Algebraic Geometry, la Rabida, Spain 1981: Proceedings by J. M. Aroca, R. Buchweitz, M. Giusti, M. Merle

By J. M. Aroca, R. Buchweitz, M. Giusti, M. Merle

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1) induced by inclusions is an isomorphism. The group on the left hand side of the above arrow is the free product with amalgamation; it is the construction for groups corresponding to U1 ∪ U2 for spaces, as we shall see later in discussing pushouts. H. Fox. One important consequence is that the fundamental group shared the same possibilities and the same difficulties of computation as general abstract groups. The problem was with the connectivity assumption on U12 , since this prevented the use of the theorem for deducing the result that the fundamental group of the circle S 1 is isomorphic to the group Z of integers.

1] Nonabelian Algebraic Topology In fact there is a range of intuitions which abstract groups are unable to express, and for which other concepts such as groupoid, pseudogroup and inverse semigroup have turned out to be more appropriate. As Mackenzie writes in [Mac87]: The concept of groupoid is one of the means by which the twentieth century reclaims the original domain of applications of the group concept. The modern, rigorous concept of group is far too restrictive for the range of geometrical applications envisaged in the work of Lie.

6] Nonabelian Algebraic Topology ‘Teichm¨uller groupoid’ (rather than a ‘Teichm¨uller group’) is a must, and part of the very crux of the matter . . 8. 1) is an isomorphism when U1 , U2 are open subsets of X = U1 ∪ U2 and A meets each path component of U1 , U2 and U12 = U1 ∩ U2 . Here we write Aλ = Uλ ∩ A for λ = 1, 2, 12. What one would expect is that the proof would construct directly an inverse to η. Alternatively, the proof would verify in turn that η is surjective and injective. The proof we give might at first seem roundabout, but in fact it follows the important procedure of verifying a universal property.

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