An isomorphism theorem of Hurewicz type in the proper homotopy category

Math 527 - Homotopy Theory Hurewicz theorem

Alternate proof. Using a bit of differential topology (or a more geometric construction along the lines of Hatcher § 4.1 Exercise 15), consider the degree of a smooth map f : S → S. Since every homotopy class [f ] contains a smooth representative, and all such maps have the same degree (i.e. degree is a homotopy invariant), this defines a function deg : πn(S )→ Z. One readily shows that deg is ...

The Hurewicz Theorem

The fundamental group and homology groups both give extremely useful information, particularly about path-connected spaces. Both can be considered as functors, so we can use these constructional invariants as convenient guides to classifying spaces. However, though homology groups are often easy to compute, the fundamental group sometimes is not. In fact, it is often not even obvious when a spa...

The Homotopy Type of the Cobordism Category

Khovanov Homotopy Type, Burnside Category, and Products

In this paper, we give a new construction of a Khovanov homotopy type. We show that this construction gives a space stably homotopy equivalent to the Khovanov homotopy types constructed in [LS14a] and [HKK] and, as a corollary, that those two constructions give equivalent spaces. We show that the construction behaves well with respect to disjoint unions, connected sums and mirrors, verifying se...

An isomorphism theorem for digraphs

A seminal result by Lovász states that two digraphs A and B (possibly with loops) are isomorphic if and only if for every digraph X the number of homomorphisms X → A equals the number of homomorphisms X → B. Lovász used this result to deduce certain cancellation properties for the direct product of digraphs. We develop an analogous result for the class of digraphs without loops, and with weak h...