# An introduction to G-convergence by Gianni Dal Maso

By Gianni Dal Maso

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Additional resources for An introduction to G-convergence

Example text

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, . .