Homotopy type of planar continua

Embedding planar compacta in planar continua with application: homotopic maps between planar Peano continua are characterized by the fundamental group homomorphism

A planar compactum with connected complement can be an embedded in a cellular continuum by attaching a null sequence of arcs. Two based maps f and g from a planar Peano continuum X to a planar set Y are homotopic iff f and g induce the same homomorphism between fundamental groups.

多面体近似中佩亚诺连续体的第一同调非局部丢失

Nonlocal loss of first homotopy in polyhedral approximations of Peano continua

研究揭示了佩亚诺连续体在多面体近似中第一同调的非局部丢失现象，指出其失败并非局部特性。

Homotopy of Planar Lie Group Equivariant Presheaves

We utilise the theory of crossed simplicial groups to introduce a collection of local Quillen model structures on the category of simplicial presheaves with a compact planar Lie group action on a small Grothendieck site. As an application, we give a characterisation of equivariant cohomology theories on a site as derived mapping spaces in these model categories.

Path Triangulation, Cycles and Good Covers on Planar Cell Complexes. Extension of J.H.C. Whitehead's Homotopy System Geometric Realization and E.C. Zeeman's Collapsible Cone Theorems

This paper introduces path triangulation of points in a bounded, simply connected surface region, replacing ordinary triangles in a Delaunay triangulation with path triangles from homotopy theory. A {\bf path triangle} has a border that is a sequence of paths $h:I\to X, I=[0,1]$. The main results in this paper are that (1) a cone $D\times I$ collapses to a path triangle $h\bigtriangleup K$, extending E.C. Zeeman's collapsible dunce hat cone theorem, (2) an ordinary path triangle with geometrically realized straight edges generalizes Veech's billiard triangle, (3) a billiard ball $K\times I$ co…

The homotopy dimension of codiscrete subsets of the 2-sphere

We characterize those planar Peano continua that are homotopy equivalent to 1-dimensional sets. While many planar Peano continua are not homotopically 1-dimensional, we prove that each has fundamental group that embeds in the fundamental group of a 1-dimensional planar Peano continuum. We leave open the following question: Is a planar Peano continuum homotopically 1- dimensional if its fundamental group is isomorphic with the fundamental group of a 1-dimensional planar Peano continuum?

Transferring homotopy commutative algebraic structures

We show that the sum over planar trees formula of Kontsevich and Soibelman transfers C-infinity structures along a contraction. Applying this result to a cosimplicial commutative algebra A^* over a field of characteristic zero, we exhibit a canonical unital C-infinity structure on Tot(A^*), which is unital if A^* is; in particular, we obtain a canonical C-infinity structure on the cochain complex of a simplicial set.