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In dealing with defining his finiteness obstruction (for a finite dimensional finitely dominated CW-complex to be homotopy finite), Wall amends the given finite-dimensional complex by a homotopy equivalence so as to make it have finite skeleta, by the price of making it infinite dimensional.

If I remember it rightly, Ranicky's Ranicki's "instant finiteness obstruction" avoids explicit infinite-dimensional complexes by taking a more computational (hence less explanatory?) approach. EDIT: some (but then it must not all) details can be "less conceptual")found here.

My own impression was, however, that this whole business of finiteness obstruction is deeply rooted in the Eilenberg swindle ($1=1-1+1-1+1-...=0$) which I somehow could not appreciate as a truly infinite (let alone infinite-dimensional) construction. EDIT: indeed, applying each instance of $1-1=0$ is a rather finite process and then we are just running its independent identical copies.

So this "answer" ends up being a question: what is Wall's finiteness obstruction - is it a perfect application of infinite-dimensional topology to finite-dimensional topology, or on the contrary a perfect example of how such apparent applications turn out to be illusory? I'm asking this in part as an attempt to clarify the meaning of the original question, which I still don't fully understand. If the original very general question is unambiguous, then surely this very specific question has an unabmiguous answer, right?

2 added 2 characters in body

In dealing with his finiteness obstruction (for a finite dimensional finitely dominated CW-complex to be homotopy finite), Wall amends the given finite-dimensional complex by a homotopy equivalence so as to make it have finite skeleta, by the price of making it infinite dimensional.

If I remember it rightly, Ranicky's "instant finiteness obstruction" avoids explicit infinite-dimensional complexes by taking a more computational approach (but then it must be "less conceptual").

My own impression was, however, that this whole business of finiteness obstruction is deeply rooted in the Eilenberg swindle ($1=1+1-1+1-...=0$) $1=1-1+1-1+1-...=0$) which I somehow could not appreciate as a truly infinite (let alone infinite-dimensional) construction.

So this "answer" ends up being a question: what is Wall's finiteness obstruction - is it a perfect application of infinite-dimensional topology to finite-dimensional topology, or on the contrary a perfect example of how such apparent applications turn out to be illusory? I'm asking this in part as an attempt to clarify the meaning of the original question, which I still don't fully understand. If the original very general question is unambiguous, then surely this very specific question has an answer, right?

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In dealing with his finiteness obstruction (for a finite dimensional finitely dominated CW-complex to be homotopy finite), Wall amends the given finite-dimensional complex by a homotopy equivalence so as to make it have finite skeleta, by the price of making it infinite dimensional.

If I remember it rightly, Ranicky's "instant finiteness obstruction" avoids explicit infinite-dimensional complexes by taking a more computational approach (but then it must be "less conceptual").

My own impression was, however, that this whole business of finiteness obstruction is deeply rooted in the Eilenberg swindle ($1=1+1-1+1-...=0$) which I somehow could not appreciate as a truly infinite (let alone infinite-dimensional) construction.

So this "answer" ends up being a question: what is Wall's finiteness obstruction - is it a perfect application of infinite-dimensional topology to finite-dimensional topology, or on the contrary a perfect example of how such apparent applications turn out to be illusory? I'm asking this in part as an attempt to clarify the meaning of the original question, which I still don't fully understand. If the original very general question is unambiguous, then surely this very specific question has an answer, right?