References:

Some fun resources:

\(A\times B, \prod X_j\) are direct products.

\(A\oplus B, \bigoplus_j X_j\) are direct sums, the subset of \(A\times B\) where only finitely many terms are nonzero.

- \({{\mathbb{Z}}}^n\) denotes the direct sum of \(n\) copies of the group \({{\mathbb{Z}}}\).
- Note that \(A \oplus B \hookrightarrow A\times B\).

\(A\ast B, \ast_j X_J\) are free products, \(F_n \mathrel{\vcenter{:}}={\mathbb{Z}}^{\ast n}\) is the free group on \(n\) generators.

- Note that the abelianization yields \(\qty{\ast_j X_j} = \bigoplus_j X_j\).

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*}

Spaces are assumed to be connected and path connected, so \(\pi_0(X) = H_0(X) = {\mathbb{Z}}\).

Graded objects like \(\pi_*, H_*, H^*\) are sometimes represented as lists. In this case, all list entries start indexing at 1. Examples: \begin{align*} \pi_*(X) &= [\pi_1(X), \pi_2(X), \pi_3(X), \cdots] \\ H_*(X) &= [H_1(X), H_2(X), H_3(X), \cdots] .\end{align*}

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.

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 \([\alpha] \in \pi_1(A)\).

- Connectedness
- Compactness
- Hausdorff Spaces
- Path-Connectedness

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.

- A set is closed if and only if its complement is open.
- A set is closed iff it contains all of its limit points.
- A closet set in a subspace: \(Y\subset X \implies \operatorname{cl}_{Y}(A) \mathrel{\vcenter{:}}=\operatorname{cl}_{X}(A)\cap Y\).

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

- \(q\) is surjective, and
- \(U \subseteq Y\) 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*}

\(T_0\): For any 2 points \(x_1\neq x_2\), at least one \(x_i\) (say \(x_1\)) admits a neighborhood not containing \(x_2\).

\(T_1\): For any 2 points,

*both*admit neighborhoods not containing the other.\(T_2\): For any 2 points, both admit

*disjoint*separating neighborhoods.\(T_{2.5}\): For any 2 points, both admit

*disjoint closed*separating neighborhoods.\(T_3\): \(T_0\) &

*regular*. Given any point \(x\) and any closed \(F\not\ni x\), there are neighborhoods separating \(F\) and \(x\).\(T_{3.5}\): \(T_0\) & completely regular. Any point \(x\) and closed \(F\not\ni x\) can be separated by a continuous function.

\(T_4\): \(T_1\) & normal. Any two disjoint closed subsets can be separated by neighborhoods.

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*}

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:

- Products
- Pullbacks
- Inverse / projective limits
- The \(p{\hbox{-}}\)adic integers \({\mathbb{Z}}_{p}\).

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

- 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 \(U\)), - \({ \left.{{p}} \right|_{{V_i}} }: V_i \to U\) is a homeomorphism for each sheet.

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

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 \(f: {\partial}B^n \to X\).

- A \(0{\hbox{-}}\)cell is a point.
- A \(1{\hbox{-}}\)cell is an interval \([-1, 1] = B^1 \subset {\mathbb{R}}^1\). Attaching requires a map from \(S^0 =\left\{{-1, +1}\right\} \to X\)
- A \(2{\hbox{-}}\)cell is a solid disk \(B^2 \subset {\mathbb{R}}^2\) in the plane. Attaching requires a map \(S^1 \to X\).
- A \(3{\hbox{-}}\)cell is a solid ball \(B^3 \subset {\mathbb{R}}^3\). Attaching requires a map from the sphere \(S^2 \to X\).

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 \(n=2k\).

- When \(k\) is odd, \(I\) is skew-symmetric and thus a
*symplectic form.* - When \(k\) is even (and thus \(n \equiv 0 \pmod 4\)) this is a symmetric form.
- Satisfies \(I(x,y) = (-1)^{k(n-k)} I(y, x)\)

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:

- \(M\) is orientable.
- The map \(W: (M, x) \to {\mathbb{Z}}_{2}\) is trivial.
- \(\tilde M_{o} = M \coprod {\mathbb{Z}}_{2}\) (two sheets).
- \(\tilde M_{o}\) is disconnected
- The projection \(\tilde M_{o} \to M\) admits a section.

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

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

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

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 \({\mathbb{R}}^n\).

- \({\left\lvert {K\cap B_\varepsilon(\sigma)} \right\rvert} < \infty\) 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.

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:

- Finite discrete sets with the discrete topology.
- Subspaces of \({\mathbb{R}}\): \((a, b), (a, b], (a, \infty)\), etc.
- Sets given by real sequences, such as \(\left\{{0}\right\} \cup\left\{{{1 \over n}{~\mathrel{\Big|}~}n\in {\mathbb{Z}}^{\geq 1}}\right\}\)

- \({\mathbb{Q}}\)
- The topologist’s sine curve
- One-point compactifications
- \({\mathbb{R}}^\omega\) for \(\omega\) the least uncountable ordinal (?)
- The Hawaiian earring
- The Cantor set

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

- \({\mathbb{HP}}^n\), quaternionic projective space
- The Dunce Cap
- The Alexander Horned sphere

The following spaces are non-Hausdorff:

- The cofinite topology on any infinite set.
- \({\mathbb{R}}/{\mathbb{Q}}\)
- The line with two origins.
- Any variety \(V(J) \subseteq {\mathbb{A}}^n_{/k}\) for \(k\) a field and \(J{~\trianglelefteq~}k[x_1, \cdots, x_{n}]\).

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

Knot complements in \(S^3\)

Covering spaces (hyperbolic geometry)

Lens spaces

Matrix groups

Prism spaces

Pair of pants

Seifert surfaces

Surgery

Simplicial Complexes

- Nice minimal example:

Operations

- Cartesian product \(A\times B\)
- Wedge product \(A \vee B\)
- Connect Sum \(A \# B\)
- Quotienting \(A/B\)
- Puncturing \(A\setminus \left\{{a_{i}}\right\}\)
- Smash product
- Join
- Cones
- Suspension
- Loop space
- Identifying a finite number of points

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

- Discrete
- Cofinite
- Discrete and Indiscrete
- Uniform

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

- Non-Hausdorff
- Compact

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

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

- Any set \(U\) can be written as \(U = \cup_{i\in I} x_I\) (for some \(I\) depending on \(U\)), and
- Unions of open sets are open.

Thus \(U\) is open.

Some facts about the discrete topology:

- Definition: every subset is open.
- Always Hausdorff
- Compact iff finite
- Totally disconnected
- If \(X\) is discrete, every map \(f:X\to Y\) for any \(Y\) is continuous (obvious!)

Some facts about the indiscrete topology:

- Definition: the only open sets are \(\emptyset, X\)
- Never Hausdorff
- If \(Y\) is indiscrete, every map \(f:X\to Y\) is continuous (obvious!)
- Always compact

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 \(P\):

- Compactness
- Separability
- If \(f\) is surjective:
- Connectedness
- Density

The following are **not preserved**:

- Openness
- Closedness

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.

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 \(X\).

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

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

- Let \(\left\{{A_i}\right\} \rightrightarrows A\) be a covering of \(A\) by sets open in \(A\).
- Each \(A_i = B_i \cap A\) for some \(B_i\) open in \(B\) (definition of subspace topology)
- Define \(V = \left\{{B_i}\right\}\), then \(V \rightrightarrows A\) is an open cover.
- Since \(A\) is closed, \(W\mathrel{\vcenter{:}}= B\setminus A\) is open
- Then \(V\cup W\) is an open cover of \(B\), and has a finite subcover \(\left\{{V_i}\right\}\)
- Then \(\left\{{V_i \cap A}\right\}\) is a finite open cover of \(A\).

The continuous image of a compact set is compact.

A closed subset of a Hausdorff space is compact.

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

Points are closed in \(T_1\) spaces.

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:

- \(f\) is continuous
- \(A\subset X \implies f(\operatorname{cl}_X(A)) \subset \operatorname{cl}_X(f(A))\)
- \(B\) closed in \(Y \implies f^{-1}(B)\) closed in \(X\).
- For each \(x\in X\) and each neighborhood \(V \ni f(x)\), there is a neighborhood \(U\ni x\) such that \(f(U) \subset V\).

See Munkres page 104.

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

- \(f\) is a closed map.
- If \(f\) is surjective, \(f\) is a quotient map.
- If \(f\) is injective, \(f\) is a topological embedding.
- If \(f\) is bijective, it is a homeomorphism.

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*}

- For each \(y\in Y\) choose neighborhoods \(A_y, B_y \subseteq Y\) such that \begin{align*} (x, y) \in A_y \times B_y \subseteq U .\end{align*}
- By compactness of \(Y\), reduce this to finitely many \(B_y \rightrightarrows Y\) so \(Y = \bigcup_{j=1}^n B_{y_j}\)
- Set \(O\mathrel{\vcenter{:}}=\cap_{j=1}^n B_{y_j}\); this works.

- Algebraic topology topics:
- Classification of compact surfaces
- Euler characteristic
- Connect sum
- Homology and cohomology groups
- Fundamental group
- Singular/cellular/simplicial homology
- Mayer-Vietoris long exact sequences for homology and cohomology
- Diagram chasing
- Degree of maps from \(S^n \to S^n\)
- Orientability, compactness
- Top-level homology and cohomology
- Reduced homology and cohomology
- Relative homology
- Homotopy and homotopy invariance
- Deformation retract
- Retract
- Excision
- Kunneth 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 maps
- Free actions
- Properly discontinuous action
- Universal cover
- Correspondence between covering spaces and subgroups of the fundamental group of the base.
- Lifting paths
- Homotopy lifting property
- Deck transformations
- The action of the fundamental group
- Normal/regular cover

\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:

- \(S^n \cong D^n / {{\partial}}D^n\): collapsing the boundary of a disc is homeomorphic to a sphere.
- \(S^n \cong D^n \displaystyle\coprod_{{{\partial}}D^n} D^n\): gluing two discs along their boundary.

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

Constructed in one of several equivalent ways:

- \(S^n/\sim\) where \(\mathbf{x} \sim -\mathbf{x}\), i.e. antipodal points are identified.
- The space of lines in \({\mathbb{R}}^{n+1}\).

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,

- Taking the unit sphere in \({\mathbb{C}}^n\) and identifying \(\mathbf{z} \sim -\mathbf{z}\).
- The space of lines in \({\mathbb{C}}^{n+1}\)

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:

- The general linear group, \(\operatorname{GL}_{n}({\mathbb{R}})\)
- The special linear group \(SL_{n}({\mathbb{R}})\)

- The orthogonal group, \(O_{n}({\mathbb{R}})\)
- The special orthogonal group, \(SO_{n}({\mathbb{R}})\)

- The real unitary group, \(U_{n}({\mathbb{C}})\)
- The special unitary group, \(SU_{n}({\mathbb{R}})\)

- The symplectic group \(Sp(2n)\)

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

- Affine \(n\)-space over a field \({\mathbb{A}}^n(k) = k^n \rtimes GL_{n}(k)\)
- The projective space \({\mathbb{P}}^n(k)\)
- The projective linear group over a ring \(R\), \(PGL_{n}(R)\)
- The projective special linear group over a ring \(R\), \(PSL_{n}(R)\)
- The modular groups \(PSL_{n}({\mathbb{Z}})\)
- Specifically \(PSL_{2}({\mathbb{Z}})\)

\(K(G, n)\) 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:

- \(K({\mathbb{Z}}, 1) = S^1\)
- \(K({\mathbb{Z}}, 2) = {\mathbb{CP}}^\infty\)
- \(K({\mathbb{Z}}/2{\mathbb{Z}}, 1) = {\mathbb{RP}}^\infty\)

\(M(G, n)\) 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:

- \(M({\mathbb{Z}}, n) = S^n\)
- \(M({\mathbb{Z}}/2{\mathbb{Z}}, 1) = {\mathbb{RP}}^2\)
- \(M({\mathbb{Z}}/p{\mathbb{Z}}, n)\) is made by attaching \(e^{n+1}\) to \(S^n\) via a degree \(p\) map.

- \({\mathcal{M}}\simeq S^1\) where \({\mathcal{M}}\) is the Mobius band.
- \({\mathbb{CP}}^n = {\mathbb{C}}^n \coprod {\mathbb{CP}}^{n-1} = \coprod_{i=0}^n {\mathbb{C}}^i\)
- \({\mathbb{CP}}^n = S^{2n+1} / S^n\)
- \(S^n / S^k \simeq S^n \vee \Sigma S^k\).

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

- \({\mathbb{RP}}^1 \cong S^1\)
- \({\mathbb{CP}}^1 \cong S^2\)
- \({\operatorname{SO}}(3) \cong {\mathbb{RP}}^2\)?

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 \(X\):

- Remove a line segment
- Remove an entire line/axis
- Remove a hole
- Quotient by a group action (e.g. antipodal map, or rotation)
- Remove a knot
- Take complement in ambient space

\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*}

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 | \(K({\mathbb{Z}}\rtimes_{-1} {\mathbb{Z}}, 1)\) | \(H_*S^1 \times H_* {\mathbb{RP}}^\infty\) | \(1 + 2x + x^2\) | ? |

\({\mathbb{R}}^n\) is a contractible space, and so \([S^m, {\mathbb{R}}^n] = 0\) for all \(n, m\) which makes its homotopy groups all zero.

\(D(k, {\mathbb{R}}^n) = {\mathbb{R}}^n - \left\{{x_{1} \ldots x_{k}}\right\} \simeq\bigvee_{i=1}^k S^1\) by a deformation retract.

\(S^n \cong B^n / {\partial}B^n\) and employs an attaching map

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

\(B^n \simeq{\mathbb{R}}^n\) by normalizing vectors.

Use the inclusion \(S^n \hookrightarrow B^{n+1}\) as the attaching map.

\({\mathbb{CP}}^1 \cong S^2\).

\({\mathbb{RP}}^1 \cong S^1\).

Use \(\left[ \pi_{1}, \prod \right]= 0\) and the universal cover \({\mathbb{R}}^1 \twoheadrightarrow S^1\) to yield the cover \({\mathbb{R}}^n \twoheadrightarrow T^n\).

Take the universal double cover \(S^n \twoheadrightarrow^{\times 2} {\mathbb{RP}}^n\) to get equality in \(\pi_{i\geq 2}\).

Use \({\mathbb{CP}}^n = S^{2n+1} / S^1\)

Alternatively, the fundamental group is \({\mathbb{Z}}\ast{\mathbb{Z}}/ bab^{-1}a\). Use the fact the \(\tilde K = {\mathbb{R}}^2\).

\(M \simeq S^1\) by deformation-retracting onto the center circle.

\(D(1, S^n) \cong {\mathbb{R}}^n\) and thus \(D(k, S^n) \cong D(k-1, {\mathbb{R}}^n) \cong \bigvee^{k-1} S^1\)

\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.

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 \(f,g\).

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

Take the set \begin{align*} L \mathrel{\vcenter{:}}=\left\{{\alpha: S^1\to X \mathrel{\Big|}\alpha(0) = \alpha(1) = x_{0}}\right\} .\end{align*}

Define an equivalence relation \(\alpha \sim \beta\) iff \(\alpha \simeq\beta\) in \(X\), so there exists a homotopy

\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

Symmetric: Follows from considering \(H(s, 1-t)\).

Reflexive: Take \(H(s, t) = \alpha (s)\) for all \(t\).

Transitive: Follows from reparameterizing.

Define \(L/\sim\), which contains elements like \([\alpha]\) and \([\operatorname{id}_{x_{0}}]\), the equivalence classes of loops after quotienting by this relation.

Define a product structure: for \([\alpha], [\beta] \in L/\sim\), define \([\alpha][\beta] = [\alpha \cdot \beta]\), where we just need to define a product structure on actual loops. Do this by reparameterizing: \begin{align*} (\alpha \cdot \beta )(s) \mathrel{\vcenter{:}}= \begin{cases} \alpha (2s) & s \in [0, 1/2] \\ \beta (2s-1) & s \in [1/2, 1] . \end{cases} \end{align*}

Check that this map is:

Continuous: by the pasting lemma and assumed continuity of \(f, g\).

Well-defined: ?

Check that this is actually a group

Identity element: The constant loop \(\operatorname{id}_{x_0}: I\to X\) where \(\operatorname{id}_{x_0}(t) = x_0\) for all \(t\).

Inverses: The reverse loop \(\mkern 1.5mu\overline{\mkern-1.5mu\alpha\mkern-1.5mu}\mkern 1.5mu(t) \mathrel{\vcenter{:}}=\alpha(1-t)\).

Closure: Follows from the fact that start/end points match after composing loops, and reparameterizing.

Associativity: Follows from reparameterizing.

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

- See Hatcher 1.19, p.28
- See Hatcher’s proof that \(\pi_{1}\) is a group
- See change of basepoint map

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-connected^{2}

, 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

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*}

- Construct a map going backwards
- Show it is surjective
- “There and back” paths

- Show it is injective
- Divide \(I\times I\) into a grid

\(A = {\mathbb{Z}}/4{\mathbb{Z}}= \left\langle{x {~\mathrel{\Big|}~}x^4}\right\rangle, B = {\mathbb{Z}}/6{\mathbb{Z}}= \left\langle{y {~\mathrel{\Big|}~}x^6}\right\rangle, Z = {\mathbb{Z}}/2{\mathbb{Z}}= \left\langle{z {~\mathrel{\Big|}~}z^2}\right\rangle\). 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*}

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

\(H_{1}\) 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*}

- A loop in \(X \times Y\) is a continuous map \(\gamma : I \xrightarrow{} X \times Y\) given by \(\gamma (t) = (f(t), g(t)\) in components.
- \(\gamma\) being continuous in the product topology is equivalent to \(f, g\) being continuous maps to \(X, Y\) respectively.
- Similarly a homotopy \(F: I^2 \to X \times Y\) is equivalent to a pair of homotopies \(f_t, g_t\) of the corresponding loops.
- So the map \([ \gamma ] \mapsto ([f], [g])\) is the desired bijection.

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

\(\Rightarrow\): 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”}

- For a graph \(G\), we always have \(\pi_{1}(G) \cong {\mathbb{Z}}^n\) where \(n = |E(G - T)|\), the complement of the set of edges in any maximal tree. Equivalently, \(n = 1-\chi(G)\). Moreover, \(X \simeq\bigvee^n S^1\) in this case.

:::

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.

- \(\pi_{k\leq n}S^n = 0\)
- \(\pi_{n}(X) \cong \pi_{n}(X^{(n)})\)

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

\(\pi_{i\geq 2}(X)\) is always abelian.

- \(X\) simply connected \(\implies \pi_{k}(X) \cong H_{k}(X)\) up to and including the first nonvanishing \(H_{k}\)

\(\pi_{k} \bigvee X \neq \prod \pi_{k} X\) (counterexample: \(S^1 \vee S^2\))

- Nice case: \(\pi_{1}\bigvee X = \ast \pi_{1} X\) by Van Kampen.

\(\pi_{i}(\widehat{X}) \cong \pi_{i}(X)\) for \(i\geq 2\) whenever \(\widehat{X} \twoheadrightarrow X\) is a universal cover.

\(\pi_{i}(S^n) = 0\) for \(i < n\), \(\pi_{n}(S^n) = {\mathbb{Z}}\)

- Not necessarily true that \(\pi_{i}(S^n) = 0\) when \(i > n\)!!!
- E.g. \(\pi_{3}(S^2) = {\mathbb{Z}}\) by Hopf fibration

- Not necessarily true that \(\pi_{i}(S^n) = 0\) when \(i > n\)!!!
\(S^n / S^k \simeq S^n \vee \Sigma S^{k}\)

- \(\Sigma S^n = S^{n+1}\)

General mantra: homotopy plays nicely with products, homology with wedge products.

^{3}\(\pi_{k}\prod X = \prod \pi_{k} X\) by LES.

^{4}In general, homotopy groups behave nicely under homotopy pull-backs (e.g., fibrations and products), but not homotopy push-outs (e.g., cofibrations and wedges). Homology is the opposite.

Constructing a \(K(\pi, 1)\): since \(\pi = \left< S \mathrel{\Big|}R\right> = F(S)/R\), take \(\bigvee^{|S|} S^1 \cup_{|R|} e^2\). In English, wedge a circle for each generator and attach spheres for relations.

:::

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