By Andrew H. Wallace
Proceeding from the view of topology as a kind of geometry, Wallace emphasizes geometrical motivations and interpretations. as soon as past the singular homology teams, although, the writer advances an knowing of the subject's algebraic styles, leaving geometry apart that allows you to learn those styles as natural algebra. a variety of workouts seem through the textual content. as well as constructing scholars' considering by way of algebraic topology, the routines additionally unify the textual content, due to the fact a lot of them function effects that seem in later expositions. broad appendixes provide useful reports of history material.
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Additional info for Algebraic topology: homology and cohomology
3 Contraction Mapping Theorem and Its Applications 35 in consequence, ||u1 − u2 ||∞ ≤ 18 || f (·, v1 ) − f (·, v2 )||∞ . At this point, we may assume, for example, that f is Lipschitz in the variable u with constant L and then, for all t, | f (t, v1 (t)) − f (t, v2 (t))| ≤ L|v1 (t) − v2(t)| ≤ L v1 − v2 ∞ . In particular, if L < 8, then T is a contraction and the problem has a unique solution. A similar result was obtained by Picard  in 1893, by the method of successive approximations. 2). 1.
We already know that L is a continuous operator, so it remains to prove the continuity of L−1 . And this is a very well-known fact: the reader who is familiar with functional analysis may invoke the open mapping theorem or the Fredholm alternative. But it is also true that, in this case, the proof is straightforward. Because L−1 is linear, it suffices to prove the continuity at the origin, so let ϕn → 0 and un = L−1 ϕn . 2, un ∞ ≤ 18 un ∞ → 0 and un ∞ ≤ un ∞ → 0. We conclude that un C2 → 0, and the proof is complete.
Suppose that f has no fixed points; then h does not vanish. Indeed, • If s ≥ 12 , then h(t, s) = λ (s)eit − f (λ (s)eit ) = 0 since f has no fixed points; • If s < 12 , then h(t, s) = eit − (1 − λ (2s)) f (eit ) = 0 since f ∞ ≤ 1 and, again, f has no fixed points. Thus, we may define 2π ∂ h (t, s) ∂t I(s) := h(t, s) 0 dt. 19) By standard results I is differentiable, and I (s) = 2π 0 ∂ ∂s ∂h ∂ t (t, s) h(t, s) dt = 2π 0 ∂ ∂t ∂h ∂ s (t, s) h(t, s) dt = 2π ∂h ∂ s (·, s) h(·, s) . 0 Now observe that h(0, s) = h(2π , s) for all s; in other words, the curve γ (t) := is closed, and hence I (s) = 0 for all s.
Algebraic topology: homology and cohomology by Andrew H. Wallace