Algebraic topology topics:
Classification of compact surfaces
Euler characteristic of a finite CW complex (Tag D-QK5BM)
Let \(X\) be a finite CW complex with \(c_n\) cells of dimension \(n\). The Euler characteristic of \(X\) is \[ \chi(X)\coloneqq\sum_{n\geq0}(-1)^nc_n. \]
For a finite CW complex \(X\), \[ \chi(X)=\sum_{n\geq0}(-1)^n\rank H_n(X;\ZZ), \] where \(H_n(X;\ZZ)\) is singular homology [@Hat02]. Consequently \(\chi(X)\) does not depend on the CW structure, and finite CW complexes that are homotopy equivalent have the same Euler characteristic.
Connected sum
Homology and cohomology groups
Fundamental group (Tag D-EBNUE)
Let \(X\) be a topological space and \(x_0\in X\), and let \(I = [0,1]\). The fundamental group \(\pi_1(X, x_0)\) is the set of classes of loops \(\gamma\colon I\to X\) based at \(x_0\) under homotopy rel endpoints, with multiplication \[ [\gamma][\eta]\coloneqq[\gamma\cdot\eta], \] where \(\gamma\cdot\eta\) is the concatenation of \(\gamma\) and \(\eta\).
The multiplication on \(\pi_1(X, x_0)\) is well defined and makes \(\pi_1(X, x_0)\) a group. Its identity is the class of the constant loop at \(x_0\), and \(\inverseof{[\gamma]} = [\bar\gamma]\), where \(\bar\gamma(s)\coloneqq\gamma(1-s)\).
Let \(h\colon I\to X\) be a path from \(x_0\) to \(x_1\), with reverse \(\bar h\). Then \(\beta_h\colon\pi_1(X, x_1)\to\pi_1(X, x_0)\), \(\beta_h[\gamma]\coloneqq[h\cdot\gamma\cdot\bar h]\), is a group isomorphism. It depends only on the homotopy class of \(h\) rel endpoints, and for a second path \(h'\) from \(x_0\) to \(x_1\), \(\beta_{h'} = c\circ\beta_h\), where \(c\) is conjugation by \([h'\cdot\bar h]\in\pi_1(X, x_0)\). In particular, if \(X\) is path connected, the groups \(\pi_1(X, x)\) for \(x\in X\) are all isomorphic.
See [@Hat02].
Mayer--Vietoris sequences for homology and cohomology
Diagram chasing
Degree of maps \(S^n \to S^n\)
Top homology and cohomology of manifolds
Reduced homology and cohomology
Relative homology
Homotopy and homotopy invariance
Deformation retraction (Tag D-6UHU7)
Let \(X\) be a topological space, \(A\subseteq X\) a subspace, and \(\iota\colon A\injects X\) the inclusion. A deformation retraction of \(X\) onto \(A\) is a homotopy \(F\colon X\cross I\to X\), written \(F_t \coloneqq F(\wait, t)\), such that for all \(t\in I\): \[ F_0 = \id_X, \qquad F_1(X) \subseteq A, \qquad F_t(a) = a \text{ for all } a\in A .\] The subspace \(A\) is a deformation retract of \(X\) if there exists a deformation retraction of \(X\) onto \(A\).
If \(F\) is a deformation retraction of \(X\) onto \(A\), then \(r\coloneqq F_1\colon X\to A\) is a retraction, \(r\circ\iota = \id_A\), and \(F\) is a homotopy from \(\id_X\) to \(\iota\circ r\). Hence \(\iota\) is a homotopy equivalence with homotopy inverse \(r\).
Two spaces \(X\) and \(Y\) are homotopy equivalent if and only if there exists a space \(Z\) containing both \(X\) and \(Y\) as deformation retracts; for a homotopy equivalence \(f\colon X\to Y\), the mapping cylinder of \(f\) is such a \(Z\).
[@Hat02].
What does it mean for $A \subset X$ to be a deformation retract of $X$?What map exhibits $A \subset X$ as a deformation retract of $X$?Retract (Tag D-NCLVD)
Let \(X\) be a topological space, \(A\subseteq X\) a subspace, and \(\iota\colon A\injects X\) the inclusion. A retraction of \(X\) onto \(A\) is a continuous map \(r\colon X\to A\) with \(r\circ\iota=\id_A\), that is, \(r(a)=a\) for every \(a\in A\):
The subspace \(A\) is a retract of \(X\) if a retraction of \(X\) onto \(A\) exists.
Let \(A\subseteq X\) be a subspace with inclusion \(\iota\colon A\injects X\).
Every retraction \(r\colon X\to A\) is surjective.
Continuous maps \(r\colon X\to A\) with \(r\circ\iota=\id_A\) correspond bijectively to continuous maps \(\rho\colon X\to X\) with \(\rho(X)=A\) and \(\rho(a)=a\) for all \(a\in A\), via \(\rho=\iota\circ r\).
If \(r\) is a retraction and \(a_0\in A\), then \(r_*\circ\iota_* = \id\) on \(\pi_1(A, a_0)\) and on \(H_n(A)\), so \(\iota_*\colon \pi_1(A, a_0)\to\pi_1(X, a_0)\) and \(\iota_*\colon H_n(A)\to H_n(X)\) are injective. For every point \(x_0\) of a topological space \(X\), the subspace \(\ts{x_0}\) is a retract of \(X\), via the constant map \(X\to\ts{x_0}\).
What does it mean for a subspace $A \subset X$ to be a retract of $X$?Excision axiom (Tag D-455S6)
Let \(h_*\) be a sequence of functors \(h_n\) from pairs of topological spaces to abelian groups. The functors \(h_*\) satisfy excision if for every pair \((X, A)\) and every subspace \(Z\subseteq A\) with \(\cl_X(Z) \subseteq A^\circ\), the inclusion \((X\sm Z, A\sm Z)\injects (X, A)\) induces isomorphisms \[ h_n(X\sm Z, A\sm Z) \mapsvia{\sim} h_n(X, A) \quad\text{for all } n .\]
Excision is one of the Eilenberg–Steenrod axioms for a homology theory. Singular homology satisfies it [@Hat02]; relative homotopy groups do not.
[@Hat02].
Künneth formula
Factoring maps
Fundamental theorem of algebra
Algebraic topology theorems:
Brouwer fixed point theorem
Poincaré lemma
Poincaré duality
de Rham theorem
Seifert–van Kampen theorem
Covering space theory topics:
Covering space (Tag D-ANO2D)
Let \(X\) be a topological space. A covering space of \(X\) is a topological space \(\tilde X\) together with a continuous map \(p\colon \tilde X\to X\) such that every \(x\in X\) has an open neighborhood \(U\) for which \(\inverseof{p}(U)\) is a union of pairwise disjoint open sets \(V_i\subseteq\tilde X\), each mapped homeomorphically onto \(U\) by \(p\). The sets \(V_i\) are the sheets of \(\tilde X\) over \(U\).
Let \(p_1\colon\tilde X_1\to X\) and \(p_2\colon\tilde X_2\to X\) be covering spaces of \(X\). An isomorphism of covering spaces is a homeomorphism \(f\colon\tilde X_1\to\tilde X_2\) with \(p_2\circ f = p_1\):
[@Hat02].
Free actions
Free and properly discontinuous group actions (Tag D-2PNEG)
Let \(G\) be a group acting on a topological space \(X\) by homeomorphisms, \(G\actson X\).
The action is free if \(g(x) = x\) for some \(x\in X\) implies \(g = e\).
The action is properly discontinuous if every \(x\in X\) has a neighborhood \(U\) such that for all \(g_1, g_2\in G\), \(g_1(U) \intersect g_2(U) \neq \emptyset\) implies \(g_1 = g_2\).
A properly discontinuous action in this sense is free. Other sources call an action properly discontinuous under the weaker condition that every \(x\in X\) has a neighborhood \(U\) with \(U \intersect g(U) \neq \emptyset\) for only finitely many \(g\in G\). An action satisfying the condition in the definition is called a covering space action in [@Hat02].
Universal cover (Tag D-BX3WD)
Let \(X\) be a topological space. A universal cover of \(X\) is a covering space \(p\colon \tilde X\to X\) with \(\tilde X\) simply connected [@Hat02].
Let \(X\) be path-connected and locally path-connected. Then \(X\) has a universal cover if and only if \(X\) is semilocally simply connected [@Hat02], and a universal cover is unique up to isomorphism of covering spaces [@Hat02]. A universal cover covers every path-connected covering space of \(X\) [@Hat02], and its deck transformation group is isomorphic to \(\pi_1(X)\) [@Hat02].
Correspondence between covering spaces and subgroups of the fundamental group of the base
Path lifting property (Tag D-OISBB)
Let \(p\colon E\to B\) be a continuous map and \(I=[0,1]\). The map \(p\) has the path lifting property if for every path \(\gamma\colon I\to B\) and every \(e_0\in \inverseof{p}(\gamma(0))\) there is a path \(\tilde\gamma\colon I\to E\) with \(p\circ\tilde\gamma=\gamma\) and \(\tilde\gamma(0)=e_0\).
Identifying \(\pt\times I\) with \(I\), this is the homotopy lifting property of \(p\) with respect to the one-point space \(Y=\pt\).
Let \(p\colon\tilde X\to X\) be a covering space. For every path \(\gamma\colon I\to X\) and every \(\tilde x_0\in \inverseof{p}(\gamma(0))\) there is a unique path \(\tilde\gamma\colon I\to\tilde X\) with \(p\circ\tilde\gamma=\gamma\) and \(\tilde\gamma(0)=\tilde x_0\) [@Hat02].
Homotopy lifting property (Tag D-HOCNK)
Let \(p\colon E\to B\) be a continuous map and \(Y\) a topological space. The map \(p\) has the homotopy lifting property with respect to \(Y\) if for every homotopy \(g_t\colon Y\to B\), \(t\in[0,1]\), and every continuous \(\tilde g_0\colon Y\to E\) with \(p\circ\tilde g_0 = g_0\), there is a homotopy \(\tilde g_t\colon Y\to E\) starting at \(\tilde g_0\) with \(p\circ\tilde g_t = g_t\) for all \(t\). The map \(p\) is a fibration if it has the homotopy lifting property with respect to every space \(Y\).
A covering space \(p\colon\tilde X\to X\) has the homotopy lifting property with respect to every space \(Y\), and the lifted homotopy \(\tilde g_t\) starting at a given \(\tilde g_0\) is unique.
See [@Hat02].
Deck transformation (Tag D-SI6OM)
Let \(p\colon\tilde X\to X\) be a covering space. A deck transformation of \(p\) is a homeomorphism \(\psi\colon\tilde X\to\tilde X\) with \(p\circ\psi=p\). The deck transformations form a group \(\Deck(\tilde X\to X)\) under composition.
In the slice category \(\mathsf{Top}_{/X}\), whose objects are continuous maps to \(X\) and whose morphisms are maps over \(X\), the group \(\Deck(\tilde X\to X)\) is the automorphism group \(\Aut_{\mathsf{Top}_{/X}}(p)\) of the object \(p\colon\tilde X\to X\). A deck transformation permutes each fiber \(\inverseof{p}(x)\). If \(\tilde X\) is connected, a deck transformation is determined by the image of one point, so a deck transformation with a fixed point is the identity [@Hat02].
The action of the fundamental group on fibers
Galois, normal, and regular covering spaces (Tag D-MWD2L)
Let \(p\colon\tilde X\to X\) be a covering space, let \(x_0\in X\), and let \(\Deck(\tilde X)\) be its group of deck transformations. The covering \(p\) is Galois, also called normal or regular, over \(x_0\) if \(\Deck(\tilde X)\) acts transitively on the fiber \(\inverseof{p}(x_0)\): for all \(\tilde x_1,\tilde x_2\in \inverseof{p}(x_0)\) there is \(\psi\in\Deck(\tilde X)\) with \(\psi(\tilde x_1)=\tilde x_2\).