By Allan J. Sieradski

This article is an creation to topology and homotopy. themes are built-in right into a coherent complete and constructed slowly so scholars are not crushed. the 1st 1/2 the textual content treats the topology of entire metric areas, together with their hyperspaces of sequentially compact subspaces. the second one 1/2 the textual content develops the homotopy classification. there are various examples and over 900 workouts, representing quite a lot of trouble. This ebook may be of curiosity to undergraduates and researchers in arithmetic.

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2 Trr(X, y) is a decreasing one, so, for El,r+l etc. example, F iii) t-s by 1 IF 2 ~ 00 The differential dr raises s by r and decreases 1. It is perhaps desirable to give an exrumple of this o "",,' spectral sequence. Let us take X = Y = S. The E;~ term Ext~,t(Z2' Z2) is then for brevity). (which is rechristened I recall that Hl'~(A) Hs,t(A) can be identified " Hi th the space of primitive elements in Ai". The primitive 2i element ~i thus gives us a generator hi. In H~H~ (A) Vie can define cup products and this allo"lrJs us to write dOHn part of a basis for s 4 h4 3 h 30 2 h 02 1 h o Hs,t(A).

These groups all become equal after a while, and we define the limit to be ~(X) , where X= (~}. Similarly for the homology maps induced by morphisms. We can define the boundary maps for a pair because our version of suspension has been chosen to commute with ~. Similar remarks apply to cohomology. We now turn to the question of Eilenberg-MacLane objects in this category. Theorem 3. is a free Suppose that graded module over the Steenrod algebra many generators in each dimension. complex (i) (ii) A with finitely Then there exists stable K such that: H*(K;Z2)~· F as an A-module, and ~r(X,K) ~ Hom~(F,H*(X;Z2» for each stable complex (The symbol lower the degree by Hom~ r.

I want next to consider products in the spectral sequence. In the E2 term of the special case X = Y = SO we have the cup products of homological algebra for E~,t = Hs,t(A) = Ext~,t(Z2,Z2) • We also have products in 2~r(SO,SO) ; the product structure is given geometri- cally by the composition of maps. It is a theorem that one can introduce products into the whole of the spectral sequence, compatible with these two products in E~ , and so that dr E2 is a derivation, of course. and This result is already in my paper in the Commentarii Mathematici Helvetici (7).