Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Noncommutative Topology and Prospects in Index Theory   This article is a tribute to the memory of Professor Enzo Martinelli, with
deep respect and reconesance. Nicolae Teleman. The index formula is a local
statement made on global and local data; for this reason we introduce local
Alexander - Spanier co-homology, local periodic cyclic homology, local Chern
character and local $T^{\ast}$-theory. Index theory should be done: Case 1: for
arbitrary rings, Case 2: for rings of functions over topo- logical manifolds.
Case 1 produces general index theorems, as for example, over pseudo-manifolds.
Case 2 gives a general treatment of classical and non- commutative index
theorems. All existing index theorems belong to the second category. The tools
of the theory would contain: local $T^{\ast}$ -theory, local peri- odic cyclic
homology, local Chern character. These tools are extended to non- commutative
topology. The index formula has three stages : Stage I is done in
$T^{loc}\_{i}$-theory, Stage II is done in the local periodic cyclic homology
and Stage III involves products of distributions, or restriction to the
diagonal. For each stage there corresponds a topological index and an
analytical index. The construction of $T^{\ast}$-theory involves the
T-completion. It involves also the need to work with half integers; this should
have important consequences.

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The categories of this paper are: cond-mat
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The categories of this paper are: nucl-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: adap-org
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: nlin
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The categories of this paper are: stat
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The categories of this paper are: plasm-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: quant-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: alg-geom