Euler class
and existence of sections, Euler # and local degrees; re-phrasing for
vector bundles: Euler class Poincare' dual to good zero section (rank
vector bundle=dimension base), Hopf index theorem; Euler class and
zero locus of a transversal section (any rank). [Bo-Tu].
Apr 4: the above continued.
Apr 9
Apr 16
Apr 18
Apr 23
Apr 25
Apr 30
May 2
May 7
May 9

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**Mark
Andrea
de Cataldo's homepage. **
**Quod non est in web non est in
mundo**