Homotopy groups, first

In this section we give a reformulation of the main results of [22, Ch. V] for the case of simplicial sheaves. In this context there are two noticeable differences between simplicial sheaves tmd simplicial sets. The first is that the weak homotopy type of a simplicial sheaf can not be recovered from the weak homotopy type of its Postnikoff tower unless some finitness assumptions are used (Example 1.30). The second is that a simplicial abelian group object is not necessarily weakly equivalent to the product of Eilenberg-MacLane objects corresponding to its homotopy groups (Theorem 1.34). 

One radical difference between homotopy and homology is illustrated by the case of spheres. Whereas it was very easy to determine the homology groups of spheres, their homotopy groups are in general unknown, and are the subject of current research. Thus our first example of is an exception rather than the rule. 

Proof — By definition BrtG = R7i,(BG) where n Sm/Sf, (5m/S) ij is the obvious morphism of sites. Since the third condition of Lemma 3.15 clearly holds for 7t so does the first and therefore it is sufficient to show that BG is A -local in A° ShVet Sm/S). Let, i/9G be a simplicially fibrant model for BG. Using Lemma 2.8(2) wc see that it is sufficient to show that for any striedy henselian local scheme S and a finite etale group scheme G over S of order prime to char S) the map of simplicial sets. G(S). G(As) is a weak equivalence. Since S is strictly henselian G is just a finite group. In particular we obviously have G(S) = G(A ). We also have H],(S,G) = and H],(As, G) = where the second equality holds because of the homotopy invariance of the completion of outside of characteristic ([13]) and therefore our map is a weak equivalence by Proposition 1.16. 

The relative quadratic signature is the obstruction for frame surgery to a homotopy equivalence of pairs. The relative quad L-groups L ( ) were first defined by Wall ( ll using geometric f odd-... 

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