Question

Let H_* be a homology theory. Using only the Eilenberg-Steenrod axioms, compute H_i(S^1 ∨ S^1) for all i

Answer 1

The Eilenberg-Steenrod axioms can be used to compute the homology groups for S^1 ∨ S^1, which is the wedge sum of two circles. By removing the basepoint and contracting the arcs to a point, we can determine that the homology groups are trivial.

Step-by-step explanation:

The Eilenberg-Steenrod axioms provide a set of properties that a homology theory must satisfy. One of these axioms is the homotopy invariance, which states that homology groups should be invariant under continuous deformations.

In this case, we are interested in computing the homology groups for the space S^1 ∨ S^1, which represents the wedge sum of two circles. Using the axioms, we can determine the homology groups.

First, we know that the basepoint of the wedge sum is a common point on both circles. Let's call it x. Then, if we remove the basepoint from each circle, we have two open arcs. We can contract each of these arcs to a point, yielding two spaces that are homotopy equivalent to a point.

Since a point has trivial homology groups, we can conclude that H_i(S^1 ∨ S^1) = 0 for all i.