# 19. The Relative Homeomorphism Theorem

In this lecture we establish that for sufficiently "nice" pairs $X’ \subset X$ the relative homology $H_n(X,X’)$ is isomorphic to the reduced homology $ \tilde{H}_n(X/X’)$ of the quotient space $X/X’$. This fulfils a promise made at the end of Lecture 12. We then show that subcomplexes of cell complexes are always "nice".

##### Remark

As mentioned in class today, there is an *additional student seminar *next semester that Jagna Wiśniewska and I are jointly giving. The topic is *Vector Bundles in Algebraic Topology*. The syllabus will (roughly speaking) be based on Hatcher's book Vector Bundles and K-Theory, and is intended to be completely accessible to people who are attending my two Algebraic Topology courses. The seminar is limited to 20 students, so sign up quickly (whenever signing up becomes possible... ) to avoid *epic disappointment! *😱

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