![Exercises 01 [1.1]](http://s1.studyres.com/store/data/008937002_1-4b328d0d5483323f64ee1f6669a2523e-300x300.png)
18.906 Problem Set 7 Due Friday, April 6 in class
... 2. Suppose G acts on a space F on the left such that the map g∗ : H∗ (F ) → H∗ (F ) is the identity for all g ∈ G. Let P → X be a principal G-bundle, and let E → X be the associated fiber bundle P ×G F → X with fiber F . Show that the action of π1 (X) on H∗ (F ) is trivial. In particular, if the gro ...
... 2. Suppose G acts on a space F on the left such that the map g∗ : H∗ (F ) → H∗ (F ) is the identity for all g ∈ G. Let P → X be a principal G-bundle, and let E → X be the associated fiber bundle P ×G F → X with fiber F . Show that the action of π1 (X) on H∗ (F ) is trivial. In particular, if the gro ...
Definition. Let X be a set and T be a family of subsets of X. We say
... (b) if Gα ∈ T for every α ∈ I, then α∈I Gα ∈ T Tn (c) if Gi ∈ T for every i ∈ {1, . . . , n}, n ∈ N, then i=1 Gi ∈ T . The sets from T are called open and their complements are called closed. Remark. Let (M, d) be a metric space. Let T be the family of all open sets in (M, d) in the sense of the the ...
... (b) if Gα ∈ T for every α ∈ I, then α∈I Gα ∈ T Tn (c) if Gi ∈ T for every i ∈ {1, . . . , n}, n ∈ N, then i=1 Gi ∈ T . The sets from T are called open and their complements are called closed. Remark. Let (M, d) be a metric space. Let T be the family of all open sets in (M, d) in the sense of the the ...
![THE HIGHER HOMOTOPY GROUPS 1. Definitions Let I = [0,1] be](http://s1.studyres.com/store/data/001160219_1-57248e6267c8b98f385fc9eb1b30c0a7-300x300.png)
THE HIGHER HOMOTOPY GROUPS 1. Definitions Let I = [0,1] be
... direction corresponding to t1 and the vertical direction to t2 . These pictures represent 5 snapshots (think of the parameter s as being “time”) of the homotopy G. As s goes from 0 to 1, the homotopy G interchanges the left and right halves of I 2 by first shrinking both (second picture), rotating t ...
... direction corresponding to t1 and the vertical direction to t2 . These pictures represent 5 snapshots (think of the parameter s as being “time”) of the homotopy G. As s goes from 0 to 1, the homotopy G interchanges the left and right halves of I 2 by first shrinking both (second picture), rotating t ...
Fibrations handout
... with respect to all finite CW complexes.) For either kind of fibration, E is called the total space and B the base space. Given a point b ∈ B, then Fb = p−1 (b) is the fiber of p at b. Proposition 0.2. Given a fibration p : E → B, if b0 and b1 are points in B which are connected by a path, then Fb0 ...
... with respect to all finite CW complexes.) For either kind of fibration, E is called the total space and B the base space. Given a point b ∈ B, then Fb = p−1 (b) is the fiber of p at b. Proposition 0.2. Given a fibration p : E → B, if b0 and b1 are points in B which are connected by a path, then Fb0 ...
PDF
... Let A be a concrete category over X. A source (A → Ai )i∈I in A is called initial provided that an X-morphism f : |B| → |A| is an A-morphism whenever each composite fi ◦ f : |B| → |Ai | is an A-morphism. The dual notion is called a final sink. A source (A, fi )I in the category of topological spaces ...
... Let A be a concrete category over X. A source (A → Ai )i∈I in A is called initial provided that an X-morphism f : |B| → |A| is an A-morphism whenever each composite fi ◦ f : |B| → |Ai | is an A-morphism. The dual notion is called a final sink. A source (A, fi )I in the category of topological spaces ...