Algebraische Topologie by Wolfgang Lück

By Wolfgang Lück

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Table of Contents

I. basic TOPOLOGY.
1. Set idea and Logic.

2. Topological areas and non-stop Functions.

three. Connectedness and Compactness.

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five. The Tychonoff Theorem.

6. Metrization Theorems and Paracompactness.

7. whole Metric areas and serve as Spaces.

eight. Baire areas and size Theory.

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This grows more more slowly than the virtual (complex) ,IhuruHion d = 4k - (3/2)(1 + b+(X» of the moduli space M" , and it follows that AI, \ a~k is dense in M Ic, for large k. doE) is Serre- dual to HO(EndoE f8J Kx). "rr iH a non-trivial section s of the bundle EndoE®Kx. Two cases arise according 'u wlu~ther the determinant of s is identically zero or not. If the determinant is zero Ull' l((~rnel of s defines a line bundle L * and a section of E ® L. Then we can fit h,tI4' th(~ first construction described above and estimate the nmnber of parameters IYAlllthle in the group Ext in terms of k.

For all large enough k the invariant qk,X satisfies qk,X (a, a, ... , a) > O. For the proof of this one considers the restriction of holomorphic bundles over X t : a hyperplane section ~ - a complex curve representing Q. (See the account of Witten's lectures in these Proceedings). ) Then vi have a restriction map r:Mk~WE. Over WE we have a basic holomorphic line bundle £, again just as considere in Witten's theory. It is easy enough to show that p(a) is the pull-back by r . e. eN define a holomorphic embedding j : WE -+ cpm: Furthermore one can easily see that r is an embedding, so the composite j 0 r give' a projective embedding of Mk, and N IJ( a) is the restriction of the hyperplane cl ' over projective space.

Finally le~ note that the group of orientation- preserving self- homotopy equivalences of X . naturally on the cohomology of 8*. For simplicity we suppose that the class ~ fixed by this action, we just call such a class an invariant class . Then to sum" we obtain THEOREM 7. Let X be a compact, smooth, oriented,and simply connecte manifold with b+(X) > 1. Let q, be an invariant class in Hr(B*, R) for r If 4k > 3(1 + b+(X) + r and the dimension s = 8k - 3(1 + b+(X) equals 2d + r t, the map q",q"X : H 2 (X; Z) x··· X H 2 (X : Z) --+ R, given by q",q"x ([E I ], ...

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