In
rational homotopy theory
In mathematics and specifically in topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored. It was founded by and . This simplification of homo ...
, the Halperin conjecture concerns the
Serre spectral sequence In mathematics, the Serre spectral sequence (sometimes Leray–Serre spectral sequence to acknowledge earlier work of Jean Leray in the Leray spectral sequence) is an important tool in algebraic topology. It expresses, in the language of homological ...
of certain fibrations. It is named after the
Canadian
Canadians (french: Canadiens) are people identified with the country of Canada. This connection may be residential, legal, historical or cultural. For most Canadians, many (or all) of these connections exist and are collectively the source of ...
mathematician
Stephen Halperin
John Stephen Halperin (born 1 February 1942 in Kingston, Ontario) is a Canadian mathematician who deals with differential geometry and algebraic topology.
A son of the mathematician Israel Halperin, Stephen Halperin studied at the University of To ...
.
Statement
Suppose that
is a fibration of
simply connected space
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
s such that
is
rationally elliptic and
(i.e.,
has non-zero
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
), then the
Serre spectral sequence In mathematics, the Serre spectral sequence (sometimes Leray–Serre spectral sequence to acknowledge earlier work of Jean Leray in the Leray spectral sequence) is an important tool in algebraic topology. It expresses, in the language of homological ...
associated to the fibration collapses at the
page.
Status
As of 2019, Halperin's conjecture is still open. Gregory Lupton has reformulated the conjecture in terms of formality relations.
Notes
Further reading
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Homotopy theory
Spectral sequences
Conjectures
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