12/15: The minimal L∞ Quillen model of a space. The
higher Whitehead products. Bonus: rational homotopy theory and
mysterious triality in M-theory and supergravity.
[https://arxiv.org/abs/2111.14810]
12/15: The minimal L∞ Quillen model of a space. The
higher Whitehead products. Bonus: rational homotopy theory and
mysterious triality in M-theory and supergravity.
[https://arxiv.org/abs/2111.14810]
12/13: More of L∞-algebras:
L∞-morphisms and quasi-isomorphisms.
12/10: Dg-Lie algebras and dg-cocommutative coalgebras. A first
glimpse on L∞-algebras.
12/08: The Hopf algebra C.(ΩX; ℚ) of chains on a loop space
and problems with the dg-Lie algebra of its primitives. The Quillen
model of λ(X) a space X: the dg-Lie algebra λ(K) = N. P
ℚ^ [GK], associated to a simplicial set K.
12/06: The Samelson and Whitehead products. The Hopf algebra of chains
on a loop space. Example: the Whitehead product for the sphere and the
corrseponding graded Lie algebra.
12/03: Remarks on the multiplicativity and Koszul signs in the
universal enveloping algebra construction. The primitives. Theorems of
Milnor-Moore and Cartan-Serre. The Pontryagin product.
12/01: Differential graded (dg) Lie algebras and Hopf
algebras.
11/29: Review of the main theorem. Rationalization.
11/26: Thanksgiving Break: no classes!
11/24: Thanksgiving Eve: no evening class! Have a happy holiday!
11/22: The fundamental theorem of rational homotopy theory: the
Quillen adjunction (F,A) induces a Quillen equivalence between the
subcategories of homologically connected algebras of finite type and
rational connected, nilpotent spaces of ℚ-finite type.
11/19: The equivalence of homotopy groups of a cofibrant augmented
dgca B and the simplicial set F(B).
11/17: The cohomology ring. Homotopy theory of the fundamental
adjunction (F,A). The Sullivan minimal models of a simplicial set and
a space. The homotopy groups.
11/15: Homotopy in the category of simplicial sets and the
contractibility of the simplicial set Ωp. Showing
that F sends cofibrations in ℚ-dgca≥ 0 to
fibrations in sSetop. This is the second half of
proving that (F,A) is a Quillen adjunction.
11/12: The adjunction of F and A from the previous lecture. Showing
that A sends fibrations in sSetop, i.e.,
cofibrations in sSet, to fibrations. This is half way of
proving that (F,A) is a Quillen adjunction.
11/10: The simplicial polynomial de Rham algebra of the cosimplicial
space of geometric simplices. The polynomial de Rham functor A on the
category of simplicial sets (or spaces) and its left adjoint F.
11/8: The model category of simplicial sets: generating cofibrations
and acyclic cofibrations. Kan complexes as fibrant objects. Quillen
equivalence of simplicial sets with compactly generated Hausdorff
spaces.
11/5: The geometric realization functor. The model category of
simplicial sets: the definition.
11/3: The uniqueness of a minimal Sullivan model. Simplicial sets
(sorry!) and their relation to topological spaces (the
singular-simplex functor Sing(X) so far).
11/1: Homotopic maps induce equal maps on homotopy groups. Weak
equivalences between minimal Sullivan algebras are isomorphisms.
10/29: The cofibrancy of Sullivan algebras. Homotopy groups. Right
homotopy in ℚ-dgca≥ 0
and dgca≥ 0/ℚ: path-space objects
in the two categories.
10/27: Finishing verification of the model category structure on
ℚ-dgca≥ 0: MC5(ii): proving that
Bf → B in A ↪ Bf → B is an
acyclic fibration; MC4(ii). Getting back to Sullivan algebras.
10/25: Verification of the model category structure on
ℚ-dgca≥ 0: MC2, MC3, MC4(i), MC5(i),
MC5(ii): given f: A → B, factored it into A ↪
Bf → B.
10/22: Back to the model category structure on
ℚ-dgca≥ 0. Some basic cofibrations. Property
MC1.
10/20: Quillen equivalence. Homotopy colimits and limits in model
categories in general. Construction of homotopy pushouts. Examples in
the category Top of spaces. Mapping cylinders, mapping cones,
and homotopy cofibers. The dual story: homotopy pullbacks, mapping
path spaces and homotopy fibers (just mentioned).
10/18: The left and right derived functors for a Quillen adjunction
are adjoint.
10/15: Proof of the existence of left and right derived functors. Ken
Brown's lemma. Example: Tor and Ext.
10/13: Lemma: left homotopic morphisms are taken to equal morphisms by
a functor which takes weak equivalences to isomorphisms. Quillen
adjunctions. Left and right derived functors.
10/11: An "economic" description of the homotopy category of a model
category via fibrant cofibrant objects. The derived category of
R-mod.
10/8: The equivalence of left and right homotopies πl
(A,B) = πr (A, B) for A cofibrant and B
fibrant. "Whitehead's" theorem in a model category.
via via
10/6: The homotopy category of a model category as a localization with
respect to weak equivalences. Modeling homotopies in model categories:
cylinder and path space objects, left and right homotopies.
10/4: No course meeting. I suggest that from 4:00 to 4:50 p.m you
attend my Zoom talk on Rational homotopy theory and mysterious
duality at (believe it or not)
the First
International Congress on Topology and Related Topics, which will
be happening on Zoom in Lima, Perú. It will be a good (?)
overview of what will be coming at the end of the course, but the
material of my talk will not be needed to go on with the course. So,
it is up to you whether you want to attend my talk. There are other
talks you may wish to attend. For example, our own Craig Westerland
will be talking there 5-5:50 p.m. The congress will be running Monday
through Thursday, 9 a.m. - 6:50 p.m. CDT. Here is the Zoom link:
https://zoom.us/j/97977182294?pwd=VTQzU0xCVU1ERjlrUWVBWDJMU2VwUT09
10/1: Model category structures on dg-R-mod and
dgcaℚ≥ 0. Stability under pullbacks and
pushouts.
9/29: Introduction to model categories.
9/27: The injectivity of H≤ n+2(fn+1) and
what fails when H1(A) ≠ 0. Homotopy theory in
dgcaℚ≥ 0.
9/24: The existence of a minimal Sullivan model of a homologically
connected DGCA. The proof in the homologically simply connected case.
9/22: Minimal Sullivan models. Examples: the spheres and the wedge of
two three-spheres.
9/20: Connected and simply connected DGCAs, Sullivan algebras, minimal
algebras. A simply connected Sullivan algebra is minimal.
9/17: Free GCAs, free DGCAs, and semifree DGCAs. The algebra of
polynomial differential forms on a simplex is a free DGCA on a certain
complex. Augmentation, the augmentation ideal, indecomposables.
9/15: The de Rham algebra as a viable guess for determining the
rational homotopy type, among other guesses discussed. Differential
graded commutative algebras (DGCAs): definitions. The Koszul rule of
signs.
9/13: The homotopy and homology groups of
S1Q. Rational equivalence and
rationalization. The rationalization of S1
and RP2. The rational homotopy type.
9/10: Review of relevant homotopy theory: CW complexes, the Whitehead
theorem along with a "counterexample," CW approximation, and an
equivalent definition of weak equivalence via the fundamental group
and higher homology groups. Rational spaces and an example:
S1Q.
9/8: Introduction. Plan of the course. Literature. Review of some
homotopy theory: homotopy groups, weak equivalence.
Last modified: (2021-12-14 20:51:17 CST)