Singular patterns of generic maps of surfaces with boundary into the plane

Cusp cobordism group of Morse functions

“…Furthermore, the number of cusps of G 1 | W is even. Hence, by Theorem 1.1 in [29], G 1 | W can be modified on a compact subset of W \ ∂W to a fold map H : W → R 2 that realizes the pattern (f, ϕ) on W . Finally, the map G :…”

“…• The case n = 2: We apply the method of singular patterns as developed in [29]. Let W be the connected compact 2-manifold obtained by removing from V suitable small open disk neighborhoods of those cusps of G 1 which lie on components of S(G 1 ) that are diffeomorphic to [0, 1].…”

“…, and ϕ to be the partition consisting of the sets of the form ∂S, where S runs through the components of S(G 1 | W ) that are diffeomorphic to [0, 1]. Then, by construction, G 1 | W is a realization of the singular pattern (f, ϕ) on W in the sense of Definition 5.2 in [29]. Furthermore, the number of cusps of G 1 | W is even.…”

“…In order to control the parity of the number of cusps on the components of the singular set we combine Levine's cusp elimination technique (see [12] and Section 3.1) with the complementary process of creating pairs of cusps along fold lines (see Section 3.2). Moreover, for generic maps between surfaces, we use the method of singular patterns developed by the author in [29].…”

The Chromatic Brauer Category and Its Linear Representations

“…Of course, the loss of information should be kept at a minimum during this linearization process, which is motivation for studying faithfulness of such linear representations Br → Vect. This knowledge is required when it comes to the explicit computation of state sum invariants (compare Section 6.3 and Section 10.5 in [27] as well as Remark 9.5 in [28]).…”