Simplex

(Def) Two sets, or topological spaces, are structurally same if there is a one-to-one function mapping one onto the other such that both this function and its inverse are continuous. Two spaces that are structurally same in this sense are homeomorphic.

(Def) oriented 0-simplex is a point P. Oriented 1-simplex is a directed line segment P_{1} P_{2} joining the points P_{1} and P_{2} and viewed as traveled in the direction from P_{1} to P_{2}. Oriented 2-simplex is a triangular region P_{1} P_{2} P_{3}.

P_{i} P_{j} P_{k} is equal to P_{1} P_{2} P_{3} if \begin{pmatrix} 1 & 2 & 3 \ i & j & k \end{pmatrix} is an even permutation. opposite to P_{1} P_{2} P_{3} if it is an odd permutation.

(Def) Oriented 3-simplex is given by an ordered sequence P_{1} P_{2} P_{3} P_{4} of four vertices of a solid tetrahedron. Simplexes are oriented, or have an orientation, meaning that we are concerned with the order of the vertices as well as actual points where vertices are located.

(Def) Boundary of a 0-sumplex P is an empty simplex. notation is “ \partial_{0} (P) = 0”

Boundary of a 1-simplex P_{1} P_{2} is defined by \partial_{1} (P_{1} P_{2}) = P_{2} – P_{1}.

Boundary of a 2-simplex is defined by \partial_{2} (P_{1} P_{2} P_{3}) = P_{2} P_{3} – P_{1} P_{3} + P_{1} P_{2}

Boundary of a 3-simplex is \partial_{3} (P_{1} P_{2} P_{3} P_{4}) = P_{2} P_{3} P_{4} – P_{1} P_{3} P_{4} + P_{1} P_{2} P_{4} – P_{1} P_{2} P_{3} . Similar definition holds for \partial_{n} for n > 3.

Each individual summand of the boundary of a simplex is a face of the simplex.

(Def) A space divided up into simplexes according to following requirements is a simplical complex.

Each point of the space belongs to at least one simplex.

Each point of the space belongs to only a finite number of simplexes.

Two different (up to orientation) simplexes either have no points in common or one is (except possibly for orientation) a face of the other or a face of a face of the other, etc. or the set of points in common is a face, or a face of a face, etc.,, of each complex.

(Def) For a simplical complex X, The group C_{n} (X) of (oriented) n-chains of X is the free abelian group generated by the (oriented) n-simplexes of X.

Every element of C_{n} (X) is a finite sum of the form \sum_{i} m_{i} \sigma_{i}, where the \sigma_{i} are n-simplexes of X and m_{i} \in \mathbf{Z}. We accomplish addition of chains by taking the algrbraic sum of the coefficients of each occurrence in the chains of a fixed simplex.

(Def) if \sigma is an n-simplex, \partial_{n} ( \sigma ) \in C_{n-1} (X) for n = 1,2,3. define C_{-1} (X) = {0} , then we will also have \partial_{0} (\sigma ) \in C_{-1} (X). Since C_{n} (X) is free abelian, \partial_{n} gives a unique boundary homomorphism \partial_{n} mapping C_{n} (X) into C_{n-1} (X) for n = 0,1,2,3.

(Def) Kernel of \partial_{n} consists of those n-chains with boundary 0. The elements of the kernel are n-cycles. The usual notation for the kernel of \partial_{n} , group of n-cycles, is ‘Z_{n} (X) ‘.

Image under \partial_{n} , the group of (n-1)-boundaries, consists of those (n-1)-chains that are boundaries of n-chains. This groups is denoted by ‘B_{n-1} (X) ’.

(Thm 41.9) Let X be a simplical complex, and let C_{n} (X) be the n-chains of X for n = 0,1,2,3. Then the composite homomorphism \partial_{n-1} \partial_{n} mapping C_{n} (X) into C_{n-2} (X) maps everything into 0 for n = 1,2,3. That is, for each c \in C_{n} (X) we have \partial_{n-1} (\partial_{n} (c)) = 0. We use the notation \partial_{n-1} \partial_{n} = 0, or \partial^{2} = 0.

(Cor 41.10) For n = 0,1,2 and 3, B_{n} (X) is a subgroup of Z_{n} (X).

(Def 41.11) The factor group H_{n} (X) = Z_{n} (X) / B_{n} (X) is the n-dimensional homology group of X.