The wedge sum of pointed spaces identifies all their basepoints. In particular, a wedge of circles has first homology .
A pinch map collapses a separating subspace to a point so that the quotient splits into a wedge sum. For a connected sum of oriented manifolds, collapsing the separating boundary sphere gives a map . The two wedge projections have degree of a continuous mapping one with compatible orientations. Consequently their pullbacks identify the two top orientation classes, while positive-degree classes from different summands have zero cup product.
The wedge of two circles is a graph with one vertex and two loop edges. Its fundamental group is the free group on the two oriented loops.
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In topology, a wedge sum is a specific way of combining two or more topological spaces into a single space. The construction involves taking a collection of spaces and identifying a single point from each space. The basic idea is as follows: 1. **Choose Spaces**: Consider two or more topological spaces, say \(X_1, X_2, \ldots, X_n\).