By Igusa K.
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In algebraic topology a few classical invariants - reminiscent of Betti numbers and Reidemeister torsion - are outlined for compact areas and finite team activities. they are often generalized utilizing von Neumann algebras and their strains, and utilized additionally to non-compact areas and endless teams. those new L2-invariants include very fascinating and novel details and will be utilized to difficulties bobbing up in topology, K-Theory, differential geometry, non-commutative geometry and spectral concept.
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If f ∈ t, then f X is a Killing vector field. Now we fix a K ∈ t. Let · (X ) be the vector space of smooth forms on X . Let L K X denote the associated Lie derivative operator acting on the de Rham complex ( · (X ), d X ). The Cartan formula asserts that L K X = [d X , i K X ]. 1), i K X denotes interior multiplication by K X . 1) is actually an anticommutator. Duistermaat–Heckman formulas and index theory 5 Put d KX = d X + i K X . 1) can be rewritten in the form d KX,2 = L K X . , of the forms α which are such that L K X α = 0.
We can identify N X K / X to the normal bundle N M K /M . By a construction given in [B86a], (T H M, g T X ) determine a unique Euclidean connection on T X . This connection restricts to the Levi-Civita connection along the fibre X . If g T M is a metric on M such that g T M restricts to g T X on T X , and moreover T H M and T X are orthogonal in T M with respect to g T M , then ∇ T X is the projection of the Levi-Civita connection ∇ T M on T M with respect to the splitting T M = T H M ⊕ T X . Of course ∇ T X is T -invariant.