Download Algebraic Models for Homotopy Types EPFL July 2013 Exercises 1

Algebraic Models for Homotopy Types
EPFL July 2013
Exercises
1. In a closed model category, show that a map that has the left lifting property
with respect to acyclic fibrations (resp., fibrations) is a cofibration (resp.,
acyclic cofibration). (Hint: Factor and apply the lifting property to the
factored map.)
2. (Homotopy pushouts I) In a closed model category, show that in a pushout
square, if one arrow is an acyclic cofibration, so is the parallel arrow. (Hint:
Use the previous exercise.)
3. Let X be an object in a closed model category and (IX, ∂0 , ∂1 , σ) a cylinder
object. Assume that X is cofibrant.
(a) Prove that the maps ∂0 : X → IX and ∂1 : X → IX are cofibrations.
(b) For a map f : X → Y , let M f = IX ∪X Y (pushout of ∂1 and f ). Show
that Y → M f is an acyclic cofibration.
(c) Assume that Y is also cofibrant and show that the map X → M f
(induced by ∂0 ) is a cofibration.
4. (K. Brown’s Lemma) Let F : M → C be a functor from a closed model
category to a category C . Suppose C has a subcategory W of “weak equivalences” that satisfies the 2-out-of-3 property (e.g., the subcategory of isomorphisms). Show that if F sends acyclic cofibrations to weak equivalences,
then it sends every weak equivalence between cofibrant objects to a weak
equivalence. (Hint: Use the previous problem.)
5. Show that if X and Y are cofibrant objects in a closed model category, then
X q Y is the coproduct of X and Y in the homotopy category.
6. For a closed model category M and an object A of M , let M\A be the
category of objects under A: An object is a map A → X in M and a map
is a commutative diagram. Show that M\A is a closed model category with
cofibrations, fibrations, and weak equivalences inherited from M . Show
that for any map f : A → B, the functor f ∗ : M\B → M\A is the right
adjoint of a Quillen adjunction.
7. Let A → B be a weak equivalence between cofibrant objects. Show that
the Quillen adjunction in the previous problem is a Quillen equivalence.
(Hint: Consider the category whose objects are the closed model categories
and whose maps are the right adjoints of Quillen adjunctions, and apply
K. Brown’s Lemma.)
8. (Homotopy Pushouts II) Given a diagram where all objects are cofibrant,
the horizontal maps are cofibrations and the vertical maps are weak equivalences
o A /
/B
Co
'
C0 o
'
o A0 /
'
/ B0
show that the induced map on pushouts B ∪A C → B 0 ∪A0 C 0 is a weak
equivalence. (Hint: Use the previous three problems.)
9. (Homotopy Pushouts III) Given a diagram where all objects are cofibrant,
the right-ward maps are cofibrations and the vertical maps are weak equivalences
/B
Co
A /
'
C0 o
'
A0 /
'
/ B0
show that the induced map on pushouts B ∪A C → B 0 ∪A0 C 0 is a weak
equivalence. (Hint: Use the previous problems and problem 6.)
10. (Stokes’ Theorem) Show that formal integration of polynomial forms on
simplices extends by linearity to a map of differential graded modules from
the polynomial De Rham complex of a simplicial set to the normalized
cochain complex. (Note that this map does not preserve the multiplication.)
11. Show that any simply connected almost free commutative differential graded
Q-algebra is isomorphic to one of the form M ⊗ PX where M is a minimal
algebra and X contractible differential graded Q-module.
12. Show that a quasi-isomorphism of minimal algebras is an isomorphism.
(Hint: Show that the functor Q (indecomposables) from commutative differential graded algebras to chain complexes preserves weak equivalences
between cofibrant objects.)
13. Use minimal models to compute the rational homotopy groups of spheres.
14. Use minimal models to compute the rational homotopy groups of CP n and
HP n .
15. Use minimal models to compute the rational homotopy groups of S m ∨ S n
for n ≥ m ≥ 2. (This is harder than the last 2 problems.)