By Andrew H. Wallace
Protecting mathematical necessities to a minimal, this undergraduate-level textual content stimulates scholars' intuitive realizing of topology whereas keeping off the tougher subtleties and technicalities. Its concentration is the tactic of round ameliorations and the learn of severe issues of features on manifolds. 1968 version.
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Additional info for Differential topology: first steps
6). 9) f ∗ Y g∗ Z ∼ = (f × g)∗ (Y Z). 10) f! W g! X ∼ = (f × g)! (W X). Now take A = A , B = B and f = g. For ex-spaces Y and Z over B, f ∗ (Y ∧B Z) ∼ = f ∗ ∆∗B (Y Z) ∼ = (∆B ◦ f )∗ (Y Z). 9), ∼ ∆∗ (f × f )∗ (Y f ∗ Y ∧A f ∗ Z = A Z) ∼ = ((f × f ) ◦ ∆A )∗ (Y Z). The right sides are isomorphic since ∆B ◦ f = (f × f ) ◦ ∆A . Similarly, ∼ f! ∆∗ (f × id)∗ (Y X) ∼ f! (f ∗ Y ∧A X) = = f! ((f × id) ◦ ∆A )∗ (Y X), A while ∼ ∆∗ (id × f )! (Y X). Y ∧B f! 11. It is illuminating conceptually to go further and consider group actions from an external point of view.
The fiber Xb is a based Gb -space with Gb -fixed basepoint s(b), where Gb is the isotropy group of b. Recall from [105, II§1] the distinction between the category KG of G-spaces and nonequivariant maps and the category GK of G-spaces and equivariant maps; the former is enriched over GK , the latter over K . We have a similar dichotomy on the ex-space level. Here we have a conflict of notation with our notation for categories of ex-spaces, and we agree to let KG,B denote the category whose objects are the ex-G-spaces over B and whose morphisms are the maps of underlying ex-spaces over B, that is, the maps f : X −→ Y such that f ◦ s = t and q ◦ f = p.
1. The category GKG/H of ex-G-spaces over G/H is equivalent to the category HK∗ of based H-spaces. Proof. The equivalence sends an ex-G-space (Y, p, s) over G/H to the Hspace p−1 (eH) with basepoint the H-fixed point s(eH). Its inverse sends a based H-space X to the induced G-space G ×H X, with the evident structure maps. More formally, recall that there are “induction” and “coinduction” functors ι! and ι∗ from H-spaces to G-spaces that are left and right adjoint to the forgetful functor ι∗ that sends a G-space Y to Y regarded as an H-space.