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Extra resources for 4-dimensional compact projective planes with a 7-dimensional collineation group
The information the lemma provides is given by the connecting homomorphism 6 : H I ( Q , s ( w-,~ Rk)) -~ H2(Q, Zk). For a given extension E, there exists a morphism f : E --+ S(W, I~k) >~Q1 as above if and only if the extension E corresponds to a cohomology class in the image of 6. So, the eventual surjectivity of the connecting homomorphism ~ guarantees the existence of a morphism f for any extension. Moreover, the number of such morphisms f, up to conjugation with an element of S ( W , •k), is measured by the kernel of the 6.
However, to decide on isomorphism types, group cohomology is not the best instrument. Consequently, the argument that res only maps a finite number of elements onto the class of N , although correct, is unsecure with respect to an isomorphism type classification. Indeed, there might be more t h a n one class in H 2 ( N / z , Z ) representing a group isomorphic to N and it is not clear why a group < E > in the inverse image under res of one class should be isomorphic to some group in the inverse image of an other class.
2. 1)). 4 Point 3. and 4. from the theorem above can be used to define the concept of a canonical type representation. Such kind of definitions were for instance used in . Having this concept of a (general) canmfical type representation, it is clear now, that to get a nicer geometric structure, one needs to use smaller subgroups of 7-/(RK). In view of the iterative set up, it is however necessary that these subgroups satisfy one crucial condition: assume we restrict AA(RK, ]~k~. to a subspace S(• K, ~k) containing the space of constant mappings R ~ and assume we restrict 7-/(IRK) to a subgroup STi(RK), then one observes that it is necessary that S(I~ K, I~k) is a (Gl(k, ~) • ST-/(Rg))-submodule such that there is a monomorphism, s(R Rk) (Gl(k, R) • SU(R )) Possible examples of such situations are S m o o t h r e p r e s e n t a t i o n s : Let C~176 k) denote the vector space of functions f : ~ g --~ ~k, which are infinitely many times differentiable and denote by C~(IRK) the group, under composition of maps, of smooth diffeomorphisms of IRg .