General Homotopies
\begin{align*} X\times{\mathbf{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 \(f,g\).
Fundamental Group
Definition
Given a pointed space \((X,x_{0})\), we define the fundamental group \(\pi_{1}(X)\) as follows:
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Take the set \begin{align*} L \coloneqq\left\{{\alpha: S^1\to X \mathrel{\Big|}\alpha(0) = \alpha(1) = x_{0}}\right\} .\end{align*}
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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*}
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Check that this relation is
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Symmetric: Follows from considering \(H(s, 1-t)\).
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Reflexive: Take \(H(s, t) = \alpha (s)\) for all \(t\).
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Transitive: Follows from reparameterizing.
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Define \(L/\sim\), which contains elements like \([\alpha]\) and \([\operatorname{id}_{x_{0}}]\), the equivalence classes of loops after quotienting by this relation.
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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) \coloneqq \begin{cases} \alpha (2s) & s \in [0, 1/2] \\ \beta (2s-1) & s \in [1/2, 1] . \end{cases} \end{align*}
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Check that this map is:
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Continuous: by the pasting lemma and assumed continuity of \(f, g\).
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Well-defined: ?
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Check that this is actually a group
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Identity element: The constant loop \(\operatorname{id}_{x_0}: I\to X\) where \(\operatorname{id}_{x_0}(t) = x_0\) for all \(t\).
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Inverses: The reverse loop \(\overline{\alpha}(t) \coloneqq\alpha(1-t)\).
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Closure: Follows from the fact that start/end points match after composing loops, and reparameterizing.
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Associativity: Follows from reparameterizing.
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Elements of the fundamental group are homotopy classes of loops, and every continuous map between spaces induces a homomorphism on fundamental groups.
Conjugacy in \(\pi_{1}\):
- See Hatcher 1.19, p.28
- See Hatcher’s proof that \(\pi_{1}\) is a group
- See change of basepoint map
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 \(U \coloneqq U_{1} \cap U_{2} \neq \emptyset\) are open and path-connected 1
, 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\vert}~}R_{A}}\right\rangle \\ \pi_1 U_2 = B = \left\langle{G_{B} {~\mathrel{\Big\vert}~}R_{B}}\right\rangle \end{cases} \implies A \ast_{Z} B \coloneqq\left\langle{ G_{A}, G_{B} {~\mathrel{\Big\vert}~}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\vert}~}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\vert}~}\alpha_{1}, \cdots, \alpha_{l}\right\rangle \\ \pi_{1}(V, w) &=\left\langle v_{1}, \cdots, v_{m} {~\mathrel{\Big\vert}~}\beta_{1}, \cdots, \beta_{n}\right\rangle \\ \pi_{1}(U \cap V, x_0) &=\left\langle w_{1}, \cdots, w_{p} {~\mathrel{\Big\vert}~}\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\vert}~}1\leq i \leq p}\right\} }\right\rangle } \end{align*}
- Construct a map going backwards
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Show it is surjective
- “There and back” paths
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Show it is injective
- Divide \(I\times I\) into a grid
\(A = {\mathbf{Z}}/4{\mathbf{Z}}= \left\langle{x {~\mathrel{\Big\vert}~}x^4}\right\rangle, B = {\mathbf{Z}}/6{\mathbf{Z}}= \left\langle{y {~\mathrel{\Big\vert}~}x^6}\right\rangle, Z = {\mathbf{Z}}/2{\mathbf{Z}}= \left\langle{z {~\mathrel{\Big\vert}~}z^2}\right\rangle\). 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\vert}~}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.
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*}
- 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.
For a graph \(G\), we always have \(\pi_{1}(G) \cong {\mathbf{Z}}^n\) 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.
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 \(S^2 \times{\mathbf{RP}}^3\) and \(S^3 \times{\mathbf{RP}}^2\) 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.
- \(\pi_{k\leq n}S^n = 0\)
- \(\pi_{n}(X) \cong \pi_{n}(X^{(n)})\)
\todo[inline]{Theorem}
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\(\pi_{i\geq 2}(X)\) is always abelian.
- \(X\) simply connected \(\implies \pi_{k}(X) \cong H_{k}(X)\) up to and including the first nonvanishing \(H_{k}\)
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\(\pi_{k} \bigvee X \neq \prod \pi_{k} X\) (counterexample: \(S^1 \vee S^2\))
- Nice case: \(\pi_{1}\bigvee X = \ast \pi_{1} X\) by Van Kampen.
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\(\pi_{i}(\widehat{X}) \cong \pi_{i}(X)\) for \(i\geq 2\) whenever \(\widehat{X} \twoheadrightarrow X\) is a universal cover.
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\(\pi_{i}(S^n) = 0\) for \(i < n\), \(\pi_{n}(S^n) = {\mathbf{Z}}\)
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Not necessarily true that \(\pi_{i}(S^n) = 0\) when \(i > n\)!!!
- E.g. \(\pi_{3}(S^2) = {\mathbf{Z}}\) by Hopf fibration
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Not necessarily true that \(\pi_{i}(S^n) = 0\) when \(i > n\)!!!
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\(S^n / S^k \simeq S^n \vee \Sigma S^{k}\)
- \(\Sigma S^n = S^{n+1}\)
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General mantra: homotopy plays nicely with products, homology with wedge products. 2
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\(\pi_{k}\prod X = \prod \pi_{k} X\) by LES. 3
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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.
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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.