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A finite connected CW complex has the edge-and-2-cell presentation

Statement

Let $K$ be a finite connected CW complex, let $v_{0}$ be a vertex, and let $T$ be a maximal tree in its 1-skeleton. Choose one orientation of each 1-cell outside $T$ . For each such edge $e : u \to v$ , let $x_{e}$ be the based loop that follows $T$ from $v_{0}$ to $u$ , traverses $e$ , and returns along $T$ from $v$ to $v_{0}$ . For every 2-cell $f$ , choose a point on its attaching circle and a path in the 1-skeleton from $v_{0}$ to its image; the resulting based attaching loop has a unique reduced word $r_{f}$ in the free basis $(x_{e})$ . Then

$π_{1} (K, v_{0}) ≅ ⟨ x_{e} (e \notin T) ∣ r_{f} (f a 2-cell) ⟩ .$

Cells of dimension at least three add no generators or relators. Changing the connecting path conjugates its relator, and reversing an edge orientation inverts the corresponding generator, so these choices do not change the presented group.

Statement references

Based loops and the fundamental group

Seifert–van Kampen identifies the fundamental group with a group pushout

Facts & Assumptions

Given: The finite connected CW complex $K$ , a vertex $v_{0}$ , a maximal tree $T$ in its 1-skeleton, and the attaching maps of its 2-cells.

[F1]

Based loops and the fundamental group defines $π_{1} (X, x_{0})$ as the set of based loops modulo endpoint-fixed path homotopy and proposes traversal-first concatenation. Loop classes form the group $π_{1} (X, x_{0})$ under concatenation proves that this product is well defined and makes that set a group, with the constant loop as identity and reversal as inverse.

[F2]

CW quotients and collapse of a contractible subcomplex: collapsing a nonempty contractible CW subcomplex with a supplied contraction gives a homotopy equivalence of the quotient map.

[F3]

The fundamental group of a finite wedge of circles is free of that rank: the fundamental group of a finite wedge of circles is free on the once-around loops of its circle summands.

[F4]

Seifert–van Kampen identifies the fundamental group with a group pushout: for an open cover by path-connected sets with path-connected overlap, the fundamental group of the union is the pushout of the two inclusion-induced groups.

[F5]

$Deg : π_{1} (R / Z, [0]) \to (Z, +)$ is an isomorphism: the based fundamental group of a circle is infinite cyclic, generated by a once-around loop.

Proof

1.1givenF2F3construct

The 1-skeleton of a connected CW complex is connected: attaching a cell of dimension at least two has connected attaching sphere, so it cannot join two components of the preceding skeleton. Since $K$ is finite, its 1-skeleton is a finite connected graph. The unique paths in $T$ contract it to $v_{0}$ , supplying the contraction required by [F2]. The quotient that collapses $T$ to a point is a wedge with one circle for each 1-cell outside $T$ , and [F3] gives the free basis represented by the loops $x_{e}$ .

1.2givenF1F4F5construct

Suppose a 2-cell is attached to a path-connected finite subcomplex $X$ along $α : S^{1} \to X$ . In the attached space, the open sets consisting of $X$ together with an outer collar of the disk and the inner open disk cover the space; the first retracts to $X$ , the second is contractible, and their overlap is an annulus retracting to $S^{1}$ . Choose a basepoint on that annulus and connect it to $v_{0}$ and to the attaching basepoint by the collar and a path in $X$ . Under the retraction to $X$ , the once-around generator of the annulus maps to the resulting based attaching loop $γ * α * \overset{ˉ}{γ}$ ; in the inner disk it maps to the identity. By [F4] and [F5], the pushout therefore imposes exactly that loop as a relation. The quotient by its normal closure has the same universal property: a homomorphism out of it is precisely a homomorphism out of $π_{1} (X, v_{0})$ that sends the based attaching loop to the identity.

2.1step 1.1step 1.2algebra

Attaching the finitely many 2-cells one at a time therefore quotients the free group from step 1.1 by the normal closure of all the words $r_{f}$ . Replacing a connecting path conjugates the corresponding based loop, so its normal closure is unchanged.

2.2F4step 1.2algebra

If an $n$ -cell with $n \geq 3$ is attached, the same collar cover has overlap homotopy equivalent to $S^{n - 1}$ . For $m \geq 2$ , the sphere $S^{m}$ is covered by the complements of two antipodal points; both sets are contractible and their overlap $S^{m - 1} \times R$ is path-connected. [F4] therefore gives $π_{1} (S^{m}) = 1$ . Applying [F4] to the cell-attachment cover shows that attaching the $n$ -cell induces an isomorphism on fundamental groups.

3.1givenstep 2.1step 2.2∎

Attaching every higher-dimensional cell preserves the group from step 2.1. The finite presentation in the Statement follows, including the cases with no non-tree edges or no 2-cells.

Depends on

Dependency tree · 2 levels

Sources

Allen Hatcher, Algebraic Topology, Chapter 1, §1.2 Applications to Cell Complexes and §1.A Graphs and Free Groups