Question

I am studying a paper of L. Lovász, ``A homology theory for spanning trees of a graph,'' but professor Babai has told me that Lovász later realized that this work is better framed in the language of homotopy theory.

Does anyone know if Lovász (or anyone else) published a homotopy-based treatment of the results in the above paper?

Answer 1

I think that this has something to do with what's called A-homotopy theory. In fact, the following two papers on A-homotopy theory reference Lovasz's paper, and from looking at the first paper below it seems like it's exactly the sort of thing you're looking for.

MR1808443 (2001k:57029) Barcelo, Hélène; Kramer, Xenia; Laubenbacher, Reinhard; Weaver, Christopher Foundations of a connectivity theory for simplicial complexes. Adv. in Appl. Math. 26 (2001), no. 2, 97–128. (Reviewer: Andrew Vince), 57Q05 (05B35 05C10 55P99 55Q05)

MR2163440 (2006f:52017) Barcelo, Hélène; Laubenbacher, Reinhard Perspectives on $A$-homotopy theory and its applications. Discrete Math. 298 (2005), no. 1-3, 39–61. (Reviewer: Jean-Louis Cathelineau), 52B40 (05B35 05E25 37F20 52C35 55R80 57Q05)