An introduction to G-convergence by Gianni Dal Maso

By Gianni Dal Maso

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P6)) follows immediately from the regular value definition of the Brouwer degree (resp. definition of the twisted degree). To prove property (P8), assume (without loss of generality) that f and g are regular normal. 12 with the classical Hopf Theorem (see, for instance, [44]) applied to appropriate fundamental domains, construct “local” homotopies between f and g around the corresponding zeros and extend the obtained homotopies by G-equivariance. 5), extend the previous “partial homotopy” over × [0, 1].

Lifting homeomorphism). 7. Let X be a G-manifold on which a compact Lie group G acts freely. Let X/G be connected. Then, there always exists a regular fundamental domain. 2 combined with the following statement applied to X/G. 8. Let M be a smooth connected n-dimensional manifold (in general noncompact). Take a smooth countable triangulation of M. Then, there always exists a subset To of M satisfying the following conditions: (i) To is open in M; (ii) To is dense in M; (iii) To is contractible; (iv) M \ To is contained in the n − 1-dimensional skeleton.

Notice that if H = K ϕ,l is a twisted subgroup and (H ) (H ), then H is also a twisted subgroup. In particular, every subgroup H ∈ (H ) is twisted. Consequently, it makes sense to talk about the lattice of the conjugacy classes of twisted subgroups in G. Moreover, if dim W (K) = 0 and Lψ,l is a twisted subgroup such that (Lψ,l ) (K ϕ,l ), then it is easy to see that dim W (L) = 0. 20. Denote by t1 (G) the set of all conjugacy classes of the ϕ-twisted lfolded subgroups H = K ϕ,l , l = 1, 2, . .

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