Homology of spaces of directed paths on Euclidean cubical complexes

Abstract

We compute the homology of the spaces of directed paths on a certain class of cubical subcomplexes of the directed Euclidean space R n by a recursive process. We apply this result to calculate the homology and cohomology of the space of directed loops on the (n − 1)-skeleton of the directed torus T n. Keywords Directed paths • Cubical complex • Path space • Homology • Cohomology Mathematics Subject Classification (2000) 55P10 • 55P15 • 55U10 • 68Q85 1 Introduction One of the most important problems of directed algebraic topology is the determination of the homotopy type of spaces of directed paths P(X) y x between two points x, y of a directed space X. This problem seems to be difficult in general; however several Dedicated to Hvedri Inassaridze on his 80th birthday Communicated by Ronald Brown. The authors gratefully acknowledge support from the European Science Foundation network "Applied and Computational Algebraic Topology" that allowed them to collaborate on this paper during mutual visits made possible by grants 4671 and 5432.

Homology of Spaces of Directed Paths in Euclidean Pattern Spaces

“…and is zero otherwise. This recovers the above mentioned Theorem 1.1 of Raussen and Ziemiański [14].…”

“…We conclude with a few remarks about possible extensions of the results of this paper that we hope to deal with in future work. One obvious challenge concerns finding maps from spheres, and more generally products of spheres, into path space such that the images of the fundamental classes may serve as generators for homology in the appropriate dimensions, aiming at a generalization of [14,Corollary 3.10] in the paper of Raussen and Ziemiański. This is work in progress.…”

“…Example: If F consists of the single set [n] then X F = R n \ Z n . Raussen and Ziemiański [14] investigated the path space P (R n \ Z n ) k+1 0 and determined its homology groups and its cohomology ring. Their result concerning homology is the following: Theorem 1.1 (Raussen and Ziemiański [14]).…”

Directed path spaces via discrete vector fields

“…Thus, the main theorem of [11] (Theorem 1.1) follows immediately from Theorem 8.10 if n = 3 since there are no critical routes having consecutive dimensions. For n = 3, the homology calculation requires, as in the previous cases, some additional calculations we do not present here.…”

“…Note that there is 1-1 correspondence between critical routes in K and cube sequences in K defined in [11,Section 1.4]: if ((a j ) q+1 j=1 , (b j ) q j=0 ) is a critical route in K, then [b 1 , . .…”