By Dmitry Kozlov
This quantity is the 1st accomplished remedy of combinatorial algebraic topology in ebook shape. the 1st a part of the publication constitutes a speedy stroll throughout the major instruments of algebraic topology. Readers - graduate scholars and dealing mathematicians alike - will most likely locate relatively valuable the second one half, which includes an in-depth dialogue of the most important learn recommendations of combinatorial algebraic topology. even if functions are sprinkled through the moment half, they're valuable concentration of the 3rd half, that is solely dedicated to constructing the topological constitution idea for graph homomorphisms.
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Extra info for Combinatorial Algebraic Topology (Algorithms and Computation in Mathematics)
Geometry of the simplicial join One can define the join of arbitrary topological spaces. Let I denote the closed unit interval. 31. Let X and Y be two topological spaces. The join of X and Y is the topological space X ∗ Y defined as follows: X ∗ Y = I × X × Y / ∼, where the equivalence relation ∼ is given by • • (0, x, y) ∼ (0, x, y˜), for all y, y˜ ∈ Y ; (1, x, y) ∼ (1, x ˜, y), for all x, x ˜ ∈ X. 9) where on the righthand side we take the simplicial join. Given geometric realizations of ∆1 in Rm and ∆2 in Rn , a geometric realization of ∆1 ∗ ∆2 in Rm+n+1 can be obtained as follows.
We start with a discrete set of points. This is our 0-skeleton, and we proceed by induction on the dimension of the attached faces. At step d we attach the d-dimensional faces, all at once. Each face is represented by some convex polytope P in the sample space Rd . To attach it we need a continuous map f : ∂P → X, where X denotes the part of the complex created in the first d − 1 steps. The attaching map must satisfy an additional condition: we request that it should be a homeomorphism between ∂P and f (∂P ), and that this homeomorphism should preserve the cell structures, where the cell structure on ∂P is simply the given polytopal structure, and the cell structure on f (∂P ) is induced from the previous gluing process.
K. We also remark that instead of taking the successive joins, one can think of X1 ∗ · · · ∗ Xk as the quotient space X1 ∗ · · · ∗ Xk = ∆[k] × X1 × · · · × Xk / ∼, where the equivalence relation ∼ is given by (α, x1 , . . , xk ) ∼ (α, x′1 , . . , x′k ) if tuples (x1 , . . , xk ) and (x′1 , . . , x′k ) coincide on the support simplex of α (where the support simplex of α is the minimal subsimplex of ∆[k] containing α). 22 2 Cell Complexes Geometry of barycentric subdivision The geometric realizations of the abstract simplicial complexes Bd ∆ and ∆ are related in a fundamental way.