1 Preface

References:

Some fun resources:

1.1 Notation

Notation Definition
\(X\times Y, \prod_{j\in J} X_j, X^{\times n}\) Products
\(X\oplus Y, \bigoplus_{j\in J} X_j, X^{\oplus n}\) Direct sums
\(X\ast Y, \ast_{j\in J} X_j, X^{\ast n}\) Free products
\(X\otimes Y, \bigotimes_{j\in J} X_j, X^{\otimes n}\) Tensor products
\({\mathbb{Z}}^n\) The free abelian group of rank \(n\)
\({\mathbb{Z}}^{\ast n}\) The free group on \(n\) generators
\(\pi_0(X)\) The set of path components of \(X\)
\(G=1\) The trivial abelian group
\(G=0\) The trivial nonabelian group

Both the product and direct sum have coordinate-wise operations. For finite index sets \({\left\lvert {J} \right\rvert}< \infty\), the direct sum and product coincide, but in general there is only an injection \(\bigoplus_j X_i \hookrightarrow\prod_j X_j\). In the direct sum \(\bigoplus_j X_j\) have only finitely many nonzero entries, while the product allows infinitely many nonzero entries. So in general, I always use the product notation.

The free group on \(n\) generators is the free product of \(n\) free abelian groups, but is not generally abelian! So we use multiplicative notation, and elements \begin{align*} x \in {\mathbb{Z}}^{\ast n} = \left< a_1, \ldots, a_n\right> \end{align*} are finite words in the noncommuting symbols \(a_i^k\) for \(k\in {\mathbb{Z}}\). E.g. an element may look like \begin{align*} x = a_1^2 a_2^4 a_1 a_2^{-2} .\end{align*}

The free abelian group of rank \(n\) is the abelianization of \({\mathbb{Z}}^{\ast n}\), and its elements are characterized by \begin{align*} x\in {\mathbb{Z}}^{\ast n} = \left\langle{ a_1, \cdots, a_n }\right\rangle \implies x = \sum_n c_i a_i \text{ for some } c_i \in {\mathbb{Z}} \end{align*} where the \(a_i\) are some generating set of \(n\) elements and we used additive notation since the group is abelian. E.g. such an element may look like \begin{align*} x = 2a_1 + 4a_2 + a_1 - a_2 = 3a_1 + 3a_2 .\end{align*}

1.2 Conventions

1.3 Some Prerequisite Algebra Facts

A group morphism \(f:X \to Y\) can not be injective if \(Y\) is trivial unless \(X\) is also trivial.

There are no nontrivial homomorphisms from finite groups into free groups. In particular, any homomorphism \({\mathbb{Z}}_n \to {\mathbb{Z}}\) is trivial.

(Click to expand)

Homomorphisms preserve torsion; the former has \(n{\hbox{-}}\)torsion while the latter does not.

This is especially useful if you have some \(f: A\to B\) and you look at the induced homomorphism \(f_*: \pi_1(A) \to\pi_1(B)\). If the former is finite and the latter contains a copy of \({\mathbb{Z}}\), then \(f_*\) has to be the trivial map \(f_*([\alpha]) = e \in \pi_1(B)\) for every \([\alpha] \in \pi_1(A)\).

2 Summary and Topics: Point-Set Topology

3 Definitions

3.1 Point-Set Topology

A set \(S\) in a metric space \((X, d)\) is bounded iff there exists an \(m\in {\mathbb{R}}\) such that \(d(x, y) < m\) for every \(x, y\in S\).

There does not exist a disconnecting set \(X = A{\coprod}B\) such that \(\emptyset \neq A, B \subsetneq\), i.e. \(X\) is the union of two proper disjoint nonempty sets. Additional condition for a subspace \(Y\subset X\): \(\operatorname{cl}_{Y}(A) \cap V = A \cap\operatorname{cl}_{Y}(B) = \emptyset\).

Equivalently, \(X\) contains no proper nonempty clopen sets.

Set \(x\sim y\) iff there exists a connected set \(U\ni x, y\) and take equivalence classes.

See .

A topological space \((X, \tau)\) is compact if every open cover has a finite subcover.

That is, if \(\left\{{U_{j} {~\mathrel{\Big|}~}j\in J}\right\} \subset \tau\) is a collection of open sets such that \(X = \cup_{j\in J} U_{j}\), then there exists a finite subset \(J' \subset J\) such that \(X \subseteq \cup_{j\in J'} U_{j}\).

A map \(f:X\to Y\) between topological spaces is continuous if and only if whenever \(U \subseteq Y\) is open, \(f ^{-1} (U) \subseteq X\) is open.

A collection of subsets \(\left\{{U_\alpha}\right\}\) of \(X\) is said to cover \(X\) iff \(X = \cup_{\alpha} U_\alpha\). If \(A\subseteq X\) is a subspace, then this collection covers \(A\) iff \(A\subseteq \cup_{\alpha} U_\alpha\).

A subset \(Q\subset X\) is dense iff \(y\in N_{y} \subset X \implies N_{y} \cap Q \neq \emptyset\) iff \(\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu = X\).

A space is first-countable iff every point admits a countable neighborhood basis.

A topological space \(X\) is Hausdorff iff for every \(p\neq q \in X\) there exist disjoint open sets \(U\ni p\) and \(V\ni q\).

A map \(\iota\) with a left inverse \(f\) satisfying \(f\circ \iota = \operatorname{id}\)

For \((X, d)\) a compact metric space and \(\left\{{U_\alpha}\right\}\rightrightarrows X\), there exist \(\delta_{L} > 0\) such that \begin{align*} A\subset X, ~ {\operatorname{diam}}(A) < \delta_{L} \implies A\subseteq U_\alpha \text{ for some } \alpha .\end{align*}

For \(A\subset X\), \(x\) is a limit point of \(A\) if every punctured neighborhood \(P_{x}\) of \(x\) satisfies \(P_{x} \cap A \neq \emptyset\), i.e. every neighborhood of \(x\) intersects \(A\) in some point other than \(x\) itself.

Equivalently, \(x\) is a limit point of \(A\) iff \(x\in \operatorname{cl}_{X}(A\setminus\left\{{x}\right\})\).

A space is locally connected at a point \(x\) iff \(\forall N_{x} \ni x\), there exists a \(U\subset N_{x}\) containing \(x\) that is connected.

A space \(X\) is locally compact iff every \(x\in X\) has a neighborhood contained in a compact subset of \(X\).

A collection of subsets \({\mathcal{S}}\) of \(X\) is locally finite iff each point of \(M\) has a neighborhood that intersects at most finitely many elements of \({\mathcal{S}}\).

A space is locally path-connected if it admits a basis of path-connected open subsets.

A neighborhood of a point \(x\) is any open set containing \(x\).

A space is normal if any two disjoint closed subsets can be separated by neighborhoods.

If \(p\in X\), a neighborhood basis at \(p\) is a collection \({\mathcal{B}}_{p}\) of neighborhoods of \(p\) such that if \(N_{p}\) is a neighborhood of \(p\), then \(N_{p} \supseteq B\) for at least one \(B\in {\mathcal{B}}_{p}\).

A map \(f:X\to Y\) is an open map (respectively a closed map) if and only if whenever \(U \subseteq X\) is open (resp. closed), \(f(U)\) is again open (resp. closed)>

A topological space \(X\) is paracompact iff every open cover of \(X\) admits an open locally finite refinement.

A map \(q:X\to Y\) is a quotient map if and only if

  1. \(q\) is surjective, and
  2. \(U \subseteq Y\) is open if and only if \(q ^{-1} (U)\) is open.

A space \(X\) is path connected if and only if for every pair of points \(x\neq y\) there exists a continuous map \(f:I \to X\) such that \(f(0) = x\) and \(f(1) = y\).

Set \(x\sim y\) iff there exists a path-connected set \(U\ni x, y\) and take equivalence classes.

A subset \(A\subseteq X\) is precompact iff \(\operatorname{cl}_{X}(A)\) is compact.

For \((X, \tau_X)\) and \((Y, \tau_Y)\) topological spaces, defining \begin{align*} \tau_{X \times Y} \mathrel{\vcenter{:}}=\left\{{U \times V {~\mathrel{\Big|}~}U\in \tau_X,\, V\in \tau_Y}\right\} \end{align*} yields the product topology on \(X \times Y\).

A cover \({\mathcal{V}}\rightrightarrows X\) is a refinement of \({\mathcal{U}}\rightrightarrows X\) iff for each \(V\in {\mathcal{V}}\) there exists a \(U\in{\mathcal{U}}\) such that \(V\subseteq U\).

A space \(X\) is regular if whenever \(x\in X\) and \(F\not\ni x\) is closed, \(F\) and \(x\) are separated by neighborhoods.

A map \(r\) in \(A\mathrel{\textstyle\substack{\hookrightarrow^{\iota}\\\textstyle\dashleftarrow_{r}}} X\) satisfying \begin{align*}r\circ\iota = \operatorname{id}_{A}.\end{align*} Equivalently \(X \twoheadrightarrow_{r} A\) and \({\left.{{r}} \right|_{{A}} } = \operatorname{id}_{A}\). If \(X\) retracts onto \(A\), then \(i_*\) is injective.

Alt: Let \(X\) be a topological space and \(A \subset X\) be a subspace, then a retraction of \(X\) onto \(A\) is a map \(r: X\to X\) such that the image of \(X\) is \(A\) and \(r\) restricted to \(A\) is the identity map on \(A\).

A subset \(U \subseteq X\) is saturated with respect to a surjective map \(p: X\twoheadrightarrow Y\) if and only if whenever \(U \cap p ^{-1} (y) = V \neq \emptyset\), we have \(V \subseteq U\), i.e. \(U\) contains every set \(p ^{-1} (y)\) that it intersects. Equivalently, \(U\) is the complete inverse image of a subset of \(Y\).

A space \(X\) is separable iff \(X\) contains a countable dense subset.

A space is second-countable iff it admits a countable basis.

For \((X, \tau)\) a topological space and \(U \subseteq X\) an arbitrary subset, the space \((U, \tau_U)\) is a topological space with a subspace topology defined by \begin{align*} \tau_U \mathrel{\vcenter{:}}=\left\{{Y \cap U {~\mathrel{\Big|}~}U \in \tau}\right\} .\end{align*}

A map \(\pi\) with a right inverse \(f\) satisfying \begin{align*}\pi \circ f = \operatorname{id}\end{align*}

Closed under arbitrary unions and finite intersections.

For \(X\) an arbitrary set, a collection of subsets \({\mathcal{B}}\) is a basis for \(X\) iff \({\mathcal{B}}\) is closed under intersections, and every intersection of basis elements contains another basis element. The set of unions of elements in \(B\) is a topology and is denoted the topology generated by \({\mathcal{B}}\).

A continuous map \(f:X \to Y\) for which \(X\cong f(X)\) are homeomorphic is called a topological embedding.

For \(f: (X, d_{x}) \to (Y, d_{Y})\) metric spaces, \begin{align*} \forall \varepsilon> 0, ~\exists \delta > 0 \text{ such that } \quad d_{X}(x_{1}, x_{2}) < \delta \implies d_{Y}(f(x_{1}), f(x_{2})) < \varepsilon .\end{align*}

3.2 Algebraic Topology

For an \(R{\hbox{-}}\)module \(M\), a basis \(B\) is a linearly independent generating set.

Points \(x\in M^n\) defined by \begin{align*} {\partial}M = \left\{{x\in M: H_{n}(M, M-\left\{{x}\right\}; {\mathbb{Z}}) = 0}\right\} \end{align*}

Denoting \(\Delta^p \xrightarrow{\sigma} X \in C_{p}(X; G)\), a map that sends pairs (\(p{\hbox{-}}\)chains, \(q{\hbox{-}}\)cochains) to \((p-q){\hbox{-}}\)chains \(\Delta^{p-q} \to X\) by \begin{align*} H_{p}(X; R)\times H^q(X; R) \xrightarrow{\frown} H_{p-q}(X; R)\\ \sigma \frown \psi = \psi(F_{0}^q(\sigma))F_{q}^p(\sigma) \end{align*} where \(F_{i}^j\) is the face operator, which acts on a simplicial map \(\sigma\) by restriction to the face spanned by \([v_{i} \ldots v_{j}]\), i.e. \(F_{i}^j(\sigma) = {\left.{{\sigma}} \right|_{{[v_{i} \ldots v_{j}]}} }\).

A map \(X \xrightarrow{f} Y\) is said to be cellular if \(f(X^{(n)}) \subseteq Y^{(n)}\) where \(X^{(n)}\) denotes the \(n{\hbox{-}}\) skeleton.

An element \(c \in C_{p}(X; R)\) can be represented as the singular \(p\) simplex \(\Delta^p \to X\).

Given two maps between chain complexes \((C_*, {\partial}_{C}) \xrightarrow{f, ~g} (D_*, {\partial}_{D})\), a chain homotopy is a family \(h_{i}: C_{i} \to B_{i+1}\) satisfying \begin{align*}f_{i}-g_{i} = {\partial}_{B, i-1}\circ h_{n} + h_{i+1}\circ {\partial}_{A, i}\end{align*} .

A map between chain complexes \((C_*, {\partial}_{C}) \xrightarrow{f} (D_*, {\partial}_{D})\) is a chain map iff each component \(C_{i} \xrightarrow{f_{i}} D_{i}\) satisfies \begin{align*} f_{i-1}\circ{\partial}_{C, i} = {\partial}_{D,i} \circ f_{i} \end{align*} (i.e this forms a commuting ladder)

A manifold that is compact, with or without boundary.

An cochain \(c \in C^p(X; R)\) is a map \(c \in \hom(C_{p}(X; R), R)\) on chains.

A constant map \(f: X\to Y\) iff \(f(X) = y_{0}\) for some \(y_{0}\in Y\), i.e. for every \(x\in X\) the output value \(f(x) = y_{0}\) is the same.

For a directed system \((X_{i}, f_{ij}\), the colimit is an object \(X\) with a sequence of projections \(\pi_{i}:X\to X_{i}\) such that for any \(Y\) mapping into the system, the following diagram commutes:

For a space \(X\), defined as \begin{align*} CX = \frac{X\times I} {X \times\left\{{0}\right\}} .\end{align*} Example: The cone on the circle \(CS^1\)

Note that the cone embeds \(X\) in a contractible space \(CX\).

A space \(X\) is contractible if \(\operatorname{id}_X\) is nullhomotopic. i.e. the identity is homotopic to a constant map \(c(x) = x_0\).

Equivalently, \(X\) is contractible if \(X \simeq\left\{{x_0}\right\}\) is homotopy equivalent to a point. This means that there exists a mutually inverse pair of maps \(f: X \to\left\{{x_0}\right\}\) and \(g:\left\{{x_0}\right\} \to X\) such that \(f\circ g \simeq\operatorname{id}_{\left\{{x_0}\right\}}\) and \(g\circ f \simeq\operatorname{id}_X\).1

A covering space of \(X\) is the data \(p: \tilde X \to X\) such that

  1. Each \(x\in X\) admits a neighborhood \(U\) such that \(p ^{-1} (U)\) is a union of disjoint open sets in \(\tilde V_i \subseteq X\) (the sheets of \(\tilde X\) over \(U\)),
  2. \({ \left.{{p}} \right|_{{V_i}} }: V_i \to U\) is a homeomorphism for each sheet.

An isomorphism of covering spaces \(\tilde X_1 \cong \tilde X_2\) is a commutative diagram

Link to diagram

A map taking pairs (\(p{\hbox{-}}\)cocycles, \(q{\hbox{-}}\)cocycles) to \((p+q){\hbox{-}}\)cocyles by \begin{align*} H^p(X; R) \times H^q(X; R) \xrightarrow{\smile} H^{p+q}(X; R)\\ (a \cup b)(\sigma) = a(\sigma \circ I_{0}^p)~b(\sigma \circ I_{p}^{p+q}) \end{align*} where \(\Delta^{p+q} \xrightarrow{\sigma} X\) is a singular \(p+q\) simplex and

\begin{align*}I_{i}^j: [i, \cdots, j] \hookrightarrow\Delta^{p+q} .\end{align*}

is an embedding of the \((j-i){\hbox{-}}\)simplex into a \((p+q){\hbox{-}}\)simplex.

On a manifold, the cup product is Poincaré dual to the intersection of submanifolds. Also used to show \(T^2 \not\simeq S^2 \vee S^1 \vee S^1\).

An \(n{\hbox{-}}\)cell of \(X\), say \(e^n\), is the image of a map \(\Phi: B^n \to X\). That is, \(e^n = \Phi(B^n)\). Attaching an \(n{\hbox{-}}\)cell to \(X\) is equivalent to forming the space \(B^n \coprod_{f} X\) where \(f: {\partial}B^n \to X\).

For a covering space \(\tilde X \xrightarrow{p} X\), self-isomorphisms \(f:\tilde X \to \tilde X\) of covering spaces are referred to as deck transformations.

A map \(r\) in \(A\mathrel{\textstyle\substack{\hookrightarrow^{\iota}\\\textstyle\dashleftarrow_{r}}} X\) that is a retraction (so \(r\circ \iota = \operatorname{id}_{A}\)) that also satisfies \(\iota \circ r \simeq\operatorname{id}_{X}\).

Note that this is equality in one direction, but only homotopy equivalence in the other.

Equivalently, a map \(F:I\times X\to X\) such that \begin{align*} F_{0}(x) &= \operatorname{id}_{X} F_{t}(x)\mathrel{\Big|}_{A} &= \operatorname{id}_{A} F_{1}(X) &= A .\end{align*}

Alt:

A deformation retract is a homotopy \(H:X\times I \to X\) from the identity on \(X\) to the identity on \(A\) that fixes \(A\) at all times: \begin{align*} H: X\times I \to X \\ H(x, 0) = \operatorname{id}_X \\ H(x, 1) = \operatorname{id}_A \\ x\in A \implies H(x, t) \in A \quad \forall t \end{align*}

Equivalently, this requires that \({\left.{{H}} \right|_{{A}} } = \operatorname{id}_A\)

A deformation retract between a space and a subspace is a homotopy equivalence, and further \(X\simeq Y\) iff there is a \(Z\) such that both \(X\) and \(Y\) are deformation retracts of \(Z\). Moreover, if \(A\) and \(B\) both have deformation retracts onto a common space \(X\), then \(A \simeq B\).

Given any \(f: S^n \to S^n\), there are induced maps on homotopy and homology groups. Taking \(f^*: H^n(S^n) \to H^n(S^n)\) and identifying \(H^n(S^n) \cong {\mathbb{Z}}\), we have \(f^*: {\mathbb{Z}}\to{\mathbb{Z}}\). But homomorphisms of free groups are entirely determined by their action on generators. So if \(f^*(1) = n\), define \(n\) to be the degree of \(f\).

For a functor \(T\) and an \(R{\hbox{-}}\)module \(A\), a left derived functor \((L_{nT})\) is defined as \(h_{n}(TP_{A})\), where \(P_{A}\) is a projective resolution of \(A\).

For \(x\in M\), the only nonvanishing homology group \(H_{i}(M, M - \left\{{x}\right\}; {\mathbb{Z}})\)

A functor \(T\) is right exact if a short exact sequence

\begin{align*}0 \to A \to B \to C \to 0 \end{align*} yields an exact sequence

\begin{align*}\ldots TA \to TB \to TC \to 0 \end{align*} and is left exact if it yields

\begin{align*}0 \to TA \to TB \to TC \to \ldots \end{align*} Thus a functor is exact iff it is both left and right exact, yielding

\begin{align*}0 \to TA \to TB \to TC \to 0 .\end{align*}

\({\,\cdot\,}\otimes_{R} {\,\cdot\,}\) is a right exact bifunctor.

An \(R{\hbox{-}}\)module is flat if \(A\otimes_{R} {\,\cdot\,}\) is an exact functor.

An action \(G\curvearrowright X\) is properly discontinuous if each \(x\in X\) has a neighborhood \(U\) such that all of the images \(g(U)\) for \(g\in G\) are disjoint, i.e. \(g_1(U) \cap g_2(U) \neq \emptyset \implies g_1 = g_2\). The action is free if there are no fixed points.

Sometimes a slightly weaker condition is used: every point \(x\in X\) has a neighborhood \(U\) such that \(U \cap G(U) \neq \emptyset\) for only finitely many \(G\).

A \({\hbox{-}}\)module \(M\) with a basis \(S = \left\{{s_{i}}\right\}\) of generating elements. Every such module is the image of a unique map \(\mathcal{F}(S) = R^S \twoheadrightarrow M\), and if \(M = \left< S \mathrel{\Big|}\mathcal{R} \right>\) for some set of relations \(\mathcal{R}\), then \(M \cong R^S / \mathcal{R}\).

For a connected, closed, orientable manifold, \([M]\) is a generator of \(H_{n}(M; {\mathbb{Z}}) = {\mathbb{Z}}\).

\(S = \left\{{s_{i}}\right\}\) is a generating set for an \(R{\hbox{-}}\) module \(M\) iff \begin{align*}x\in M \implies x = \sum r_{i} s_{i}\end{align*} for some coefficients \(r_{i} \in R\) (where this sum may be infinite).

Let \(X, Y\) be topological spaces and \(f,g: X \to Y\) continuous maps. Then a homotopy from \(f\) to \(g\) is a continuous function

\(F: X \times I \to Y\)

such that

\(F(x, 0) = f(x)\) and \(F(x,1) = g(x)\)

for all \(x\in X\). If such a homotopy exists, we write \(f\simeq g\). This is an equivalence relation on \(\text{Hom}(X,Y)\), and the set of such classes is denoted \([X,Y] \mathrel{\vcenter{:}}=\hom (X,Y)/\simeq\).

Let \(f: X \to Y\) be a continuous map, then \(f\) is said to be a homotopy equivalence if there exists a continuous map \(g: X \to Y\) such that

\(f\circ g \simeq\operatorname{id}_Y\) and \(g\circ f \simeq\operatorname{id}_X\).

Such a map \(g\) is called a homotopy inverse of \(f\), the pair of maps is a homotopy equivalence.

If such an \(f\) exists, we write \(X \simeq Y\) and say \(X\) and \(Y\) have the same homotopy type, or that they are homotopy equivalent.

Note that homotopy equivalence is strictly weaker than homeomorphic equivalence, i.e., \(X\cong Y\) implies \(X \simeq Y\) but not necessarily the converse.

For a manifold \(M\), a map on homology defined by \begin{align*} H_{\widehat{i}}M \otimes H_{\widehat{j}}M \to H_{\widehat{i+j}}X\\ \alpha\otimes\beta \mapsto \left< \alpha, \beta \right> \end{align*} obtained by conjugating the cup product with Poincaré Duality, i.e. 

\begin{align*}\left< \alpha, \beta \right> = [M] \frown ([\alpha]^\vee\smile [\beta]^\vee) .\end{align*}

Then, if \([A], [B]\) are transversely intersecting submanifolds representing \(\alpha, \beta\), then \begin{align*}\left<\alpha, \beta\right> = [A\cap B]\end{align*} . If \(\widehat{i} = j\) then \(\left< \alpha, \beta \right> \in H_{0} M = {\mathbb{Z}}\) is the signed number of intersection points.

Alt: The pairing obtained from dualizing Poincare Duality to obtain \begin{align*}\mathrm{F}(H_{i} M) \otimes\mathrm{F}(H_{n-i}M) \to {\mathbb{Z}}\end{align*} Computed as an oriented intersection number between two homology classes (perturbed to be transverse).

The nondegenerate bilinear form cohomology induced by the Kronecker Pairing: \begin{align*}I: H^k(M_{n}) \times H^{n-k}(M^n) \to {\mathbb{Z}}\end{align*} where \(n=2k\).

A map pairing a chain with a cochain, given by \begin{align*} H^n(X; R) \times H_{n}(X; R) \to R \\ ([\psi, \alpha]) \mapsto \psi(\alpha) \end{align*} which is a nondegenerate bilinear form.

At a point \(x \in V \subset M\), a generator of \(H_{n}(V, V-\left\{{x}\right\})\). The degree of a map \(S^n \to S^n\) is the sum of its local degrees.

A generating \(S\) for a module \(M\) is linearly independent if \(\sum r_{i} s_{i} = 0_M \implies \forall i,~r_{i} = 0\) where \(s_{i}\in S, r_{i} \in R\).

\(H_{n}(X, X-A; {\mathbb{Z}})\) is the local homology at \(A\), also denoted \(H_{n}(X \mathrel{\Big|}A)\)

At a point \(x\in M^n\), a choice of a generator \(\mu_{x}\) of \(H_{n}(M, M - \left\{{x}\right\}) = {\mathbb{Z}}\).

An \(n{\hbox{-}}\)manifold is a Hausdorff space in which each neighborhood has an open neighborhood homeomorphic to \({\mathbb{R}}^n\).

A manifold in which open neighborhoods may be isomorphic to either \({\mathbb{R}}^n\) or a half-space \(\left\{{\mathbf{x} \in {\mathbb{R}}^n \mathrel{\Big|}x_{i} > 0}\right\}\).

A covering space is normal if and only if for every \(x\in X\) and every pair of lifts \(\tilde x_1, \tilde x_2\), there is a deck transformation \(f\) such that \(f(\tilde x_1) = \tilde x_2\).

A map \(X\xrightarrow{f} Y\) is nullhomotopic if it is homotopic to a constant map \(X \xrightarrow{g} \left\{{y_{0}}\right\}\); that is, there exists a homotopy \begin{align*} F: X\times I &\to Y \\ {\left.{{F}} \right|_{{X\times\left\{{0}\right\}}} } &= f \quad F(x, 0) = f(x) \\ {\left.{{F}} \right|_{{X\times\left\{{1}\right\}}} } &= g \quad F(x, 1) = g(x) = y_{0}\\ .\end{align*}

Alt:

If \(f\) is homotopic to a constant map, say \(f: x \mapsto y_0\) for some fixed \(y_0 \in Y\), then \(f\) is said to be nullhomotopic. In other words, if \(f:X\to Y\) is nullhomotopic, then there exists a homotopy \(H: X\times I \to Y\) such that \(H(x, 0) = f(x)\) and \(H(x, 1) = y_0\).

Note that constant maps (or anything homotopic) induce zero homomorphisms.

For a group action \(G\curvearrowright X\), the orbit space \(X/G\) is defined as \(X/\sim\) where \(\forall x,y\in X, x\sim y \iff \exists g\in G \mathrel{\Big|}g.x = y\).

A manifold for which an orientation exists, see “Orientation of a Manifold.”

For any manifold \(M\), a two sheeted orientable covering space \(\tilde M_{o}\). \(M\) is orientable iff \(\tilde M\) is disconnected. Constructed as \begin{align*} \tilde M = \coprod_{x\in M}\left\{{\mu_{x} \mathrel{\Big|}\mu_{x}~ \text{is a local orientation}}\right\} .\end{align*}

A family of \(\left\{{\mu_{x}}\right\}_{x\in M}\) with local consistency: if \(x,y \in U\) then \(\mu_{x}, \mu_{y}\) are related via a propagation.

Formally, a function \begin{align*}M^n \to \coprod_{x\in M} H(X \mathrel{\Big|}\left\{{x}\right\})\\ x \mapsto \mu_{x}\end{align*} such that \(\forall x \exists N_{x}\) in which \(\forall y\in N_{x}\), the preimage of each \(\mu_{y}\) under the map \(H_{n}(M\mathrel{\Big|}N_{x}) \twoheadrightarrow H_{n}(M\mathrel{\Big|}y)\) is a single generator \(\mu_{N_{x}}\).

TFAE:

A pairing alone is an \(R{\hbox{-}}\)bilinear module map, or equivalently a map out of a tensor product since \(p: M\otimes_{R} N \to L\) can be partially applied to yield \(\phi: M \to L^N = \hom_{R}(N, L)\). A pairing is perfect when \(\phi\) is an isomorphism.

For a closed, orientable \(n{\hbox{-}}\)manifold, following map \([M] \frown {\,\cdot\,}\) is an isomorphism: \begin{align*} D: H^k(M; R) \to H_{n-k}(M; R) \\ D(\alpha) = [M] \frown \alpha\end{align*}

A space \(X\) is semilocally simply connected if every \(x\in X\) has a neighborhood \(U\) such that \(U\hookrightarrow X\) induces the trivial map \(\pi_1(U;x) \to \pi_1(X, x)\).

Given a simplex \(\sigma = [v_1 \cdots v_n]\), define the face map \begin{align*} {\partial}_i:\Delta^n &\to \Delta^{n-1} \\ \sigma &\mapsto [v_1 \cdots \widehat{v}_i \cdots v_n] \end{align*}

A simplicial complex is a set \(K\) satisfying

  1. \(\sigma \in K \implies {\partial}_i\sigma \in K\).

  2. \(\sigma,\tau\in K \implies \sigma\cap\tau = \emptyset,~ {\partial}_i\sigma,~\text{or}~{\partial}_i\tau\).

This amounts to saying that any collection of \((n-1)\)-simplices uniquely determines an \(n\)-simplex (or its lack thereof), or that that map \(\Delta^k \to X\) is a continuous injection from the standard simplex in \({\mathbb{R}}^n\).

  1. \({\left\lvert {K\cap B_\varepsilon(\sigma)} \right\rvert} < \infty\) for every \(\sigma\in K\), identifying \(\sigma \subseteq {\mathbb{R}}^n\).

For a map \begin{align*}K\xrightarrow{f} L\end{align*} between simplicial complexes, \(f\) is a simplicial map if for any set of vertices \(\left\{{v_{i}}\right\}\) spanning a simplex in \(K\), the set \(\left\{{f(v_{i})}\right\}\) are the vertices of a simplex in \(L\).

A space \(X\) is simply connected if and only if \(X\) is path-connected and every loop \(\gamma : S^1 \xrightarrow{} X\) can be contracted to a point.

Equivalently, there exists a lift \(\widehat{\gamma }: D^2 \xrightarrow{} X\) such that \({ \left.{{\widehat{\gamma}}} \right|_{{{{\partial}}D^2}} } = \gamma\).

Equivalently, for any two paths \(p_1, p_2: I \xrightarrow{} X\) such that \(p_1(0) = p_2(0)\) and \(p_1(1) = p_2(1)\), there exists a homotopy \(F: I^2 \xrightarrow{} X\) such that \({ \left.{{F}} \right|_{{0}} } = p_1,\, { \left.{{F}} \right|_{{0}} } = p_2\).

Equivalently, \(\pi _1 X = 1\) is trivial.

\begin{align*}x \in C_{n}(x) \implies X = \sum_{i} n_{i} \sigma_{i} = \sum_{i} n_{i} (\Delta^n \xrightarrow{\sigma_{i}} X) .\end{align*}

\begin{align*}x \in C^n(x) \implies X = \sum_{i} n_{i} \psi_{i} = \sum_{i} n_{i} (\sigma_{i} \xrightarrow{\psi_{i}} X) .\end{align*}

Compact represented as \(\Sigma X = CX \coprod_{\operatorname{id}_{X}} CX\), two cones on \(X\) glued along \(X\). Explicitly given by

\begin{align*}\Sigma X = \frac{X\times I}{(X\times\left\{{0}\right\}) \cup(X\times\left\{{1}\right\}) \cup(\left\{{x_{0}}\right\} \times I)} .\end{align*}

For an \(R{\hbox{-}}\)module \begin{align*} \operatorname{Tor}_{R}^n({\,\cdot\,}, B) = L_{n}({\,\cdot\,}\otimes_{R} B) ,\end{align*} where \(L_{n}\) denotes the \(n\)th left derived functor.

4 Examples

4.1 Point-Set

4.1.1 Common Spaces and Operations

The following are some standard “nice” spaces: \begin{align*} S^n, {\mathbb{D}}^n, T^n, {\mathbb{RP}}^n, {\mathbb{CP}}^n, \mathbb{M}, \mathbb{K}, \Sigma_{g}, {\mathbb{RP}}^\infty, {\mathbb{CP}}^\infty .\end{align*}

The following are useful spaces to keep in mind to furnish counterexamples:

Examples of some more exotic spaces that show up less frequently:

The following spaces are non-Hausdorff:

The following are some examples of ways to construct specific spaces for examples or counterexamples:

Operations

4.1.2 Alternative Topologies

The following are some nice examples of topologies to put on familiar spaces to produce counterexamples:

The cofinite topology on any space \(X\) is always

A topology \((X, \tau)\) is the discrete topology iff points \(x\in X\) are open.

(Click to expand)

If \(\left\{{x}\right\}_i\) is open for each \(x_i \in X\), then

Thus \(U\) is open.

Some facts about the discrete topology:

Some facts about the indiscrete topology:

5 Theorems

The following properties are “pushed forward” through continuous maps, in the sense that if property \(P\) holds for \(X\) and \(f:X\to Y\), then \(f(X)\) also satisfies \(P\):

The following are not preserved:

See more here.

5.1 Metric Spaces and Analysis

A bounded collection of nested closed sets \(C_1 \supset C_2 \supset \cdots\) in a metric space \(X\) is nonempty \(\iff X\) is complete.

If \(\left\{{[a_n, b_n] {~\mathrel{\Big|}~}n\in {\mathbb{Z}}^{\geq 0}}\right\}\) is a nested sequence of closed and bounded intervals, then their intersection is nonempty. If \({\operatorname{diam}}([a_n, b_n]) \overset{n\to\infty}0\), then the intersection contains exactly one point.

A continuous function on a compact set is uniformly continuous.

(Click to expand)

Take \(\left\{{B_{\varepsilon\over 2}(y) {~\mathrel{\Big|}~}y\in Y}\right\}\rightrightarrows Y\), pull back to an open cover of \(X\), has Lebesgue number \(\delta_L > 0\), then \(x' \in B_{\delta_L}(x) \implies f(x), f(x') \in B_{\varepsilon\over 2}(y)\) for some \(y\).

Lipschitz continuity implies uniform continuity (take \(\delta = \varepsilon/C\))

Counterexample to the converse: \(f(x) = \sqrt x\) on \([0, 1]\) has unbounded derivative.

For \(f:X \to Y\) continuous with \(X\) compact and \(Y\) ordered in the order topology, there exist points \(c, d\in X\) such that \(f(x) \in [f(c), f(d)]\) for every \(x\).

A metric space \(X\) is sequentially compact iff it is complete and totally bounded.

A metric space is totally bounded iff every sequence has a Cauchy subsequence.

A metric space is compact iff it is complete and totally bounded.

If \(X\) is a complete metric space, \(X\) is a Baire space: the intersection of countably many dense open sets in \(X\) is again dense in \(X\).

5.2 Compactness

\(U\subset X\) a Hausdorff spaces is closed \(\iff\) it is compact.

A closed subset \(A\) of a compact set \(B\) is compact.

(Click to expand)

The continuous image of a compact set is compact.

A closed subset of a Hausdorff space is compact.

5.3 Separability

A retract of a Hausdorff/connected/compact space is closed/connected/compact respectively.

Points are closed in \(T_1\) spaces.

5.4 Maps and Homeomorphism

A continuous bijection \(f: X\to Y\) where \(X\) is compact and \(Y\) is Hausdorff is an open map and hence a homeomorphism.

Every space has at least one retraction - for example, the constant map \(r:X \to\left\{{x_0}\right\}\) for any \(x\_0 \in X\).

A continuous bijective open map is a homeomorphism.

For \(f:X\to Y\), TFAE:

(Click to expand)

See Munkres page 104.

If \(f:X\to Y\) is continuous where \(X\) is compact and \(Y\) is Hausdorff, then

5.5 The Tube Lemma

Let \(X, Y\) be spaces with \(Y\) compact. For each \(U \subseteq X \times Y\) and each slice \(\left\{{x}\right\} \times Y \subseteq U\), there is an open \(O \subseteq X\) such that \begin{align*} \left\{{x}\right\} \times Y \subseteq O \times Y \subseteq U .\end{align*}

(Click to expand)

6 Summary of Standard Topics

7 Examples: Algebraic Topology

7.1 Standard Spaces and Modifications

\begin{align*} {\mathbb{D}}^n = \mathbb{B}^n &\mathrel{\vcenter{:}}=\left\{{ \mathbf{x} \in {\mathbb{R}}^{n} {~\mathrel{\Big|}~}{\left\lVert {\mathbf{x}} \right\rVert} \leq 1}\right\} {\mathbb{S}}^n &\mathrel{\vcenter{:}}=\left\{{ \mathbf{x} \in {\mathbb{R}}^{n+1} {~\mathrel{\Big|}~}{\left\lVert {\mathbf{x}} \right\rVert} = 1}\right\} = {{\partial}}{\mathbb{D}}^n \\ .\end{align*}

Note: I’ll immediately drop the blackboard notation, this is just to emphasize that they’re “canonical” objects.

The sphere can be constructed in several equivalent ways:

Note the subtle differences in dimension: \(S^n\) is a manifold of dimension \(n\) embedded in a space of dimension \(n+1\).

Low Dimensional Discs/Balls vs Spheres

Constructed in one of several equivalent ways:

One can also define \({\mathbb{RP}}^ \infty \mathrel{\vcenter{:}}=\varinjlim_{n} {\mathbb{RP}}^n\). Fits into a fiber bundle of the form

Defined in a similar ways,

Can similarly define \({\mathbb{CP}}^ \infty \mathrel{\vcenter{:}}=\varinjlim_n {\mathbb{CP}}^n\). Fits into a fiber bundle of the form

The \(n{\hbox{-}}\)torus, defined as \begin{align*} T^n \mathrel{\vcenter{:}}=\prod_{j=1}^n S^1 = S^1 \times S^1 \times \cdots .\end{align*}

The real Grassmannian, \({\operatorname{Gr}}(n, k)_{/{\mathbb{R}}}\), i.e. the set of \(k\) dimensional subspaces of \({\mathbb{R}}^n\). One can similar define \({\operatorname{Gr}}(n, k)_{{\mathbb{C}}}\) for complex subspaces. Note that \({\mathbb{RP}}^n = {\operatorname{Gr}}(n, 1)_{{\mathbb{R}}}\) and \({\mathbb{CP}}^n = {\operatorname{Gr}}(n, 1)_{/{\mathbb{C}}}\).

The Stiefel manifold \(V_{n}(k)_{{\mathbb{R}}}\), the space of orthonormal \(k{\hbox{-}}\)frames in \({\mathbb{R}}^n\)?

Lie Groups:

Some other spaces that show up, but don’t usually have great algebraic topological properties:

\(K(G, n)\) is an Eilenberg-MacLane space, the homotopy-unique space satisfying \begin{align*} \pi_{k}(K(G, n)) = \begin{cases} G & k=n, \\ 0 & \text{else} \end{cases} \end{align*}

Some known examples:

\(M(G, n)\) is a Moore space, the homotopy-unique space satisfying \begin{align*} H_{k}(M(G, n); G) = \begin{cases} G & k=n, \\ 0 & k\neq n. \end{cases} \end{align*}

Some known examples:

In low dimensions, there are some “accidental” homeomorphisms:

7.2 Modifying Known Spaces

Write \(D(k, X)\) for the space \(X\) with \(k\in {\mathbb{N}}\) distinct points deleted, i.e. the punctured space \(X - \left\{{x_{1}, x_{2}, \ldots x_{k}}\right\}\) where each \(x_{i} \in X\).

The “generalized uniform bouquet?” \(\mathcal{B}^n(m) = \bigvee_{i=1}^n S^m\). There’s no standard name for this, but it’s an interesting enough object to consider!

Possible modifications to a space \(X\):

8 Low Dimensional Homology Examples

\begin{align*} \begin{array}{cccccccccc} S^1 &= &[&{\mathbb{Z}}, &{\mathbb{Z}}, &0, &0, &0, &0\rightarrow & ]\\ {\mathcal{M}}&= &[&{\mathbb{Z}}, &{\mathbb{Z}}, &0, &0, &0, &0\rightarrow & ]\\ {\mathbb{RP}}^1 &= &[&{\mathbb{Z}}, &{\mathbb{Z}}, &0, &0, &0, &0\rightarrow & ]\\ {\mathbb{RP}}^2 &= &[&{\mathbb{Z}}, &{\mathbb{Z}}_{2}, &0, &0, &0, &0\rightarrow & ]\\ {\mathbb{RP}}^3 &= &[&{\mathbb{Z}}, &{\mathbb{Z}}_{2}, &0, &{\mathbb{Z}}, &0, &0\rightarrow & ]\\ {\mathbb{RP}}^4 &= &[&{\mathbb{Z}}, &{\mathbb{Z}}_{2}, &0, &{\mathbb{Z}}_{2}, &0, &0\rightarrow & ]\\ S^2 &= &[&{\mathbb{Z}}, &0, &{\mathbb{Z}}, &0, &0, &0\rightarrow & ]\\ {\mathbb{T}}^2 &= &[&{\mathbb{Z}}, &{\mathbb{Z}}^2, &{\mathbb{Z}}, &0, &0, &0\rightarrow & ]\\ {\mathbb{K}}&= &[&{\mathbb{Z}}, &{\mathbb{Z}}\oplus {\mathbb{Z}}_{2}, &0, &0, &0, &0\rightarrow & ]\\ {\mathbb{CP}}^1 &= &[&{\mathbb{Z}}, &0, &{\mathbb{Z}}, &0, &0, &0\rightarrow & ]\\ {\mathbb{CP}}^2 &= &[&{\mathbb{Z}}, &0, &{\mathbb{Z}}, &0, &{\mathbb{Z}}, &0\rightarrow & ]\\ \end{array} .\end{align*}

9 Table of Homotopy and Homology Structures

The following is a giant list of known homology/homotopy.

\(X\) \(\pi_*(X)\) \(H_*(X)\) CW Structure \(H^*(X)\)
\({\mathbb{R}}^1\) \(0\) \(0\) \({\mathbb{Z}}\cdot 1 + {\mathbb{Z}}\cdot x\) 0
\({\mathbb{R}}^n\) \(0\) \(0\) \(({\mathbb{Z}}\cdot 1 + {\mathbb{Z}}\cdot x)^n\) 0
\(D(k, {\mathbb{R}}^n)\) \(\pi_*\bigvee^k S^1\) \(\bigoplus_{k} H_* M({\mathbb{Z}}, 1)\) \(1 + kx\) ?
\(B^n\) \(\pi_*({\mathbb{R}}^n)\) \(H_*({\mathbb{R}}^n)\) \(1 + x^n + x^{n+1}\) 0
\(S^n\) \([0 \ldots , {\mathbb{Z}}, ? \ldots]\) \(H_*M({\mathbb{Z}}, n)\) \(1 + x^n\) or \(\sum_{i=0}^n 2x^i\) \({\mathbb{Z}}[{}_{n}x]/(x^2)\)
\(D(k, S^n)\) \(\pi_*\bigvee^{k-1}S^1\) \(\bigoplus_{k-1}H_*M({\mathbb{Z}}, 1)\) \(1 + (k-1)x^1\) ?
\(T^2\) \(\pi_*S^1 \times \pi_* S^1\) \((H_* M({\mathbb{Z}}, 1))^2 \times H_* M({\mathbb{Z}}, 2)\) \(1 + 2x + x^2\) \(\Lambda({}_{1}x_{1}, {}_{1}x_{2})\)
\(T^n\) \(\prod^n \pi_* S^1\) \(\prod_{i=1}^n (H_* M({\mathbb{Z}}, i))^{n\choose i}\) \((1 + x)^n\) \(\Lambda({}_{1}x_{1}, {}_{1}x_{2}, \ldots {}_{1}x_{n})\)
\(D(k, T^n)\) \([0, 0, 0, 0, \ldots]\)? \([0, 0, 0, 0, \ldots]\)? \(1 + x\) ?
\(S^1 \vee S^1\) \(\pi_*S^1 \ast \pi_* S^1\) \((H_*M({\mathbb{Z}}, 1))^2\) \(1 + 2x\) ?
\(\bigvee^n S^1\) \(\ast^n \pi_* S^1\) \(\prod H_* M({\mathbb{Z}}, 1)\) \(1 + x\) ?
\({\mathbb{RP}}^1\) \(\pi_* S^1\) \(H_* M({\mathbb{Z}}, 1)\) \(1 + x\) \({}_{0}{\mathbb{Z}}\times {}_{1}{\mathbb{Z}}\)
\({\mathbb{RP}}^2\) \(\pi_*K({\mathbb{Z}}/2{\mathbb{Z}}, 1)+ \pi_* S^2\) \(H_*M({\mathbb{Z}}/2{\mathbb{Z}}, 1)\) \(1 + x + x^2\) \({}_{0}{\mathbb{Z}}\times {}_{2}{\mathbb{Z}}/2{\mathbb{Z}}\)
\({\mathbb{RP}}^3\) \(\pi_*K({\mathbb{Z}}/2{\mathbb{Z}}, 1)+ \pi_* S^3\) \(H_*M({\mathbb{Z}}/2{\mathbb{Z}}, 1) + H_*M({\mathbb{Z}}, 3)\) \(1 + x + x^2 + x^3\) \({}_{0}{\mathbb{Z}}\times {}_{2}{\mathbb{Z}}/2{\mathbb{Z}}\times {}_{3}{\mathbb{Z}}\)
\({\mathbb{RP}}^4\) \(\pi_*K({\mathbb{Z}}/2{\mathbb{Z}}, 1)+ \pi_* S^4\) \(H_*M({\mathbb{Z}}/2{\mathbb{Z}}, 1) + H_*M({\mathbb{Z}}/2{\mathbb{Z}}, 3)\) \(1 + x + x^2 + x^3 + x^4\) \({}_{0}{\mathbb{Z}}\times ({}_{2}{\mathbb{Z}}/2{\mathbb{Z}})^2\)
\({\mathbb{RP}}^n, n \geq 4\) even \(\pi_*K({\mathbb{Z}}/2{\mathbb{Z}}, 1)+ \pi_*S^n\) \(\prod_{\text{odd}~i < n} H_*M({\mathbb{Z}}/2{\mathbb{Z}}, i)\) \(\sum_{i=1}^n x^i\) \({}_{0}{\mathbb{Z}}\times \prod_{i=1}^{n/2}{}_{2}{\mathbb{Z}}/2{\mathbb{Z}}\)
\({\mathbb{RP}}^n, n \geq 4\) odd \(\pi_*K({\mathbb{Z}}/2{\mathbb{Z}}, 1)+ \pi_*S^n\) \(\prod_{\text{odd}~ i \leq n-2} H_*M({\mathbb{Z}}/2{\mathbb{Z}}, i) \times H_* S^n\) \(\sum_{i=1}^n x^i\) \(H^*({\mathbb{RP}}^{n-1}) \times {}_{n}{\mathbb{Z}}\)
\({\mathbb{CP}}^1\) \(\pi_*K({\mathbb{Z}}, 2) + \pi_* S^3\) \(H_* S^2\) \(x^0 + x^2\) \({\mathbb{Z}}[{}_{2}x]/({}_2x^{2})\)
\({\mathbb{CP}}^2\) \(\pi_*K({\mathbb{Z}}, 2) + \pi_* S^5\) \(H_*S^2 \times H_* S^4\) \(x^0 + x^2 + x^4\) \({\mathbb{Z}}[{}_{2}x]/({}_2x^{3})\)
\({\mathbb{CP}}^n, n \geq 2\) \(\pi_*K({\mathbb{Z}}, 2) + \pi_*S^{2n+1}\) \(\prod_{i=1}^n H_* S^{2i}\) \(\sum_{i=1}^n x^{2i}\) \({\mathbb{Z}}[{}_{2}x]/({}_2x^{n+1})\)
Mobius Band \(\pi_* S^1\) \(H_* S^1\) \(1 + x\) ?
Klein Bottle \(K({\mathbb{Z}}\rtimes_{-1} {\mathbb{Z}}, 1)\) \(H_*S^1 \times H_* {\mathbb{RP}}^\infty\) \(1 + 2x + x^2\) ?

\begin{align*} \phi: (D^n, {\partial}D^n) &\to S^n \\ (D^n, {\partial}D^n) &\mapsto (e^n, e^0) .\end{align*}

10 Theorems: Algebraic Topology

10.1 General Homotopies

\begin{align*} X\times{\mathbb{R}}^n \simeq X \times{\{\operatorname{pt}\}}\cong X .\end{align*}

The ranks of \(\pi_{0}\) and \(H_{0}\) are the number of path components.

Any two continuous functions into a convex set are homotopic.

(Click to expand)

The linear homotopy. Supposing \(X\) is convex, for any two points \(x,y\in X\), the line \(tx + (1-t)y\) is contained in \(X\) for every \(t\in[0,1]\). So let \(f, g: Z \to X\) be any continuous functions into \(X\). Then define \(H: Z \times I \to X\) by \(H(z,t) = tf(z) + (1-t)g(z)\), the linear homotopy between \(f,g\). By convexity, the image is contained in \(X\) for every \(t,z\), so this is a homotopy between \(f,g\).

10.2 Fundamental Group

10.2.1 Definition

Given a pointed space \((X,x_{0})\), we define the fundamental group \(\pi_{1}(X)\) as follows:

\begin{align*} H: &S^1 \times I \to X \\ & \begin{cases} H(s, 0) = \alpha(s)\\ H(s, 1) = \beta(s) , \end{cases} \end{align*} - Check that this relation is

Elements of the fundamental group are homotopy classes of loops, and every continuous map between spaces induces a homomorphism on fundamental groups.

10.2.2 Conjugacy in \(\pi_{1}\):

10.2.3 Calculating \(\pi_1\)

If \(\tilde X \to X\) the universal cover of \(X\) and \(G\curvearrowright\tilde X\) with \(\tilde X/G = X\) then \(\pi_1(X) = G\).

\(\pi_1 X\) for \(X\) a CW-complex only depends on the 2-skeleton \(X^{2}\), and in general \(\pi_k(X)\) only depends on the \(k+2\)-skeleton. Thus attaching \(k+2\) or higher cells does not change \(\pi_k\).

Suppose \(X = U_{1} \cup U_{2}\) where \(U_1, U_2\), and \(U \mathrel{\vcenter{:}}= U_{1} \cap U_{2} \neq \emptyset\) are open and path-connected2

, and let \(x_0 \in U\).

Then the inclusion maps \(i_{1}: U_{1} \hookrightarrow X\) and \(i_{2}: U_{2} \hookrightarrow X\) induce the following group homomorphisms: \begin{align*} i_{1}^*: \pi_{1}(U_{1}, x_0) \to\pi_{1}(X, x_0) \\ i_{2}^*: \pi_{1}(U_{2}, x_0) \to\pi_{1}(X, x_0) \end{align*}

There is a natural isomorphism \begin{align*} \pi_{1}(X) \cong \pi_{1} U \ast_{\pi_{1}(U \cap V)} \pi_{1} V ,\end{align*}

where the amalgamated product can be computed as follows: A pushout is the colimit of the following diagram

Example of a pushout of spaces

For groups, the pushout is realized by the amalgamated free product: if \begin{align*} \begin{cases} \pi_1 U_1 = A = \left\langle{G_{A} {~\mathrel{\Big|}~}R_{A}}\right\rangle \\ \pi_1 U_2 = B = \left\langle{G_{B} {~\mathrel{\Big|}~}R_{B}}\right\rangle \end{cases} \implies A \ast_{Z} B \mathrel{\vcenter{:}}=\left\langle{ G_{A}, G_{B} {~\mathrel{\Big|}~}R_{A}, R_{B}, T}\right\rangle \end{align*} where \(T\) is a set of relations given by \begin{align*} T = \left\{{\iota_{1}^*(z) \iota_{2}^* (z) ^{-1} {~\mathrel{\Big|}~}z\in \pi_1 (U_1 \cap U_2)}\right\} ,\end{align*} where \(\iota_2^*(z) ^{-1}\) denotes the inverse group element. If we have presentations

\begin{align*} \pi_{1}(U, x_0) &= \left\langle u_{1}, \cdots, u_{k} {~\mathrel{\Big|}~}\alpha_{1}, \cdots, \alpha_{l}\right\rangle \\ \pi_{1}(V, w) &=\left\langle v_{1}, \cdots, v_{m} {~\mathrel{\Big|}~}\beta_{1}, \cdots, \beta_{n}\right\rangle \\ \pi_{1}(U \cap V, x_0) &=\left\langle w_{1}, \cdots, w_{p} {~\mathrel{\Big|}~}\gamma_{1}, \cdots, \gamma_{q}\right\rangle \end{align*}

then \begin{align*} \pi_{1}(X, w) &= \left\langle u_{1}, \cdots, u_{k}, v_{1}, \cdots, v_{m} \middle\vert \begin{cases} \alpha_{1}, \cdots, \alpha_{l} \\ \beta_{1}, \cdots, \beta_{n} \\ I\left(w_{1}\right) J\left(w_{1}\right)^{-1}, \cdots, I\left(w_{p}\right) J\left(w_{p}\right)^{-1} \\ \end{cases} \right\rangle \\ \\ &= \frac{ \pi_{1}(U_1) \ast \pi_{1}(U_2) } { \left\langle{ \left\{{\iota_1^*(w_{i}) \iota_2^*(w_{i})^{-1}{~\mathrel{\Big|}~}1\leq i \leq p}\right\} }\right\rangle } \end{align*}

(Click to expand)

\(A = {\mathbb{Z}}/4{\mathbb{Z}}= \left\langle{x {~\mathrel{\Big|}~}x^4}\right\rangle, B = {\mathbb{Z}}/6{\mathbb{Z}}= \left\langle{y {~\mathrel{\Big|}~}x^6}\right\rangle, Z = {\mathbb{Z}}/2{\mathbb{Z}}= \left\langle{z {~\mathrel{\Big|}~}z^2}\right\rangle\). Then we can identify \(Z\) as a subgroup of \(A, B\) using \(\iota_{A}(z) = x^2\) and \(\iota_{B}(z) = y^3\). So \begin{align*}A\ast_{Z} B = \left\langle{x, y {~\mathrel{\Big|}~}x^4, y^6, x^2y^{-3}}\right\rangle\end{align*} .

\begin{align*} \pi_1(X \vee Y) = \pi_1(X) \ast \pi_1(Y) .\end{align*}

(Click to expand)

By van Kampen, this is equivalent to the amalgamated product over \(\pi_1(x_0) = 1\), which is just a free product.

10.2.4 Facts

\(H_{1}\) is the abelianization of \(\pi_{1}\).

If \(X, Y\) are path-connected, then \begin{align*} \pi_1 (X \times Y) = \pi_1(X) \times\pi_2(Y) .\end{align*}

(Click to expand)

\(\pi_{1}(X) = 1\) iff \(X\) is simply connected.

(Click to expand)

\(\Rightarrow\): Suppose \(X\) is simply connected. Then every loop in \(X\) contracts to a point, so if \(\alpha\) is a loop in \(X\), \([\alpha] = [\operatorname{id}_{x_{0}}]\), the identity element of \(\pi_{1}(X)\). But then there is only one element in in this group.

\(\Leftarrow\): Suppose \(\pi_{1}(X) = 0\). Then there is just one element in the fundamental group, the identity element, so if \(\alpha\) is a loop in \(X\) then \([\alpha] = [\operatorname{id}_{x_{0}}]\). So there is a homotopy taking \(\alpha\) to the constant map, which is a contraction of \(\alpha\) to a point.

:::{.fact “Unsorted facts”}

:::

10.3 General Homotopy Theory

A map \(X \xrightarrow{f} Y\) on CW complexes that is a weak homotopy equivalence (inducing isomorphisms in homotopy) is in fact a homotopy equivalence.

Individual maps may not work: take \(S^2 \times{\mathbb{RP}}^3\) and \(S^3 \times{\mathbb{RP}}^2\) which have isomorphic homotopy but not homology.

The Hurewicz map on an \(n-1{\hbox{-}}\)connected space \(X\) is an isomorphism \(\pi_{k\leq n}X \to H_{k\leq n} X\).

I.e. for the minimal \(i\geq 2\) for which \(\pi_{iX} \neq 0\) but \(\pi_{\leq i-1}X = 0\), \(\pi_{iX} \cong H_{iX}\).

Any continuous map between CW complexes is homotopy equivalent to a cellular map.

:::{.fact title="Unsorted facts about higher homotopy groups}

:::

11 Covering Spaces

Some pictures to keep in mind when it comes to covers and path lifting: