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I would like to ask the same question that was already asked here: loop homology product for oriented compact manifolds with boundary.

I hope it is okay to open a new question. The original question seems to be edited in a way such that the current version of the question has not much to do with the given answers.

The question: All the references I found work with closed manifolds to define string operation on the free loop space. Which operations (like product and BV structure) are expected to be still well-defined on the free loop space of a manifold with boundary? And if not, what are the fundamental issues and are there attempts to define weaker structures?

Any references also not directly answering but leading me on the right track would be great.

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  • $\begingroup$ You ?probably? don't mean to say "loop space". Perhaps you mean the free loop space, i.e. the space $Map(S^1, M)$? The loop space is usually a space of pointed maps. $\endgroup$
    – Ryan Budney
    Commented Apr 12, 2022 at 9:53
  • $\begingroup$ Yes, you are right, I am referring to the free loop space. I edited the "free" into the question. $\endgroup$
    – Valentin
    Commented Apr 12, 2022 at 12:05

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