Algebraic Reflections of Topological Reality

An informal introduction to algebraic homotopy theory

Topology

A topological space is a set X together with a collection T of subsets of X that are deemed to be open. The collection T must satisfy certain axioms.

Example: X = RR (the real line)T = { unions of open intervals }

Topology

A couple of attractive topological spaces…

TopologyA continuous map from (X, T ) to (Y, U )is a function

f : X Ysuch that the pre-image of any open set of Y is an open set of X.

Example: f(x)= sin x-3cos x is a continuous map from RR to RR.

Homotopy

Homotopy = continuous deformation

Allowed: shrinking, stretching, bendingForbidden: cutting and pasting

Angles and distances are NOT preserved!

Homotopy

A complicated unknot…

Homotopy

A well-known example of homotopic topological spaces

Homotopy

And another, somewhat less well-known…

Homotopy invariants

H : {topological spaces} {algebraic gadgets}such that if (X,T) homotopic to (Y,U), then H(X,T)=H(Y,U).

Example: the number of holes in a surfaceis a homotopy invariant (coffee

cups and donuts…)

Homotopy invariants

Warning! In general, H(X,T)=H(Y,U) does NOT imply that

(X,T) is homotopic to (Y,U).

The better a homotopy invariant is at distinguishing between topological spaces that are not homotopic, the harder it is to calculate.

Algebraic models

The algebraic gadgets are “sets with extra structure” (e.g., vector spaces) and their “special maps” are functions preserving this extra structure (e.g., linear transformations).

Topological spaces&

Continuous maps

Algebraic gadgets&

Their special maps

F

Algebraic models

We work with algebraic gadgets for which there is a reasonable notion of “homotopy” of their special maps.

Topological spaces&

Continuous maps

Algebraic gadgets&

Their special maps

F

Algebraic models

F is an algebraic model of topological spaces ifF(X,T) and F(Y,U) are homotopic whenever (X,T) and (Y,U) are homotopic.

Topological spaces&

Continuous maps

Algebraic gadgets&

Their special maps

F

My research

Developing new, more precise algebraic models that capture as much homotopy information as possible, while remaining calculable.

My research

Using new and known models to solve topological problems, similar to:

(X,T) (Z,V)

(Y,U)?

My research

Applying new and known algebraic models to solving problems in concurrency theory, e.g., to the classification of transition systems and higher dimensional automata.