By Matthias Kreck
This booklet provides a geometrical creation to the homology of topological areas and the cohomology of delicate manifolds. the writer introduces a brand new type of stratified areas, so-called stratifolds. He derives simple thoughts from differential topology equivalent to Sard's theorem, walls of cohesion and transversality. in response to this, homology teams are built within the framework of stratifolds and the homology axioms are proved. this means that for great areas those homology teams believe traditional singular homology. along with the traditional computations of homology teams utilizing the axioms, ordinary structures of vital homology periods are given. the writer additionally defines stratifold cohomology teams following an concept of Quillen. back, sure very important cohomology periods ensue very obviously during this description, for instance, the attribute periods that are built within the ebook and utilized in a while. probably the most primary effects, Poincare duality, is sort of a triviality during this method. a few basic invariants, resembling the Euler attribute and the signature, are derived from (co)homology teams. those invariants play an important function in one of the most outstanding leads to differential topology. specifically, the writer proves a distinct case of Hirzebruch's signature theorem and offers as a spotlight Milnor's unique 7-spheres. This booklet is predicated on classes the writer taught in Mainz and Heidelberg. Readers could be acquainted with the fundamental notions of point-set topology and differential topology. The ebook can be utilized for a mixed creation to differential and algebraic topology, in addition to for a fast presentation of (co)homology in a path approximately differential geometry.
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Additional info for Differential Algebraic Topology: From Stratifolds to Exotic Spheres (Graduate Studies in Mathematics, Volume 110)
Depending on the choice of L, this may be a very strange space. , L = T. Then we can consider the inclusion of ∂T into X = Y − L as our net. We say that this inclusion detects the hole ◦ obtained by deleting T if we cannot extend the inclusion from ∂T to X to a map from T into X. We now weaken our knowledge of X by assuming that it is obtained from Y by deleting the interior of some compact c-stratifold, but we do not know which one. We only know the boundary S of the deleted c-stratifold. Then the only way to test if we have a hole with boundary—the boundary of the deleted stratifold—is to consider all compact c-stratifolds T having the same boundary S and to try to extend the inclusion of the boundary to a continuous map from T to X.
Then we consider the bordism class αi := [pt, xi ], where the latter means the 0-dimensional manifold pt together with the map mapping this point to xi . We claim that the bordism classes αi form a basis of SH0 (X; Z/2). This follows from the deﬁnition of path components and bordism classes once we know that for points x and y in X, we have [pt, x] = [pt, y] if and only if there is a path joining x and y. If x and y can be joined by a path then the path is a bordism from (pt, x) to (pt, y) and so [pt, x] = [pt, y].
We call our objects stratifolds because on the one hand they are stratiﬁed spaces, while on the other hand they are in a certain sense very close to smooth manifolds even though stratifolds are much more general than smooth manifolds. As we will see, many of the fundamental tools of diﬀerential topology are available for stratifolds. In this respect smooth manifolds and stratifolds are not very diﬀerent and deserve a similar name. 18 2. Stratifolds Remark: It’s a nice property of smooth manifolds that once an algebra C ⊂ C 0 (M ) is given for a locally compact Hausdorﬀ space M with countable basis, the question, whether (M, C) is a smooth manifold is a local question.