Smoothings of Piecewise Linear Manifolds
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A01=Barry Mazur
A01=Morris W. Hirsch
Affine transformation
Approximation
Associative property
Author_Barry Mazur
Author_Morris W. Hirsch
Bijection
Bundle map
Category=PBP
Classification theorem
Codimension
Coefficient
Cohomology
Commutative property
Computation
Convex cone
Convolution
Corollary
Counterexample
Diffeomorphism
Differentiable function
Differentiable manifold
Differential structure
Dimension
Direct proof
Division by zero
Embedding
Empty set
eq_isMigrated=1
eq_isMigrated=2
eq_nobargain
Equivalence class
Equivalence relation
Euclidean space
Existential quantification
Exponential map (Lie theory)
Fiber bundle
Fibration
Functor
Grassmannian
H-space
Homeomorphism
Homotopy
Integral curve
Inverse problem
Isomorphism class
K0
Linearization
Manifold
Mathematical induction
Milnor conjecture
Natural transformation
Neighbourhood (mathematics)
Normal bundle
Obstruction theory
Open set
Partition of unity
Piecewise linear
Polyhedron
Reflexive relation
Regular map (graph theory)
Sheaf (mathematics)
Smoothing
Smoothness
Special case
Submanifold
Tangent bundle
Tangent vector
Theorem
Topological manifold
Topological space
Topology
Transition function
Transitive relation
Vector bundle
Vector field
Product details
- ISBN 9780691081458
- Weight: 227g
- Dimensions: 152 x 229mm
- Publication Date: 21 Oct 1974
- Publisher: Princeton University Press
- Publication City/Country: US
- Product Form: Paperback
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The intention of the authors is to examine the relationship between piecewise linear structure and differential structure: a relationship, they assert, that can be understood as a homotopy obstruction theory, and, hence, can be studied by using the traditional techniques of algebraic topology. Thus the book attacks the problem of existence and classification (up to isotopy) of differential structures compatible with a given combinatorial structure on a manifold. The problem is completely "solved" in the sense that it is reduced to standard problems of algebraic topology. The first part of the book is purely geometrical; it proves that every smoothing of the product of a manifold M and an interval is derived from an essentially unique smoothing of M. In the second part this result is used to translate the classification of smoothings into the problem of putting a linear structure on the tangent microbundle of M. This in turn is converted to the homotopy problem of classifying maps from M into a certain space PL/O. The set of equivalence classes of smoothings on M is given a natural abelian group structure.
