Deformation retracts
If \(A\subseteq X\) is a deformation retract, then inclusion induces an isomorphism of fundamental groups \(\pi_1(A,a) \cong \pi_1(X,a)\) for \(a\in A\).
- \(\RR^n \sm \ts{0}\) deformation retracts onto \(S^{n-1}\).
- A torus minus a point deformation retracts onto \(S^1\vee S^1\).
- \(\RR^3\) minus a line deformation retracts onto a circle, and \(\RR^3\) minus a point onto \(S^2\).
- The Möbius band deformation retracts onto its core circle.
Van Kampen’s theorem
(Van Kampen)
Let \(X = U\union V\) with \(U, V\) open, and \(U, V, U\intersect V\) path connected, and let \(x_0\in U\intersect V\). Then the maps induced by inclusion give \[ \pi_1(X,x_0) \cong \pi_1(U,x_0) *_{\pi_1(U\intersect V,x_0)} \pi_1(V,x_0). \] If \(U\intersect V\) is simply connected, this is the free product \(\pi_1(U,x_0)*\pi_1(V,x_0)\). If \(V\) is simply connected, this is \(\pi_1(U,x_0)\) modulo the normal closure of the image of \(\pi_1(U\intersect V,x_0)\).
(Standard decompositions)
- A CW complex. For a CW complex \(X\) with \(2\)-skeleton \(X^{(2)}\), \(\pi_1(X)\cong\pi_1(X^{(2)})\), and attaching one \(2\)-cell \(e^2\) along \(\varphi\colon S^1\to X^{(1)}\) takes \(U\) a neighborhood of \(X^{(1)}\) deformation retracting onto it and \(V\) the open cell; the result is \(\pi_1(X^{(1)})\) modulo the normal closure of \([\varphi]\). Inductively, \(\pi_1(X)\) is presented by generators from a graph \(X^{(1)}\) and one relation for each \(2\)-cell.
- A wedge. If \(x_0\in X\) and \(y_0\in Y\) have contractible open neighborhoods deformation retracting onto them, then \(\pi_1(X\vee Y)\cong\pi_1(X)*\pi_1(Y)\). In particular, \(\pi_1\) of a wedge of \(n\) circles is free on \(n\) generators.
- A surface from a polygon. If all vertices of the polygon are identified to one point, the surface has one \(0\)-cell, one \(1\)-cell for each edge pair, and one \(2\)-cell, so \(\pi_1\) has one generator for each edge pair and one relation, the boundary word.
Covering spaces
Let \(p\colon \tilde X \to X\) be a covering map with \(\tilde X\) path connected and \(X\) path connected and locally path connected.
- \(p_*\) is injective, and \(p_*\pi_1(\tilde X)\) has index in \(\pi_1(X)\) equal to the number of sheets.
- If \(\tilde X\) is simply connected, then \(\pi_1(X)\) is isomorphic to the deck group.
- If the cover is normal, then \(\pi_1(X)/p_*\pi_1(\tilde X)\) is isomorphic to the deck group, which acts simply transitively on each fiber.
The universal covers \(\RR \to S^1\), \(\RR^2\to T^2\), and \(S^n \to \RP^n\) for \(n\geq 2\) give \(\pi_1(S^1)\cong\ZZ\), \(\pi_1(T^2)\cong\ZZ^2\), and \(\pi_1(\RP^n)\cong\ZZ/2\).
Products and quotients
For path-connected spaces \(X\) and \(Y\), the projections induce \(\pi_1(X\times Y) \cong \pi_1(X)\times\pi_1(Y)\).
The fundamental groups of quotients such as projective spaces and closed surfaces are listed on The standard spaces.
Choosing a method
| The space is given as | Method |
|---|---|
| a subspace of \(\RR^n\) with something deleted | deformation retract |
| a CW complex, or a polygon with identifications | van Kampen’s theorem |
| a quotient by a free, properly discontinuous group action | covering spaces and the deck group |
| a product | the product formula |
| a wedge or connected sum | van Kampen’s theorem |
Relation to homology
For a path-connected space \(X\), the Hurewicz map induces \(\pi_1(X)^{\mathrm{ab}} \cong H_1(X)\). A computation of \(\pi_1(X)\) therefore determines \(H_1(X)\).