By Matthias Kreck
This publication provides a geometrical advent to the homology of topological areas and the cohomology of delicate manifolds. the writer introduces a brand new classification of stratified areas, so-called stratifolds. He derives simple strategies from differential topology akin to Sard's theorem, walls of cohesion and transversality. in keeping with 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 usual singular homology. in addition to the traditional computations of homology teams utilizing the axioms, elementary structures of significant homology periods are given. the writer additionally defines stratifold cohomology teams following an idea of Quillen. back, yes very important cohomology sessions take place very clearly during this description, for instance, the attribute periods that are built within the e-book and utilized afterward. some of the most primary effects, Poincare duality, is nearly a triviality during this process. a few primary invariants, similar to the Euler attribute and the signature, are derived from (co)homology teams. those invariants play an important function in probably the most marvelous ends up in differential topology. particularly, the writer proves a unique case of Hirzebruch's signature theorem and offers as a spotlight Milnor's unique 7-spheres. This booklet relies on classes the writer taught in Mainz and Heidelberg. Readers might be conversant in the elemental notions of point-set topology and differential topology. The publication 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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Extra info for Differential Algebraic Topology: From Stratifolds to Exotic Spheres (Graduate Studies in Mathematics, Volume 110)
Note that one can use the local retractions to characterize elements of C, namely a continuous function f : S → R is in C if and only if its restriction to all strata is smooth and it commutes with appropriate local retractions. This implies that if f : S → R is a nowhere zero morphism then 1/f is in C. 3. Examples The ﬁrst class of examples is given by the smooth k-dimensional manifolds. These are the k-dimensional stratifolds with Si = ∅ for i < k. It is clear that such a stratifold gives a smooth manifold and in turn a k-dimensional manifold gives a stratifold.
1. Let T and T be k-dimensional c-stratifolds and let g : ∂T → ∂T be an isomorphism. Then (T ∪g T , C(T ∪g T )) is a k-dimensional stratifold. Of course, if g is an isomorphism between some components of the boundary of T and some components of the boundary of T , we can glue as above via g to obtain a c-stratifold, whose boundary is the union of the complements of these boundary components (see Appendix B, §2). 38 3. Stratifolds with boundary: c-stratifolds Finally we note that if f : T → R is a smooth function and s is a regular value of f | ◦ and f |∂T , then f −1 (s) is a c-stratifold with collar given by T restriction.
We say that T+ and T− are obtained from S by cutting along a codimension-1 stratifold, namely along g −1 (t). S g −1 (t) g R T+ T− Now we describe the reverse process and introduce gluing of stratifolds along a common boundary. Let T and T be c-stratifolds with the 3. Stratifolds with boundary: c-stratifolds 37 same boundary, ∂T = ∂T . By passing to the minimum of and we can assume that the domains of the collars are equal: c : ∂T × [0, ) → V ⊂ T and c : ∂T × [0, ) → V ⊂ T . Then we consider the topological space T ∪∂T=∂T T obtained from the disjoint union of T and T by identifying the boundaries.