By Matthias Kreck
This publication provides a geometrical creation to the homology of topological areas and the cohomology of soft manifolds. the writer introduces a brand new category of stratified areas, so-called stratifolds. He derives simple recommendations from differential topology similar to Sard's theorem, walls of solidarity and transversality. according to this, homology teams are developed within the framework of stratifolds and the homology axioms are proved. this suggests that for great areas those homology teams accept as true with 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 idea of Quillen. back, yes very important cohomology sessions ensue very evidently during this description, for instance, the attribute periods that are developed within the publication and utilized in a while. the most basic effects, Poincare duality, is nearly a triviality during this technique. a few basic invariants, resembling the Euler attribute and the signature, are derived from (co)homology teams. those invariants play an important function in essentially the most unbelievable 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 booklet relies on classes the writer taught in Mainz and Heidelberg. Readers will be accustomed to the elemental notions of point-set topology and differential topology. The publication can be utilized for a mixed advent to differential and algebraic topology, in addition to for a fast presentation of (co)homology in a direction 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
We summarize these considerations. 3. 1. (Local retractions) Let (S, C) be a stratifold. Then for x ∈ Si there is an open neighborhood U of x in S, an open neighbourhood V of x in Si and a morphism r:U →V such that U ∩ Si = V and r|V = id. Such a morphism is called a local retraction near x. If r : U → V is a local retraction near x, then r induces an isomorphism C ∞ (Si )x → Cx , [h] → [hr], the inverse of i∗ : Cx → C ∞ (Si )x . , if r : U → V is another local retraction near x, then there is a U ⊂ U ∩ U such that r|U = r |U .
It has two non-empty strata: Sk+1 = M × (0, 1) and S0 = pt. Remark: One might wonder if every smooth function on a stratum extends to a smooth function on S. This is not the case as one can see from the open ◦ cone CM , where M is a compact non-empty smooth manifold. A smooth function on the top stratum M × (0, 1) can be extended to the open cone if and only if it is constant on M × (0, ) for an appropriate > 0. Example 2: Let M be a non-compact m-dimensional manifold. The onepoint compactification of M is the space M + consisting of M and an additional point +.
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. We have a bicollar (in the world of topological spaces), a homeomorphism ϕ : ∂T × (− , ) → V ∪ V by mapping (x, t) ∈ ∂T × (− , 0] to c(x, −t) and (x, t) ∈ ∂T × [0, ) to c (x, t).