Title: Theory / Unimportant / Math / Topology / Classical topology and combinatorial group theory 2nd
Tags: Copy

Name: A face of \(\Delta^n\)
Definition: A \(n-1\)-simplex generated by \(n\) of the \(n+1\) points of \(\Delta^n\)
Page: 2
Chapter: 1
Variables: \(\Delta^n\) \(n\)-simplex

Name: The boundary of \(\Delta^n\)
Definition: The union of the faces of \(\Delta^n\)
Page: 2
Chapter: 1
Variables: \(\Delta^n\) \(n\)-simplex

Question: The distance between disjoint closed set
Answer: is positive
Page: 7
Chapter: 10
Section: 1

Name: a multiple point of \(m\)
Definition: A point \(x\in T\) such that \(|m^{-1}(x)|>1\)
Page: 10
Section: 1
Variables: \(T\) a topological space
\(m: [0,1]\to T\) an arc or closed curve
Context: Topology: curve
Name2: A singularity of \(m\)

Page: 11
Name: An identification space of \(T\)
Set: A partition of \(T\)
Chapter: 10
Section: 1
Variables: \((T,\mathcal T)\) a topological space
Base: For each element \(X\) of the partition, its neighborhood \(N_\epsilon=\bigcup_{x\in X}N_{\epsilon, T}(x)\)
Context: Topology

Name: A \(n\)-dimensional manifold
Definition: A topological space where each point has a neighbourhood homeomorphic to the interior of a \(\mathbf B^n\) the open ball of dimension \(n\)
Page: 13
Chapter: 0
Section: 6
Example: \(\mathbf B^n\), \(\mathbb R^n\)
Name2: A \(n\)-manifold
Example2: A surface is a 2-manifold

Question: What is the topology of the space of position of
Answer: The torus, because it is represented by \(\theta,\phi\in[0,2\pi)\)
Page: 13
Chapter: 0
Section: 6

Name: A bounded \(n\)-manifold
Definition: A topological figure in which every point has either an open \(n\)-ball or half \(n\)-ball neighbourhood
Page: 15
Chapter: 0
Section: 7
Example: The \(n\)-simplex
Name2: A \(n\)-manifold with boundary
Intuition: It's generally a \(n\)-manifold everywhere except on some boundary

Name: The dimension of \(C\)
Definition: The maximum dimension of the simplex of \(C\)
Page: 15
Chapter: 0
Section: 7
Variables: \(C\) a simplicial complex

Name: The boundary of \(\mathcal C\)
Definition: The \(n-1\) faces of the simplex of \(\mathcal C\) that occurs an odd number of time
Page: 15
Chapter: 0
Section: 7
Variables: \(\mathcal C\) a simplicial complex
Notation: \(\partial\mathcal C\)
Intuition: faces which occurs twice are gluing two simplexes together and so are not really part of the boundry of \(\mathcal C\)

Name: \(c\) is null-homotopic
Definition: \(c\) is homotopic to a constant function
Page: 17
Chapter: 0
Section: 1.9
Variables: \(T,\mathcal T\) a topological space
\(c:[0,1]-> T\) a curve

Name: An isotopy
Definition: An homotopy \(h:[0:1]\times T_1\to T_2\) where each \(h_t(x)=h(t,x)\) is an homeomorphism
Page: 18
Chapter: 0
Section: 1.9
Variables: \((T_1,\mathcal T_1)\), \((T_2,\mathcal T_2)\)  a topological space

Definition: \(x\mapsto h(t,x)\)
Page: 18
Chapter: 0
Section: 1.9
Variables: \((T_1,\mathcal T_1)\), \((T_2,\mathcal T_2)\)  a topological space
An homotopy \(h:[0:1]\times T_1\to T_2\)
\(t\in[0,1]\)
Notation: \(h_t\)

Smaller0: \(\mathcal K\) is finite
Greater0: \(\mathcal K\) is compact
Page: 19, 20
Chapter: 0
Section: 2.1
Variables: \(\mathcal K\) a complex
Index: , 1
Kind: , ex
Smaller to greater: iff

Smaller0: \(\mathcal K\) is locally finite
Greater0: \(\mathcal K\) is locally compact
Page: 19
Chapter: 0
Section: 2.1
Variables: \(\mathcal K\) a complex

Name: The schema of \(\mathcal K\)
Definition: The set of set of vertices of \(\mathcal K\) such that there union is a simplex belonging to \(\mathcal K\)
Page: 19
Chapter: 0
Section: 2.1
Variables: \(\mathcal K\) a complex
Example: \(\left\{
\begin{array}{lll}
\{P_1, P_2,
P_3\},\\
\{P_0\}, \{P_0, P_1\},  \\
\{P_1\}, \{P_1, P_2\},\\
\{P_2\}, \{P_2, P_3\},\\
\{P_3\}, \{P_3, P_1\}
\end{array}\right\}\)

Name: A neighbourhood star of \(P\)
Definition: The union of the simplexes containing \(P\)
Page: 19
Chapter: 0
Section: 2.1
Variables: \(\mathcal K\) a complex
\(P\) a vertex of \(\mathcal K\)

Smaller0: \(\mathcal K\) is a \(n\)-manifold
Greater0: a topological \(\mathbf B^n\)
Page: 19
Chapter: 0
Section: 2.1
Variables: \(\mathcal K\) a complex
\(P\) a vertex of \(\mathcal K\)
Suffix Smaller: the neighborhood star of \(P\) is

Name: An orientation of \(\mathcal S\)
Definition: An order \(P_0,\dots, P_{n+1}\) of the vertex of \(\mathcal S\), up to even permutation of the elements
Page: 21
Chapter: 0
Section: 2.1
Variables: \(\mathcal S=\{P_0,\dots,P_{n+1}\}\) a \(n\)-simplex
Example2: 3 dimension
Applied to2: 2 dimension

Name: An orientation of \(\mathcal K\)
Definition: An orientation of each simplex of \(\mathcal K\)
Page: 21
Chapter: 0
Section: 2.1
Variables: \(\mathcal K\) a \(n\)-complex

Name: A coherent orientation of \(\mathcal K\)
Definition: An orientation of each simplex of \(\mathcal K\) such that two simplexes sharing a face have an opposite orientation on it
Page: 21
Chapter: 0
Section: 2.1
Variables: \(\mathcal K\) a \(n\)-complex

Name: \(\mathcal K\) is orientable
Definition: \(\mathcal K\) has a coherent orientation
Page: 22
Chapter: 0
Section: 2.1
Variables: \(\mathcal K\) a \(n\)-complex
Counterexample: the mobius strip

Question: Property of \(n\)-complex
Answer: It can be embedded into \(\mathbb R^{2n+1}\)
Page: 22
Chapter: 0
Section: 2.3
Variables: \(\mathcal K\) a \(n\)-complex
Construction: Create the curve \(\left(x_i=t^i\right)_{i=1}^{2n+1}\) and places all vertices on it. Then any hyperplan contains at most \(2n+1\) vertices, so it can't contains two complexes. In particular, intersection of complexes is necessarily a face

Name: An elementary subdivision of \(\mathcal S\)
Definition: The introduction of a \((n-1)\)-cell in its interior and an \((n-2)\)-shere in its boundary
Page: 24
Chapter: 0
Section: 2.3
Variables: \(\mathcal S\) a \(n\)-simplex

Name: The barycentric subdivision of \(\Delta^n\)
Definition: The subdivision of \(\Delta^n\) obtained by adding a point \(P\) in the center of \(\Delta^n\) and one simplex by face, containing this face and \(P\)
Page: 25
Chapter: 0
Section: 2.3
Variables: \(\Delta^n\) a \(n\)-simplex
Index: 2
Kind: Ex

Name: An \(n\)-dimensional simplicial complex
Definition: each simplex meets only finitely many others
Page: 19
Chapter: 0
Section: 2.1
Variables: \(n\in\mathbb N\)
Definition2: two simplexes are either disjoint or their intresection is a common face
Conjdef: A union of simplexes of dimension \(\le n\) such that
Name2: A \(n\)-complex

Name: \(S_1\) is isomorphic to \(S_2\)
Definition: Up to renaming, \(S_1=S_2\)
Page: 25
Chapter: 0
Section: 2.5
Variables: \(\mathcal K_1,\mathcal K_2\) complexes
\(S_1\) and \(S_2\) their schemas

Name: \(\mathcal K_1\) is combinatorially homeomorphic to \(\mathcal K_2\)
Definition: After finite sequences of elementary subdivision, \(\mathcal K_1\) has a schema isomorphic to the schema of \(\mathcal K_2\)
Page: 25
Chapter: 0
Section: 2.5
Variables: \(\mathcal K_1,\mathcal K_2\) complexes

Smaller0: \(P,Q\) are the endpoints of a polygonal arc in \(\mathcal O\)
Greater0: \(P,Q\) are the endpoints of an arc in \(\mathcal O\)
Page: 26
Chapter: 0
Section: 2.5
Variables: \(\mathcal O\subseteq\mathbb R^n\) open
\(P,Q\in\mathcal O\)
Smaller to greater: iff
Equivalent: \(P, Q\) lie in a connected open set

Name: \(S\) separates \(P,Q\)
Definition: Any arc from \(P\) to \(Q\) meets \(S\)
Page: 26
Chapter: 0
Section: 2.5
Variables: \(T,\mathcal T\) a topology
\(S\subseteq T\)
\(P,Q\in T\subseteq S\)
Definition2: \(P\) and \(Q\) lie in distinct components of \(T\setminus S\) (assuming \(S\)) closed

Name: The inside of \(c\)
Definition: The set of point \(P\) such that there exists a semi-line \(L\), not parallel to any segment of \(c\), not passing through an edge of \(c\), that crosses \(c\) an odd number of time
Page: 27
Chapter: 0
Section: 2.5
Variables: \(c:[0,1]\to\mathbb R^2\) a polygonal curve

Name: The outside of \(c\)
Definition: The set of point \(P\) such that there exists a semi-line \(L\), not parallel to any segment of \(c\), not passing through an edge of \(c\), that crosses \(c\) an even number of time
Page: 27
Chapter: 0
Section: 2.5
Variables: \(c:[0,1]\to\mathbb R^2\) a polygonal curve

Name: A polygon
Definition: A region of \(\mathbb R^2\) consistiting of a polygonal curve \(p\) and its inside
Page: 28
Chapter: 0
Section: 2.5