By Botvinnik B.
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This self-contained remedy of Morse thought specializes in purposes and is meant for a graduate path on differential or algebraic topology. The publication is split into 3 conceptually precise components. the 1st half comprises the principles of Morse concept. the second one half comprises functions of Morse idea over the reals, whereas the final half describes the fundamentals and a few purposes of complicated Morse conception, a.
Traditionally, functions of algebraic topology to the examine of topological transformation teams have been originated within the paintings of L. E. 1. Brouwer on periodic adjustments and, a bit later, within the attractive mounted element theorem ofP. A. Smith for high periodic maps on homology spheres. Upon evaluating the mounted element theorem of Smith with its predecessors, the fastened aspect theorems of Brouwer and Lefschetz, one unearths that it really is attainable, at the least for the case of homology spheres, to improve the realization of mere life (or non-existence) to the particular decision of the homology kind of the fastened element set, if the map is believed to be major periodic.
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E. p ◦ Ft (A) = x0 . It follows that p ◦ Ft = ht ◦ p, where ht : X/A −→ X/A is some homotopy, such that h0 = IdX/A and h1 = p ◦ q ; it means that p ◦ q ∼ IdX/A . 3. Let X be a CW -complex and A ⊂ X be its subcomplex. Then X/A is homotopy equivalent to the complex X ∪ C(A), where C(A) is a cone over A. 2. 3. 2. Cellular Approximation Theorem. Let X and Y be CW -complexes. Recall that a map f : X −→ Y is a cellular map if f (X (n) ) ⊂ Y (n) for every n = 0, 1, . .. We emphasize that it is not required that the image of n-cell belongs to a union of n-cells.
11. 6. 12. For a group π , we let [π, π] be its commutator. Compute the group π/[π, π] for π = π1 (Mg ). Remark. We note that in particular π1 (T 2 ) ∼ = Z ⊕ Z, which is obvious from the product formula π1 (X × Y ) ∼ = π1 (X) × π1 (Y ). Recall that a non-oriented two-dimensional manifold of genus g is heomeomorphic either to Mg2 (1), a connective sum of a projective plane RP2 and g tori T 2 # · · · #T 2 , or to Mg2 (2), a connective sum of the Klein bottle Kl2 and g tori T 2 # · · · #T 2 . 7. 1.
10. Prove that a covering space p : T −→ X is regular if and only if there is no loop in X which is covered by a loop and a path (starting and ending in different points) in the same time. 11. Prove that if a covering space p : T −→ X is regular then there exists a free action of the group G = π1 (X, x0 )/π1 (T, x0 ) on the space T such that X ∼ = T /G. 12. Prove that a two-folded covering space p : T −→ X is always a regular one. 7. Let X be a “good” path-connected space (in particular, CW -complexes are “good” spaces), x0 ∈ X .