Results on cell complexes, fundamental groups, covering spaces and cellular chains, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Cell complexes are Hausdorff — Lemma 1.2. Every complex built by attaching cells is Hausdorff and closure-finite, so it is a CW complex with the weak topology of its characteristic maps. No hypothesis.
• Collapsing a subcomplex — Proposition 2.1. Y/A is a cell complex with one vertex for A and the cells outside A; if A is contractible, the quotient map is a homotopy equivalence. Hypothesis: A a nonempty subcomplex.
• Attaching cells along homotopic maps — Proposition 3.1. Attaching n-cells to a space W along homotopic families of maps gives homotopy equivalent spaces, by maps that are the identity on W and homotopies constant on W. No further hypothesis.
• Reduced walks in trees — Lemma 4.1. A nonempty graph is path-connected exactly when it is connected; in a tree each vertex is the end of exactly one reduced walk from a base vertex, and each edge joins a parent to a child. No hypothesis.
• Trees are contractible — Proposition 4.2. Every tree contracts to any of its vertices by a homotopy fixing that vertex. No hypothesis.
• Spanning trees — Lemma 4.3. Every nonempty connected graph contains a spanning tree (Zorn's lemma). No hypothesis.
• Collapsing a tree — Corollary 4.4. Collapsing a tree in the 1-skeleton is a homotopy equivalence onto a cell complex; a spanning tree leaves one vertex. No hypothesis.
• Triangulating two-complexes — Proposition 5.1. A finite two-complex whose 2-cells are attached along closed walks is homeomorphic to a finite simplicial complex in which each edge of the 1-skeleton is divided into three. No further hypothesis.
• Simplicial approximation — Proposition 5.2. A map from a finite simplicial complex to a simplicial complex is homotopic, by a straight-line homotopy inside carrier simplices, to a map simplicial for an iterated barycentric subdivision. No further hypothesis.
• The fundamental group of a graph — Theorem 2.3. The tree of reduced walks is a covering of a connected graph, and the fundamental group is the group of closed walks up to backtracking. No hypothesis.
• Free bases from a spanning tree — Corollary 2.4. The fundamental group of a connected graph is free on the classes of the closed walks through the edges outside a spanning tree. No hypothesis.
• Paths in graphs — Corollary 2.5. Paths between vertices are homotopic to paths of walks, homotopic exactly when the walks are; every circle in a graph is homotopic to a closed walk. No hypothesis.
• Attaching cells — Theorem 3.3. Attaching 2-cells to a path-connected Hausdorff space divides the fundamental group by the normal subgroup generated by the attaching loops; cells of dimension at least three do not change it. No further hypothesis.
• The two-skeleton — Corollary 3.4. A connected cell complex has a connected 1-skeleton, and its fundamental group is that of its 2-skeleton, a quotient of that of the 1-skeleton. No hypothesis.
• Presentation complexes — Corollary 4.2. The fundamental group of the presentation complex of <A | R> is the presented group. No hypothesis.
• Complexes with cells along closed walks — Corollary 4.3. A spanning tree gives a presentation of the fundamental group by the edges outside the tree and the words of the 2-cells. No hypothesis.
• Two-complexes up to homotopy — Corollary 4.4. Every connected two-complex is homotopy equivalent to one with cells along closed walks, to a presentation complex, and, if finite, to a finite simplicial complex of dimension at most two. No hypothesis.
• Universal covers of cell complexes — Theorem 1.1. A connected cell complex has a simply connected cover whose cells are the lifts of its cells; the fundamental group acts by deck transformations, freely, transitively on fibres and on the lifts of each cell; higher homotopy groups are unchanged. No hypothesis.
• Lifting criterion — Proposition 2.1. A map from a path-connected, locally path-connected space lifts through a covering exactly when its fundamental group maps into that of the covering; the lift is unique. No further hypothesis.
• Uniqueness of lifts — Proposition 2.2. Two lifts of a map from a connected space that agree at one point are equal. No hypothesis.
• Covers for normal subgroups — Proposition 3.1. For a subgroup N the orbit space of the universal cover is a covering with fundamental group N, a cell complex with lifted cells; for normal N the quotient group acts freely and transitively on fibres, and maps lift exactly when their fundamental group maps into N. No further hypothesis.
• Subcomplexes and simplices of covers — Proposition 3.2. Preimages of subcomplexes are subcomplexes, and covers of simplicial complexes are simplicial complexes with lifted simplices. No further hypothesis.
• Lifting homotopy equivalences — Proposition 4.1. A homotopy equivalence of connected cell complexes induces an isomorphism of fundamental groups and lifts to an equivariant homotopy equivalence of universal covers. No hypothesis.
• Paths as chains — Lemma 5.1. Concatenation of paths is a sum of singular simplices up to boundaries, homotopic paths are homologous, and a reversed path is the negative. No hypothesis.
• Cellular boundaries — Proposition 5.2. An edge has boundary its head minus its tail; a 2-cell attached along a closed walk has boundary the signed sum of its edges; a cell attached to a lower-dimensional complex has boundary the cycle of its attaching map. No hypothesis.
• Chains of covers — Proposition 5.3. The cellular chains of the universal cover form a complex of free modules over the group ring, with one basis element for each cell of the base, and the augmentation is onto with kernel the image of d_1. No hypothesis.
• Subcomplexes and their universal covers — Proposition 5.4. For a connected subcomplex A, the universal cover of A maps equivariantly and cellularly to that of the whole complex, so the boundary coefficients of the cells of A in the big cover are the images of those in the small one. No hypothesis.
• The Cayley complex — Proposition 6.1. The universal cover of a presentation complex is the Cayley complex; its chain complex is exact in degrees 0 and 1, with boundary maps given by a - 1 and the Fox derivatives, and its second homology is pi_2. No hypothesis.

Reading order

Read the lessons in order. The first is point-set topology of cell complexes and graphs, the second computes fundamental groups, the third treats covers and cellular chains; all three rely on the core course Algebraic Topology for the Seifert-van Kampen theorem, covering spaces and cellular homology.