Algebraic Topology¶

Job: homotopy-invariant functors $\pi_1$, $H_*$, $H^*$ — compute via covers, Seifert–van Kampen, Mayer–Vietoris; distinguish spaces up to homotopy; orient spectral sequences and ring structure.

Core¶

Homotopy and $\pi_1$¶

Paths $\gamma:I\to X$ up to homotopy rel endpoints; fundamental group $\pi_1(X,x_0)$ with concatenation. Basepoint change: isomorphism if $X$ path-connected (canonical up to conjugacy). Functorial: $f:X\to Y$ induces $f_*:\pi_1(X)\to\pi_1(Y)$.

Space	$\pi_1$
$\mathbb{R}^n$, convex, contractible	$0$
$S^1$	$\mathbb{Z}$ (degree / winding)
$S^n$, $n\ge 2$	$0$
$\mathbb{RP}^n$	$\mathbb{Z}/2$ ($n\ge 2$)
$\Sigma_g$ (genus $g$ surface)	$\langle a_i,b_i\mid \prod[a_i,b_i]=1\rangle$
$X\vee Y$ (path-conn., nice)	$\pi_1(X)*\pi_1(Y)$

Theorem (Seifert–van Kampen). $X=U\cup V$, $U,V,U\cap V$ path-connected open. Then $$ \pi_1(X)\cong\pi_1(U)*_{\pi_1(U\cap V)}\pi_1(V) $$ (amalgamated product; precise with induced maps on $\pi_1$).

Covering spaces¶

$p:\widetilde{X}\to X$ covering: evenly covered nbhds. Path lifting, homotopy lifting. For nice $X$ (path-conn., loc. path-conn., semilocally simply connected):

Theorem. Connected coverings $\leftrightarrow$ conjugacy classes of subgroups of $\pi_1(X)$. Universal cover $\leftrightarrow$ trivial subgroup; $\operatorname{Aut}(\widetilde{X}/X)\cong\pi_1(X)$.

Deck transformations act freely properly discontinuously; $X\cong\widetilde{X}/\Gamma$.

Singular homology¶

$C_n(X)=$ free abelian on singular $n$-simplices $\sigma:\Delta^n\to X$; $\partial_n=\sum (-1)^i d_i$. $H_n(X)=\ker\partial_n/\operatorname{im}\partial_{n+1}$.

Property	Statement
Homotopy invariance	$f\simeq g\Rightarrow f_=g_$ on $H_*$
Dimension	$H_n(\mathrm{pt})=\mathbb{Z}$ if $n=0$, else $0$
Additivity	wedge (nice) $\leftrightarrow$ direct sum on reduced $\widetilde{H}$
LES of pair	$\cdots\to H_n(A)\to H_n(X)\to H_n(X,A)\to H_{n-1}(A)\to\cdots$
Excision	$H_(X,A)\cong H_(X\setminus U,A\setminus U)$ under interior hypotheses
Mayer–Vietoris	LES for $X=U\cup V$ relating $H_(U\cap V)$, $H_(U)\oplus H_(V)$, $H_(X)$
Hurewicz	$\pi_1^{\mathrm{ab}}\cong H_1$ (path-conn.); higher if $\pi_{<n}=0$

Cellular homology for CW complexes: $H_*^{\mathrm{cell}}\cong H_*^{\mathrm{sing}}$; often computable from attaching maps.

Standard computations¶

Space	Homology
$S^n$	$\widetilde{H}_n\cong\mathbb{Z}$, else $0$
$\mathbb{CP}^n$	$\mathbb{Z}$ in even degrees $0,2,\ldots,2n$
$\mathbb{RP}^n$	$\mathbb{Z}/2$ pattern in $H_*$ (orientability at top for $n$ odd)
Surfaces	$H_1(\Sigma_g)\cong\mathbb{Z}^{2g}$

Degree of $f:S^n\to S^n$: $f_*:H_n\to H_n$ multiplication by $\deg f\in\mathbb{Z}$. Borsuk–Ulam, hairy ball as corollaries.

Cohomology ring¶

$H^*(-;R)$ via cochains or Ext/universal coefficients. Cup product $H^p\times H^q\to H^{p+q}$ makes $H^*(X;R)$ a graded-commutative ring (signs).
Example: $H^*(\mathbb{CP}^\infty;\mathbb{Z})\cong\mathbb{Z}[c]$ with $|c|=2$; truncations distinguish $\mathbb{CP}^2$ from $S^2\vee S^4$ (same homology, different cup structure: $c^2\neq0$ on $\mathbb{CP}^2$).
Poincaré duality: closed oriented $n$-manifold $\Rightarrow H^k(M)\cong H_{n-k}(M)$ (cap with fundamental class).

Whitehead theorem (one-liner)¶

Whitehead. A weak homotopy equivalence $f:X\to Y$ between CW complexes is a homotopy equivalence. (Weak: isomorphisms on all $\pi_n$ for all basepoints.) Consequently, for CW spaces, weak $\simeq$ $\Rightarrow$ iso on $H_*$, but the converse fails — homology equivalences need not be homotopy equivalences (e.g. Poincaré homology sphere vs $S^3$ after killing $\pi_1$ issues; classic: acyclic spaces with nontrivial $\pi_1$).

Spectral sequences (pointer)¶

A spectral sequence is a page-by-page algebraic machine $E^r_{p,q}$ with differentials $d_r$ and $E^{r+1}=H(E^r,d_r)$, abutting to a graded object associated to a filtration on a target (e.g. $H_*(X)$).

SS	Typical use
Serre	$H_(F)\Rightarrow H_(E)$ for fibration $F\to E\to B$ (hyp. on local coeffs)
Atiyah–Hirzebruch	ordinary homology of spectrum / generalized cohomology of space
Leray / Grothendieck	sheaf cohomology of composition $R^p f_\circ R^q g_$
Lyndon–Hochschild–Serre	group cohomology of extensions

One-liner: $E_2$ page alone does not determine the abutment — differentials may kill classes; convergence needs hypotheses. See homological algebra for double complexes $\Rightarrow$ SS.

Higher homotopy (orientation)¶

$\pi_n(X)=[(S^n,*),(X,x_0)]$. $\pi_n(S^n)\cong\mathbb{Z}$; $\pi_i(S^n)$ for $i>n$ are hard (stable homotopy, Serre SS). Long exact sequence of a fibration: $\cdots\to\pi_n(F)\to\pi_n(E)\to\pi_n(B)\to\pi_{n-1}(F)\to\cdots$.

See it¶

Configuration spaces / braid groups. $\pi_1(\operatorname{Conf}_n(\mathbb{C}))$ is the pure braid group; full braid group is $\pi_1$ of unordered configurations. Appears in knot theory, mapping class groups, and topological quantum computing.

Degree and oriented intersection. In differential topology, $\deg(f)$ equals signed count of preimages of a regular value (Sard + inverse function theorem) — matches algebraic degree on $H_n$.

Cup products detect attaching. The Hopf map $S^3\to S^2$ and $\mathbb{CP}^2$ cell structure: $c\cup c$ generates $H^4(\mathbb{CP}^2)$ — pure ring data.

Check¶

Compute $\pi_1(S^1\vee S^1)$ and $H_1(S^1\vee S^1)$; reconcile via Hurewicz.

$\pi_1=F_2$ free on two generators; $H_1\cong\mathbb{Z}^2\cong F_2^{\mathrm{ab}}$. Hurewicz abelianizes $\pi_1$.

Why does van Kampen not compute $\pi_2$? Give a space with $\pi_1=0$ but $\pi_2\neq 0$.

$S^2$: simply connected, $\pi_2\cong\mathbb{Z}$. SVK is about $\pi_1$ only; higher $\pi_n$ need fibrations / spectral sequences / cellular approximation.

State Whitehead’s theorem and a caveat about homology.

Weak homotopy equivalence of CW complexes $\Rightarrow$ homotopy equivalence. A map inducing iso on all $H_*$ need not be a homotopy equivalence (homology is coarser than $\pi_*$).

Pitfalls¶

Basepoints matter for $\pi_1$ morphisms; free homotopy classes of loops $\leftrightarrow$ conjugacy classes.
Homology ignores $\pi_1$ nonabelian information beyond $H_1$.
Excision / MV require open covers (or carefully saturated CW decompositions).
Spectral sequence differentials — reading only $E_2$ is a common error.
Coefficients matter: $H_*(-;\mathbb{Z})$ vs $\mathbb{Z}/2$ changes torsion and orientability stories.

Refs¶

Hatcher, Algebraic Topology
May, A Concise Course in Algebraic Topology
Spanier, Algebraic Topology
McCleary, A User’s Guide to Spectral Sequences

Next¶

Topology foundations · Homological algebra · Category theory · Group theory · Modules · Topology overview · Higher Teichmüller theory — surface groups and classifying spaces as the source groups for character varieties · Hyperbolic dynamics — mapping-class / surface-dynamics connection

Space	\(\pi_1\)
\(\mathbb{R}^n\), convex, contractible	\(0\)
\(S^1\)	\(\mathbb{Z}\) (degree / winding)
\(S^n\), \(n\ge 2\)	\(0\)
\(\mathbb{RP}^n\)	\(\mathbb{Z}/2\) (\(n\ge 2\))
\(\Sigma_g\) (genus \(g\) surface)	\(\langle a_i,b_i\mid \prod[a_i,b_i]=1\rangle\)
\(X\vee Y\) (path-conn., nice)	\(\pi_1(X)*\pi_1(Y)\)

Space	Homology
\(S^n\)	\(\widetilde{H}_n\cong\mathbb{Z}\), else \(0\)
\(\mathbb{CP}^n\)	\(\mathbb{Z}\) in even degrees \(0,2,\ldots,2n\)
\(\mathbb{RP}^n\)	\(\mathbb{Z}/2\) pattern in \(H_*\) (orientability at top for \(n\) odd)
Surfaces	\(H_1(\Sigma_g)\cong\mathbb{Z}^{2g}\)