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Algebraic Topology

Seifert–van Kampen Theorem

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The Seifert–van Kampen Theorem implies that if $X = U \cup V$ where $U, V,$ and $U \cap V$ are open and path-connected, then $\pi_1(X)$ is isomorphic to the free product $\pi_1(U) * \pi_1(V)$ modulo some relations.

True, the theorem indeed states that under these conditions, the fundamental group of $X$ is the free product of the fundamental groups of $U$ and $V$ amalgamated over the fundamental group of $U \cap V$ .

The Seifert–van Kampen Theorem can be used to determine the fundamental group of a wedge sum of two spaces, $X\vee Y$ .

True, the wedge sum $X\vee Y$ can often be written as the union of two open sets that satisfy the hypotheses of the Seifert–van Kampen Theorem, with one of the sets deformation retracting onto one of the spaces.

If a space $X$ can be decomposed into two open sets $U$ and $V$ such that $U\cap V$ is simply connected, the fundamental group of $X$ is always the direct product of the fundamental groups of $U$ and $V$ .

False, the Seifert–van Kampen Theorem yields the free product of the fundamental groups, not the direct product, unless additional structure exists (e.g., when $U$ and $V$ are both simply connected, in which case, the direct product and the free product coincide).

The Seifert–van Kampen Theorem is only applicable to spaces that are manifolds.

False, the theorem applies to topological spaces that are not necessarily manifolds. The key conditions are that the open sets must be path-connected and locally path-connected.

The Seifert–van Kampen Theorem can be used to calculate the fundamental group of the sphere $S^2$ .

False, the Seifert–van Kampen Theorem cannot be applied in this case because the sphere $S^2$ cannot be expressed as the union of two open sets that are both simply connected.

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