By Matthias Kreck

This ebook provides a geometrical creation to the homology of topological areas and the cohomology of tender manifolds. the writer introduces a brand new category of stratified areas, so-called stratifolds. He derives simple options from differential topology corresponding to Sard's theorem, walls of team spirit and transversality. according to this, homology teams are built within the framework of stratifolds and the homology axioms are proved. this suggests that for excellent areas those homology teams accept as true with traditional singular homology. in addition to the traditional computations of homology teams utilizing the axioms, easy buildings of vital homology periods are given. the writer additionally defines stratifold cohomology teams following an idea of Quillen. back, sure very important cohomology periods happen very evidently during this description, for instance, the attribute periods that are developed within the publication and utilized afterward. the most primary effects, Poincare duality, is nearly a triviality during this process. a few basic invariants, corresponding to the Euler attribute and the signature, are derived from (co)homology teams. those invariants play an important function in the most wonderful ends up in differential topology. specifically, the writer proves a different case of Hirzebruch's signature theorem and provides as a spotlight Milnor's unique 7-spheres. This e-book relies on classes the writer taught in Mainz and Heidelberg. Readers may be conversant in the elemental notions of point-set topology and differential topology. The e-book 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)**

**Example text**

What is the relation to smooth manifolds equipped with a collar? If W is a smooth manifold and c a collar, then we obviously obtain all the ingredients of the deﬁnition above by considering W as a topological space. In turn, if (W, ∂W, c) is given as in the deﬁnition above, we can in an obvi◦ ous way extend the smooth structure of W to a smooth manifold W with boundary. The smooth structure on W is characterized by requiring that c is not only a homeomorphism but a diﬀeomorphism. The advantage of the deﬁnition above is that it can be given using only the language of manifolds without boundary.

2. Let S be a stratifold and fi : S → R be a family of smooth fi maps such that supp fj is a locally ﬁnite family of subsets of S. Then is a smooth map. Proof: The local ﬁniteness implies that for each x ∈ S, there is a neighbourhood U of x such that supp fi ∩ U = ∅ for all but ﬁnitely many i1 , . . , fi |U = fi1 |U + · · · + fik |U . Since fi1 + · · · + fik is ik . Then it is clear that smooth, we conclude from the fact that the algebra of smooth functions on S is locally detectable, that the map is smooth.

For a space X, the collection of pairs (S, g), where S is an m-dimensional stratifold and g : S → X a continuous map, does not form a set. To see this, start with a ﬁxed pair (S, g) and consider the pairs (S × {i}, g), where 4. Z/2-homology 42 i is an arbitrary index. For example, we could take i to be any set. Thus, there are at least as many pairs as sets and the class of all sets is not a set. 1. The isomorphism classes of pairs (S, g) form a set. The proof of this proposition does not help with the understanding of homology.