Bonus
The simplest group named for these functions is denoted “pi-sub-one of X.” For 10 points each:
[10m] Name these functions that continuously morph one continuous map between topological spaces into another. These functions are used to define equivalence classes of loops in the fundamental group.
ANSWER: homotopies [accept homotopy or homotopy groups]
[10e] Since the fundamental group is homotopy invariant, a coffee cup and this mathematical space both have a fundamental group isomorphic to Z squared. This doughnut-shaped space is the Cartesian product of two circles.
ANSWER: toruses [or tori]
[10h] Different fundamental groups imply that the Hawaiian earring is not homeomorphic to performing this operation on countably many circles. The fundamental group of this operation applied to two spaces is the free product of their fundamental groups.
ANSWER: wedge sum [or wedge product; prompt on sum; prompt on product; reject “exterior product”]
<Other Science>
Answerlines and category may not exactly match the version played at all sites
Conversion
| Team | Opponent | Part 1 | Part 2 | Part 3 | Total | Parts |
|---|---|---|---|---|---|---|
| British Columbia | NYU | 0 | 10 | 0 | 10 | E |
| Chicago B | Cornell A | 0 | 10 | 0 | 10 | E |
| Iowa State | Johns Hopkins | 10 | 10 | 0 | 20 | ME |
| Penn State | Ottawa | 0 | 10 | 0 | 10 | E |
| Stanford | WUSTL A | 10 | 10 | 0 | 20 | ME |
| Vanderbilt | UCF | 0 | 10 | 0 | 10 | E |
Summary
| Tournament | Exact Match? | Heard | PPB | Easy % | Medium % | Hard % |
|---|---|---|---|---|---|---|
| 2025 ACF Nationals | Yes | 6 | 13.33 | 100% | 33% | 0% |