By Iain T. Adamson

This paintings goals to provide ordinary topology in an unconventional means. It supplies a assessment of the fundamental definitions including workouts with no options or proofs of the theorems partly 1, after which provides the suggestions partially 2, permitting the scholar to check solutions with their very own.

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We may consider the p-biharmonic problem with the Navier boundary conditions AP u = f (x, u) in 52 u=0u=0 ona52 by working in the space W = W2'P(Q) n Wo''(5l) with the same norm, etc. 8. ,Apum), and F e C'(Q x R') satisfies F(x, 0) = 0 and IV F(x,u)l,C(lulr-1+1) V(x,u)E[1xIl8"`, 1- ABSTRACT FORMULATION AND EXAMPLES 26 where V. F = (aF/aul,... , OF/au,,,), for some r e (1, p*) and C > 0. iIP hull (Apu,v) VuiIP-2VU,. 3. , CHAPTER 2 Background Material In this preliminary chapter we collect some of the necessary background material on algebraic topology.

Is) Stability: If SX is the suspension of X # 0, then i(SX) = i(X) + 1. (i7) Piercing property: If X, A E F, X0 and XI are closed invariant X is subsets of X such that X = X0 u X1, and ep : A x [0,1] a continuous mapping satisfying y(-x, t) = -V(x, t) for all (x, t), yp(Ax [0, 1]) is closed in X, w(Ax {0}) c X0, and cp(Ax {1}) c XI, then i(p(A x [0, 1]) n Xe n XI) i(A). (i8) Neighborhood of zero: If U is a bounded closed symmetric neighborhood of the origin in a normed linear space W, then i(dU) = dim W.

Thus, k = i(X) S i(SX) 5 k + 1. 2. 14 (iv) below (whose proof is independent of (io)), the rank of io : Hk-1(SX) Hl-'(X) is at least 1 + 8k1, contradicting the fact that io is homotopic to the constant map i1 : X SX, x (x, 1) via the imbedding F : X x [0, 1] SX, (x, t) -. (x, t) and hence rankio* = rankii = 6k1. So i(SX) = k + 1. (i7) Let Yo = cp-1(Xo), Y1 = y1(X1), and Y = A x [0,1]. Then Yo and Y1 are closed invariant subsets of Y such that Y = Yo u Y1. Since y maps B = Yo nY1 into C = co(Y) n Xo n X1i (i2) gives i(C) > i(B).