656 research outputs found
The Seifert-van Kampen Theorem in Homotopy Type Theory
Homotopy type theory is a recent research area connecting type theory with homotopy theory by interpreting types as spaces. In particular, one can prove and mechanize type-theoretic analogues of homotopy-theoretic theorems, yielding "synthetic homotopy theory". Here we consider the Seifert-van Kampen theorem, which characterizes the loop structure of spaces obtained by gluing. This is useful in homotopy theory because many spaces are constructed by gluing, and the loop structure helps distinguish distinct spaces. The synthetic proof showcases many new characteristics of synthetic homotopy theory, such as the "encode-decode" method, enforced homotopy-invariance, and lack of underlying sets
Path spaces of higher inductive types in homotopy type theory
The study of equality types is central to homotopy type theory.
Characterizing these types is often tricky, and various strategies, such as the
encode-decode method, have been developed.
We prove a theorem about equality types of coequalizers and pushouts,
reminiscent of an induction principle and without any restrictions on the
truncation levels. This result makes it possible to reason directly about
certain equality types and to streamline existing proofs by eliminating the
necessity of auxiliary constructions.
To demonstrate this, we give a very short argument for the calculation of the
fundamental group of the circle (Licata and Shulman '13), and for the fact that
pushouts preserve embeddings. Further, our development suggests a higher
version of the Seifert-van Kampen theorem, and the set-truncation operator maps
it to the standard Seifert-van Kampen theorem (due to Favonia and Shulman '16).
We provide a formalization of the main technical results in the proof
assistant Lean.Comment: v1: 23 pages; v2: 24 pages, small reformulations and reorganization
Algebraic and combinatorial codimension-1 transversality
The Waldhausen construction of Mayer-Vietoris splittings of chain complexes
over an injective generalized free product of group rings is extended to a
combinatorial construction of Seifert-van Kampen splittings of CW complexes
with fundamental group an injective generalized free product.Comment: Published by Geometry and Topology Monographs at
http://www.maths.warwick.ac.uk/gt/GTMon7/paper6.abs.htm
Noncommutative localization in topology
A survey of the applications of the noncommutative Cohn localization of rings
to the topology of manifolds with infinite fundamental group, with particular
emphasis on the algebraic K- and L-theory of generalized free products.Comment: 20 pages, LATEX. To appear in the Proceedings of the Conference on
Noncommutative Localization in Algebra and Topology, ICMS, Edinburgh, 29-30
April, 2002. v2 is a minor revision of v
Systoles of 2-complexes, Reeb graph, and Grushko decomposition
Let X be a finite 2-complex with unfree fundamental group. We prove lower
bounds for the area of a metric on X, in terms of the square of the least
length of a noncontractible loop in X. We thus establish a uniform systolic
inequality for all unfree 2-complexes. Our inequality improves the constant in
M. Gromov's inequality in this dimension. The argument relies on the Reeb graph
and the coarea formula, combined with an induction on the number of freely
indecomposable factors in Grushko's decomposition of the fundamental group.
More specifically, we construct a kind of a Reeb space ``minimal model'' for X,
reminiscent of the ``chopping off long fingers'' construction used by Gromov in
the context of surfaces. As a consequence, we prove the agreement of the
Lusternik-Schnirelmann and systolic categories of a 2-complex.Comment: 29 pages; to appear in Int. Math. Res. Notice
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