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Geometry And Topology

Algebraic and Analytic Geometry by Amnon Neeman

By Amnon Neeman

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Extra info for Algebraic and Analytic Geometry

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200 (1974), 327–344. J. A. Hilmann, The Algebraic Characterization of Geometric 4-manifolds, London Math. Soc. Lecture Note Ser. 198 (1994), Cambridge University Press, Cambridge– New York–Melbourne. ¨ H. Hopf, Uber die algebraische Anzahl von Fixpunkten, Math. Z. 29 (1929), 493–524. W. Hurewicz and H. Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton N. , 1941. J. Jezierski, One codimensional Wecken type theorems, Forum Math. 5 (1993), 421–439.

Note that the respective proofs are straightforward by reduction to the ordinary trace. 13) Properties. 1) If (in V ) the following diagram is commutative: / E1 ~ ~~ L L 1 ~~S ~   ~~ ~ E T / E1 E T then L is a Leray endomorphism if and only if L1 is a Leray endomorphism and in that case we get Tr(L) = Tr(L1 ). 2. 2) Assume that (in V ) the following diagram 0 / E1 L1 0  / E1 /E L  /E / E2 /0 L2  / E2 /0 with exact rows is commutative. Then, if two of the three endomorphisms L1 , L, L2 are Leray endomorphisms, then so is the third one and in such a case we have: Tr(L) = Tr(L1 ) + Tr(L2 ).

In the next section, we shall describe results when the two manifolds are surfaces. Note that the previous result only covers the case where the surface is the two dimensional tori. To conclude this section we may consider our three main questions stated in Section 2, where we replace Coin(f, g) by a Nielsen coincidence class. With this formulation, these questions seem very natural to us, and suitable for the study of coincidences in positive codimension, a topic which shall be treated in Section 7.