process

278 U. Bunke, T. Schick, M. Spitzweck, and A. Thomwhere dt is the normalized Haar measure. Then we have db D c. The maps.h p / p>0 induce a chain contraction .h p / p>0 of the complex C .H; R/. ThereforeH i Map.A; C .H; R// Š H i C .H; R/.A/ for i 1, too.The spectral sequence thus degenerates from the second page on. We conclude thatH i Hom ShAb S .U ;I / Š 0; i 1:We now take the cohomology of the double complex Hom i Sh Ab S .U ;I / in the otherorder, first in the U -direction and then in the I -direction. In order to calculate thecohomology of U in degree 2 we use the fact 4.15. We get again a spectral sequencewith second term (for q 2, using 4.2.11)F p;q2Š Ext p Sh Ab S ..ƒq Z H /] ; R/:We know by Corollary 3.28, 2., thatF p;02Š Ext p Sh Ab S.Z; R/ Š 0; p 1(note that we can write Z D Z.¹º/ for a one-point space). Furthermore note thatF 0;12Š Hom ShAb S .H ; R/ Š Hom top-Ab.H; R/ Š 0since there are no continuous homomorphisms H ! R. The second page of thespectral sequence thus has the following structure.2 Hom ShAb S ..ƒ2 Z H /] ; R/1 0 Ext 1 Sh Ab S .H ; R/ Ext2 Sh Ab S .H ; R/0 R 0 0 0 00 1 2 3 4Since the spectral sequence must converge to zero in positive degrees we see thatExt 1 Sh Ab S .H ; R/ Š 0:We claim that Hom ShAb S ..ƒ2 Z H /] ; R/ Š 0. Note that for A 2 S we haveHom ShAb S ..ƒ2 Z H /] ; R/.A/ Š Hom PrAb S .ƒ2 Z H ; R/.A/ Š Hom Pr Ab S=A.ƒ 2 Z H jA ; R jA /:An element 2 Hom PrAb S=A.ƒ 2 Z H jA ; R jA / induces a family of a biadditive (antisymmetric)maps W W H .W / H .W / ! R.W /for .W ! A/ 2 S=A which is compatible with restriction. Restriction to points givescontinuous biadditive maps H H ! R. Since H is compact the only such map isthe constant map to zero. Therefore W vanishes for all .W ! A/. This proves theclaim.Again, since the spectral sequence .F r ;d r / must converge to zero in higher degreeswe see thatF 2;12Š Ext 2 Sh Ab S .H ; R/ Š 0:This finishes the proof of the lemma.