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If one set in a van Kampen cover is simply connected, the other fundamental group surjects with overlap-generated kernel

Statement

Assume the hypotheses of Seifert–van Kampen identifies the fundamental group with a group pushout and suppose that $V$ is simply connected. Let

$k_{*} : π_{1} (U \cap V, x_{0}) \to π_{1} (U, x_{0})$

be induced by inclusion. Then $(j_{U})_{*} : π_{1} (U, x_{0}) \to π_{1} (X, x_{0})$ is surjective and

$\ker (j_{U})_{*} = ⟨ ⁣ ⟨ im k_{*} ⟩ ⁣ ⟩_{π_{1} (U, x_{0})} .$

Facts & Assumptions

Given: The van Kampen cover in the Statement, with $V$ simply connected.

[L1]

The fundamental group of $X$ is the pushout of the two inclusion-induced maps from the overlap group (Seifert–van Kampen identifies the fundamental group with a group pushout).

[F1]

For arbitrary homomorphisms $f : K \to G$ and $h : K \to H$ , the quotient of $G * H$ by the normal closure of $j_{G} (f (k)) j_{H} (h (k))^{- 1}$ is their pushout (A group pushout is the quotient of a free product by the amalgamating relations).

[F2]

The normal closure of a subset is the smallest normal subgroup containing it (The normal closure of a subset of a group).

[F3]

A free product is characterized by the universal property for homomorphisms from its factors (The free product of an arbitrary family of groups).

[F4]

A simply connected space has a one-element fundamental group at every basepoint (Simply connected topological spaces).

Proof

technique · direct

1.1

By [F4], $π_{1} (V, x_{0})$ is trivial. Thus [L1] identifies $π_{1} (X, x_{0})$ with the pushout of $k_{*}$ and the unique homomorphism from $π_{1} (U \cap V, x_{0})$ to the trivial group.

L1F4

2.1

By [F3], the free product of $π_{1} (U, x_{0})$ with the trivial group is canonically $π_{1} (U, x_{0})$ . Under this identification, [F1] says that the pushout in step 1.1 is $π_{1} (U, x_{0}) / ⟨ ⁣ ⟨ im k_{*} ⟩ ⁣ ⟩ .$

step 1.1F1F2F3

3.1

The canonical map from $π_{1} (U, x_{0})$ to this quotient is exactly $(j_{U})_{*}$ under [L1]. A quotient map is surjective and has the quotienting normal subgroup as its kernel, so the asserted surjectivity and kernel formula follow.

step 2.1L1∎

Depends on

Used by

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Sources

J. Peter May, A Concise Course in Algebraic Topology, Chapter 2, Section 8 (standard reference, not scraped)